Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

Abstract

Spectral and pseudospectral methods based on classical and nonclassical polynomial basis sets are used for the solution of the Fokker-Planck and Schrödinger equations. Fokker-Planck equations describe many different processes in chemistry and physics, and their study has attracted considerable attention by researchers in many different fields including astrophysics, finance and biology. Pseudospectral methods of solution of the Fokker-Planck equation are presented for several systems such as the Ornstein-Uhlenbeck model for Brownian motion, electron thermalization in atomic moderators, charged particle relaxation in plasmas and models for chemical reactions based on Kramers’ equation. A Fokker-Planck equation can be transformed to a Schrödinger equation with a potential that belongs to the class of potentials in supersymmetric quantum mechanics and expressed in terms of the superpotential. The quantum harmonic oscillator and the Morse potential belong to this class of Schrödinger equations. The pseudospectral methods developed for the solution of the Fokker-Planck equation based on nonclassical basis sets are also applied to a large number of the Schrödinger equations including the Henon-Heles potential. Fundamental aspects of different pseudospectral methods such as the Discrete Variable Representation, the Quadrature Discretization method, the Lagrange mesh method and Fourier grid methods are discussed.

6.1 The Fokker-Planck Equation in Chemistry, Physics, Astrophysics and Other Fields

The Fokker-Planck equation is a partial differential equation for a probability density function, $P(\mathbf{v},\mathbf{r},t)$, analogous to a distribution function of kinetic theory discussed in Chap. 5. The linear integral Boltzmann equation for a binary gas of test particles of mass $m$ dilutely dispersed in bath particles of mass $M$ at $T_b$ can be approximated by Fokker-Planck equations in the disparate mass limits ($\gamma = M/m \rightarrow 0$ or $\gamma \rightarrow \infty$) as a consequence of the small energy transfers in particle collisions (Andersen and Shuler 1964). A similar approximation is used in plasma physics for which charged particle Coulomb collisions involve predominantly large impact parameter grazing collisions (Rosenbluth et al. 1957; Spitzer 1962; Mitchner and Kruger 1973; Hinton 1983). These approximations are examples of a large class of Master equations for which a Fokker-Planck equation can be derived with the Kramers-Moyal expansion (Gillespie 1980; Knessl et al. 1984; Kuczka et al. 1995; Risken 1996; Frank 2007; van Kampen 2007).

An alternate derivation of the Fokker-Planck equation is based on stochastic differential equations such as the Langevin equation to model “Brownian” motion as discussed in the next section and in greater detail in several references (Chandrasekhar 1949; Risken 1996; Gardiner 2003; van Kampen 2007; Reif 2008; Paul and Baschnagel 2013). For most of the applications in this chapter, we will consider a Fokker-Planck equation in two variables and a probability density function, $P(x,t)$, where $t$ is the time and $x$ is an independent variable that represents the reduced speed or energy of a particle, the particle position or some other independent variable.

The equation is named after Adrian Fokker¹ and Max Planck.² Fokker (1914) studied the relationship between the fluctuations of the rotational motion of dipoles in an electric field and the steady state probability density function. Planck (1917) developed the time dependent equation and provided the relationship between the drift and diffusion coefficients and the random fluctuations inherent in the system.

Fokker-Planck equations are used to model numerous systems in physics, astrophysics, chemistry, biology, engineering, finance and other research fields. Fokker-Planck equations have also been used to model processes in space science, notably the solar and polar wind expansions (Lie-Svendsen and Rees 1996; Pierrard and Lemaire 1998; Marsch 2006; Echim et al. 2011). A large number of chemically reactive systems can be modelled with a Fokker-Planck equation proposed by Kramers (1940). Many aspects of turbulence are modelled as stochastic processes leading to a Fokker-Planck equation (Pope 2000). The applications of the Fokker-Planck equation to stellar dynamics and astrophysics (Chandrasekhar 1942; Spitzer 1998; Chavanis 2006; Binney and Tremaine 2008) overlap applications to plasma physics (Rosenbluth et al. 1957; Spitzer 1962; Boyd and Sanderson 2003). The set of coupled rate equations for the growth of a cluster in nucleation theory is often modelled with a Fokker-Planck equation (Shizgal and Barrett 1989; Demeio and Shizgal 1993a). The Black-Scholes model in mathematical finance (Black and Scholes 1973; Paul and Baschnagel 2013) is based on a Fokker-Planck equation. These are just a few examples of the many different Fokker-Planck equations that arise in diverse applications. Additional discussion of these topics can be found in several textbooks (Risken 1996; Gardiner 2003; Reif 2008) and review papers (Chandrasekhar 1949; Lightman and Shapiro 1978).

In Chap. 5, we expressed the solutions of the linearized and linear Boltzmann equations in terms of the eigenvalues of the collision operators involved. For the linearized collision operator and the linear collision operator with unit mass ratio ($\gamma = 1$), the integral operator can be transformed to a Schrödinger equation (Kuščer and Williams 1967; Bobylev and Mossberg 2008). The Fokker-Planck equations discussed in the sections that follow can be transformed to Schrödinger equations with well defined potential functions (Risken 1996). The potentials belong to the class of Schrödinger equations in supersymmetric quantum mechanics (Bernstein and Brown 1984; Comtet et al. 1985; Dutt et al. 1988; Cooper et al. 1995).

6.1.1 From the Langevin Equation to the Fokker-Planck Equation; Brownian Motion

We begin the discussion with the classic treatment of Brownian motion. We consider a subsystem of particles of mass $m$ that interact solely with the particles of a background medium at equilibrium at temperature $T_b$. The origin of this approach is the work of the botanist Robert Brown³ who observed the random movement of a pollen grain in a fluid at some temperature $T_b$. The movement of the so-called “Brownian” particle is random owing to the multitude of collisions of the molecules of the background fluid with the grain. Thus the scalar force, $F(t)$, on the Brownian particle is random in time. However, there is also a steady component that corresponds to the friction involved in the steady movement of the Brownian particle through the fluid. Thus we write $F(t) = F_s(t) + F_r(t)$ where $F_s(t)$ is the steady component related to the viscosity of the fluid and $F_r(t)$ is a largely unknown random or “stochastic” force. The steady component of the force is $F_s(t) = - \alpha v(t)$ where $v$ is the particle velocity in one dimension and $\alpha$ is the friction coefficient that slows the particle as it moves through the fluid. We write the “stochastic” differential equation of motion for the Brownian particle as Newton’s law with a random force, that is,

$$\begin{aligned} m\frac{dv}{dt} = - \alpha v(t) + F_r(t). \end{aligned}$$

(6.1)

Equation (6.1) is known as the Langevin⁴ equation for Brownian motion that was treated previously by Einstein (1906). The main difficulty with Eq. (6.1) is that the detailed time variation of $F_r(t)$ is largely unknown. What is remarkable with this approach is that the friction coefficient $\alpha$ in Eq. (6.1) is related to the properties of $F_r(t)$ (Reif 2008). Alternative methods are based on computer simulations that follow the time history of the particle positions and velocities (Gunther and Weaver 1978; Gillespie 1996). Monte Carlo simulations have become common in statistical physics and chemistry for the study of multidimensional complex systems in equilibrium and nonequilibrium situations (Bird 1994; Landau and Binder 2009).

An alternative approach to computer simulations is one based on the probability density of the random variable leading to a deterministic Fokker-Planck equation. The stochastic force, $F_r(t)$, is assumed to satisfy two important relations, (1) that the time (or “ensemble”) average is zero and (2) the correlation in time has a definite strength, that is

$$\begin{aligned} \overline{F_r(t)}&= 0,\nonumber \\ \overline{F_r(t)F_r(t')}&= 2\alpha k_BT_b\delta (t-t'), \end{aligned}$$

(6.2)

where $k_B$ is the Boltzmann constant and $T_b$ is the temperature of the background. The overbars indicate time or ensemble averages; see Sect. 15.5 in Reif (2008). In addition to the textbooks referenced earlier, excellent discussions of the historical development of stochastic processes are available (Uhlenbeck and Ornstein 1930; Hänggi et al. 1990; Risken 1996; Abbott 2001; Gardiner 2003; van Kampen 2007; Dunkel and Hänggi 2009; Paul and Baschnagel 2013).

Brownian motion and other stochastic processes are modelled with a probability density, $P(v,t)$, corresponding to the values of $v(t)$ sampled in a sufficiently long sequences of realizations of $v(t)$. Thus $P(v,t)$ is similar to the velocity distribution function $f(v,t)$ in kinetic theory in Chap. 5. It has been shown (Uhlenbeck and Ornstein 1930; Chandrasekhar 1949; Reif 2008) that the Fokker-Planck equation for the probability density (equivalently the velocity distribution) of the Brownian particle is given by the Ornstein–Uhlenbeck equation,

$$\begin{aligned} \frac{\partial P(v,t)}{\partial t} = \nu \frac{\partial }{\partial v} \left[ vP(v,t) + \frac{k_BT_b}{m}\frac{\partial P(v,t)}{\partial v} \right] ,\;\;\;\;\;v\in (-\infty ,\infty ), \end{aligned}$$

(6.3)

where the drift and diffusion coefficients, defined in what follows are $\nu v$ and $\nu k_BT_b/m$, respectively, and $\nu = \alpha /m$ is a collision frequency.

A stochastic process which includes multiplicative as well as additive noise (Chandrasekhar 1949; Lax 1966; Brey et al. 1987; Gitterman 1999; Biró and Jakovác 2005) yields a Fokker-Planck equation with a velocity dependent diffusion coefficient. For this more general stochastic process we have the Langevin equation of the form

$$\begin{aligned} \frac{dv}{dt} = f(v) + g(v)\xi (t) + \eta (t), \end{aligned}$$

(6.4)

with $f(v)$ and $g(v)$ are known but unspecified functions. The additive and multiplicative Gaussian random variables, $\eta (t)$ and $\xi (t)$ have zero mean,

$$\begin{aligned} \overline{\eta (t)} = 0 \;\;\;\;\;\;\; \mathrm{and} \;\;\;\;\; \overline{\xi (t)} = 0, \end{aligned}$$

(6.5)

and correlations given by,

$$\begin{aligned} \overline{\eta (t)\eta (t')} = 2D \delta (t-t') \;\;\;\;\; \overline{\xi (t)\xi (t')} = 2\beta \delta (t-t'). \end{aligned}$$

(6.6)

The Fokker-Planck equation for $P(v,t)$ corresponding to this stochastic process is of the form

$$\begin{aligned} \frac{\partial P(v,t)}{\partial t} = \frac{\partial }{\partial v} \left[ A(v)P(v,t) + \frac{\partial B(v)P(v,t)}{\partial v} \right] , \end{aligned}$$

(6.7)

where $A(v)$ and $B(v)$ are the time-independent drift and diffusion coefficients, respectively, given by,

$$\begin{aligned} A(v)&= f(v) + \beta g(v)\frac{dg(v)}{dv},\nonumber \\ B(v)&= D + \beta g^2(v). \end{aligned}$$

(6.8)

where $f(v)$, $g(v)$, $D$ and $\beta$ are defined by Eqs. (6.4) and (6.6). Additional details of this derivation are in the references cited.

Given some initial condition, $P(v,0)$, the distribution $P(v,t)$ varies in time as deduced with Eq. (6.7) and attains a steady distribution at infinite time for which $\partial P(v,t)/\partial t = 0$ and denoted by $P_0(v)$. For most of the applications to be discussed, the reduced speed, $x = \sqrt{mv^2/2k_BT_b}$ is used and generally $x \in (-\infty ,\infty )$. From Eq. (6.7), this equilibrium probability density is

$$\begin{aligned} P_0(v) = \frac{1}{B(v)}\exp \left( -\int \limits _{-\infty }^v\frac{A(v')}{B(v')}dv'\right) , \end{aligned}$$

(6.9)

and is not in general a Maxwellian.

6.1.2 Spectral Solution of the Ornstein-Uhlenbeck Fokker-Planck Equation

The equilibrium distribution analogous to Eq. (6.9) for the Brownian motion Fokker-Planck equation, Eq. (6.3), is defined by

$$\left[ vP_0(v) + \frac{k_BT_b}{m}\frac{\partial P_0(v)}{\partial v}\right] = 0,$$

and the steady state distribution of Eq. (6.3) is a Maxwellian,

$$\begin{aligned} P_0(v) = \sqrt{\frac{m}{2\pi k_BT_b}}e^{-mv^2/2k_BT_b}. \end{aligned}$$

(6.10)

normalized such that

$$\int \limits _{-\infty }^\infty P_0(v)dv = 1.$$

We consider an initial condition whereby all the particles start with a specific speed $v_0$ that is

$$\begin{aligned} P(v,0) = \delta (v-v_0). \end{aligned}$$

(6.11)

The solution of this Fokker-Planck equation is readily determined with the transformation to a strict diffusion equation where the diffusion coefficient is $D = \mu k_BT_b/m$. We redefine the variables so as to remove the term in $\partial P/\partial v$ in Eq. (6.3) in a manner analogous to the transformation of the Fokker-Planck equation to a Schrödinger equation discussed in Sect. 6.3.2. We make the change of variable $u = ve^{\nu t}$ and set $P(v,t) = e^{\nu t}Q(u,t)$. With these substitutions, the Fokker-Planck equation can be written in terms of $Q(u,t)$, that is

$$\begin{aligned} \frac{\partial Q(u,t)}{\partial t} = De^{2\nu t} \frac{\partial ^2 Q(u,t)}{\partial u^2}. \end{aligned}$$

(6.12)

With the change in the time variable to $\tau = (e^{2\nu t}-1)/\nu$, we transform the Fokker-Planck equation to the diffusion equation, that is

$$\begin{aligned} \frac{\partial Q(u,\tau )}{\partial \tau } = D \frac{\partial ^2 Q(u,\tau )}{\partial u^2}. \end{aligned}$$

(6.13)

This equation could be considered as a Fokker-Planck equation without drift which is referred to as a Weiner process (Risken 1996; Gillespie 1996).

We have solved the diffusion equation with a Fourier transform method in Chap. 4, Sect. 4.6.5 and the solution of Eq. (6.13) is

$$\begin{aligned} Q(u,\tau ) = \frac{1}{\sqrt{4\pi D\tau }}e^{-(u-u_0)^2/4D\tau }. \end{aligned}$$

(6.14)

With this result, the solution to the Ornstein–Uhlenbeck, Eq. (6.13), in the reduced speed, $x = \sqrt{mv^2/2k_BT_b}$, is

$$\begin{aligned} P(x,t) = \left[ \frac{1}{\pi (1-e^{-2\nu t})}\right] ^{1/2}\exp \left[ -\frac{(x-x_0e^{-\nu t})^2}{(1-e^{-2\nu t})}\right] . \end{aligned}$$

(6.15)

A spectral solution of Eq. (6.3) can be expressed in Hermite polynomials in $x$ by substituting $P(x,t) = e^{-x^2}g(x,t)$ into Eq. (6.3) written in terms of $x$. The result is the differential equation

$$\begin{aligned} \frac{\partial g(x,t)}{\partial t}&= \nu e^{x^2}\frac{\partial }{\partial x}\bigg [e^{-x^2}\frac{\partial g(x,t)}{\partial x}\bigg ],\nonumber \\&= \nu \bigg [-2x\frac{\partial g}{\partial x} + \frac{\partial ^2 g}{\partial x^2}\bigg ]. \end{aligned}$$

(6.16)

With the expansion in Hermite polynomials

$$\begin{aligned} g(x,t) = \sum _{n=0}^\infty c_n(t)H_n(x), \end{aligned}$$

(6.17)

Eq. (6.16) can be written as

$$\begin{aligned} \sum _{n=0}^\infty H_n(x)\frac{dc_n}{dt} = \nu \sum _{n=0}^\infty c_n(t)\bigg [-2xH_n^\prime (x) + H_n^{\prime \prime }(x)\bigg ]. \end{aligned}$$

(6.18)

With the relation $-2xH_n^\prime + H_n^{\prime \prime } = -2nH_n$, the time dependence of the coefficients is given by

$$\begin{aligned} \frac{dc_n(t)}{dt} = -2n\nu c_n(t). \end{aligned}$$

(6.19)

With the expansion of the initial condition in the Hermite polynomials, the spectral solution is given by

$$\begin{aligned} P(x,t) = e^{-x^2}\sum _{n=0}^\infty \frac{1}{2^nn!\sqrt{\pi }}H_n(x_0)H_n(x)e^{-2n\nu t}. \end{aligned}$$

(6.20)

Equations (6.15) and (6.20) can be used to study the rate of convergence of the expansion in Hermite polynomials (Wei et al. 1997). A study of the use of orthogonal expansions for the solution of Fokker-Planck equations was reported by Cukier et al. (1973). An eigenfunction analysis of the three-dimensional Ornstein-Uhlenbeck process with expansions in associated Laguerre polynomials and spherical harmonics was reported recently by Wilkinson and Pumir (2011).

6.1.3 Rayleigh and Lorentz Fokker-Planck Equations from the Boltzmann Equation; The Kramers-Moyal Expansion

In Chap. 3 (Eq. (3.49)) and Chap. 5 (Eq. (5.110)), we discussed the Wigner-Wilkins kernel (Wigner 1943; Wigner and Wilkins 1944; Hoare and Kaplinsky 1970; Hoare 1971) for the linear Boltzmann integral operator for a hard sphere cross section. This Boltzmann equation models the relaxation of an ensemble of test particles of mass $m$ dilutely dispersed in a second gas of particles of mass $M$ which is taken to be at equilibrium with a Maxwellian distribution at temperature $T_b$.

The hard sphere collision operator in the Boltzmann equation for a binary gas mixture can be approximated by Fokker-Planck equations in the mass ratio limits $\gamma = M/m \rightarrow \infty$ (the “Lorentz” limit) or $\gamma \rightarrow 0$ (the “Rayleigh” limit), respectively as shown by Andersen and Shuler (1964). The derivation is based on the expansion of the kernel in the small energy transfer on collision for these disparate mass ratios. The integral operator is expanded up to second order in the small energy transfer and this transformation of the integral equation to a Fokker-Planck equation is known as the Kramers-Moyal expansion (Risken 1996; Knessl et al. 1984; van Kampen 2007).

For the disparate mass ratio limit, $\gamma \rightarrow 0$, the Rayleigh Fokker-Planck equation is

$$\begin{aligned} \frac{\partial P(y,t)}{\partial t} = \frac{\partial }{\partial y}\left[ (y-3)P(y,t)+ \frac{\partial }{\partial y }[yP(y,t)]\right] , \end{aligned}$$

(6.21)

where $y = mv^2/k_BT_b$ is the reduced energy and $t$ is in units of $\tau$ given by $1/\tau = K_R = \frac{16}{3}Mmn_b\sigma _0\sqrt{2k_BT_b/\pi M}$. For $\gamma \rightarrow \infty$, the Lorentz Fokker-Planck equation is

$$\begin{aligned} \frac{\partial P(x,t)}{\partial t} = \frac{1 }{4}\frac{\partial }{\partial x}\left[ (2x^2-3)P(x,t)+ \frac{\partial }{\partial x }[xP(x,t)]\right] , \end{aligned}$$

(6.22)

where $x=\sqrt{mv^2/2k_BT_b}$ is the reduced speed and $1/\tau = K_L = 2\sqrt{m/M}n_b\sigma _0$ $\sqrt{2k_BT_b/M}$. The hard sphere cross section is denoted by $\sigma _0$ and $n_b$ is the density of the background gas. A spectral solution of the Rayleigh Fokker-Planck equation in terms of Hermite polynomials is described in the next section. There is no known spectral solution of the Lorentz Fokker-Planck equation in terms of classical polynomials.

6.1.4 Spectral Solution of the Rayleigh Fokker-Planck Equation

We consider an analysis similar to the one in Sect. 6.1.2. If we set $P(y,t)=P_0(y)g(y,t)$ where $P_0(y)=(2/\sqrt{\pi })\sqrt{y}e^{-y}$ in dimensionless units and we get the differential equation

$$\begin{aligned} \frac{\partial g(y,t)}{\partial t} = \frac{1}{P_0(y)}\frac{\partial }{\partial y}\left[ yP_0(y)\frac{\partial g(y,t)}{\partial y}\right] . \end{aligned}$$

(6.23)

The evaluation of the partial derivative in the square bracket gives

$$\begin{aligned} \frac{\partial g(y,t)}{\partial t} = y\frac{\partial ^2 g(y,t)}{\partial y^2} +(\frac{3}{2} -y)\frac{\partial g(y,t)}{\partial y}. \end{aligned}$$

(6.24)

The differential operator on the right hand side is related to the eigenvalue problem for the associated Laguerre polynomials, $L_n^{(\frac{1}{2})}(y)$, that is

$$\begin{aligned} y\frac{d^2L_n^{(\frac{1}{2})}(y)}{dy^2} +(\frac{3}{2} - y)\frac{dL_n^{(\frac{1}{2})}(y)}{dy} = -ny. \end{aligned}$$

(6.25)

Thus, the spectral solution is represented by the expansion in the associated Laguerre polynomials

$$\begin{aligned} g(y,t)=\sum _{n=0}^\infty c_nL_n^{(\frac{1}{2})}(y)e^{-nt}, \end{aligned}$$

(6.26)

where the coefficients for a $\delta$-function initial condition, $P(y,0)= \delta (y-y_0)$, are

$$\begin{aligned} c_n = \frac{\varGamma (n+1)}{\varGamma (n+\frac{3}{2})}L_n^{(\frac{1}{2})}(y_0). \end{aligned}$$

(6.27)

Analogous to the expansion of the Ornstein-Uhlenbeck Fokker-Planck equation in Hermite polynomials, the solution of Eq. (6.21) can be expanded in Laguerre polynomials, that is

$$\begin{aligned} P(y,t) = \sqrt{y}e^{-y}\sum _{n=0}^\infty \frac{\varGamma (n+1)}{\varGamma (n+3/2)}L_n^{(\frac{1}{2})}(y_0)L_n^{(\frac{1}{2})}(y)e^{-nt}. \end{aligned}$$

(6.28)

Andersen and Shuler (1964) summed this series and found the analytic result,

$$\begin{aligned} P(y,t) = \frac{e^{t/2}}{2\sqrt{\pi y_0(1-e^{-t})}}&\left\{ \exp \left[ -\frac{(\sqrt{y}-\sqrt{y_0e^{-t}})^2}{1-e^{-t}}\right] \right. \nonumber \\&\quad \left. - \exp \left[ -\frac{(\sqrt{y}+\sqrt{y_0e^{-t}})^2}{1-e^{-t}}\right] \right\} , \end{aligned}$$

(6.29)

analogous to Eq. (6.15) for the Ornstein-Uhlenbeck equation. This provides another opportunity to study the rate of convergence of the Laguerre expansions.

For an initial Maxwellian distribution at temperature $T(0) > T_b$, the expansion in Laguerre polynomials can be summed in closed form (Andersen and Shuler 1964) to give

$$\begin{aligned} P(y,t)=\frac{2}{\sqrt{\pi }}\left[ \frac{T_b}{T(t)}\right] ^{3/2}\sqrt{y}\exp [-\frac{T_b}{T(t)}y], \end{aligned}$$

(6.30)

which is a Maxwellian distribution with the time dependent temperature $T(t)$. Equation (6.30) also defines the intial Maxwellian at $T(0)$. An important property of this Fokker-Planck equation is that if the initial distribution function is a Maxwellian at some temperature $T(0) > T_b$, the time dependent distribution remains Maxwellian with a time dependent temperature. This property of the Rayleigh Fokker-Planck equation is referred to as canonical invariance (Andersen et al. 1964; Andersen and Shuler 1964). This is a consequence of the fact that for this Rayleigh Fokker-Planck equation the moments of the distribution function in the Laguerre basis set are uncoupled and the temperature relaxation is a pure exponential given by

$$\begin{aligned} \frac{T(t) - T_b}{T(0)-T_b} = e^{-t}. \end{aligned}$$

(6.31)

6.2 Numerical Methods for the Solution of the Fokker-Planck Equation

We have expressed the spectral solutions of the Ornstein-Uhlenbeck Fokker-Planck equation in Hermite polynomials and of the Rayleigh Fokker-Planck equation in Laguerre polynomials. Analogous pseudospectral solutions can also be derived which provide identical numerical results. We present in the next section a formalism for the use of nonclassical basis functions for the solution of the general Fokker-Planck equation in Eq. (6.7) with arbitrary drift and diffusion coefficients, $A(v)$ and $B(v)$, respectively. In Sect. 6.2.2, an equivalent pseudospectral formalism is presented.

6.2.1 Spectral Methods with Nonclassical Basis Functions

We consider a solution of the Fokker-Planck equation with a spectral method and with a basis set analogous to the solution of the Ornstein-Uhlenbeck equation in Sect. 6.1.2. The polynomial basis set is defined with the steady distribution $P_0(x)$ as the weight function. If we set $P(x,t) = P_{0}(x)g(x,t)$ in the Fokker-Planck equation, Eq. (6.7), where $P_0(x)$ is given by Eq. (6.9), then the equation for $g(x,t)$ becomes

$$\begin{aligned} \frac{\partial g(x,t)}{\partial t}&= \frac{1}{P_0(x)} \frac{\partial }{\partial x}\left[ B(x)P_0(x)\frac{\partial g(x,t) }{\partial x}\right] ,\;\;\;\; x \in [0,\infty )\nonumber \\&= -A(x)\frac{ \partial g(x,t)}{\partial x} + B(x) \frac{\partial ^2 g(x,t)}{\partial x^2},\nonumber \\&= -Lg(x,t), \end{aligned}$$

(6.32)

where the definition of $P_0(x)$ has been used. The term in square brackets in Eq. (6.32) can be considered as a flux. With the form of the operator, $L$, in the first line of Eq. (6.32), $L$ is self-adjoint with respect to $P_0(x)$ as the weight function provided that the zero flux boundary condition

$$\begin{aligned} \left. P_0(x)B(x)\frac{\partial g(x)}{\partial x}\right| _0^\infty = 0, \end{aligned}$$

(6.33)

is imposed. The linear time dependent Fokker-Planck equation, Eq. (6.32), admits a solution in terms of the eigenfunctions, $\psi _n(x)$, and eigenvalues, $\lambda _n$, defined by

$$\begin{aligned} L\psi _n(x) = \lambda _n\psi _n(x), \end{aligned}$$

(6.34)

and the solution can be written as

$$\begin{aligned} P(x,t) = P_0(x)\sum _{n=0}^\infty a_ne^{-\lambda _n t}\psi _n(x), \end{aligned}$$

(6.35)

where the $a_n$ coefficients are determined from the initial distribution, $P(x,0)$. The self-adjoint property of $L$ can be verified by calculating the matrix element $\langle \phi _1|L|\phi _2\rangle$ with $L$ defined as in Eq. (6.32). With an integration by parts, it is easily shown that $\langle \phi _1|L|\phi _2\rangle = \langle \phi _2|L|\phi _1\rangle$ provided Eq. (6.33) is satisfied.

The eigenvalues and eigenfunctions are determined with a Galerkin spectral method and nonclassical basis sets and we use the basis set $\{S_n(x)\}$ orthonormal with $P_0(x)$ as weight function, that is

$$\begin{aligned} \int \limits _0^\infty P_0(x)S_n(x)S_m(x)dx=\delta _{nm}. \end{aligned}$$

(6.36)

The set $\{S_n(x)\}$, introduced in this chapter, is used to denote a general basis set of nonclassical polynomials orthonormal with respect to different equilibrium density functions, $P_0(x)$, defined by a specific physical problem. The matrix elements of the Fokker-Planck operator $L$ in this basis set are given by

$$\begin{aligned} L^{(sp)}_{nm}&= \int \limits _0^\infty P_0(x)S_n(x)LS_m(x)dx\nonumber \\&= -\int \limits _0^\infty P_0(x)S_n(x)\frac{1}{P_0(x)}\frac{d}{dx}\left[ P_0(x)B(x)\frac{dS_m(x)}{dx}\right] dx, \end{aligned}$$

(6.37)

where the superscript (sp) denotes the polynomial spectral representation. With an integration by parts, we get the symmetric matrix representation of the Fokker-Planck operator,

$$\begin{aligned} \begin{boxed}{L^{(sp)}_{nm} = - \int \limits _0^\infty P_0(x)B(x)S_n'(x)S_m'(x)dx,\;\;\;\;\; (n,m) = 0,1, \ldots ,N-1}\end{boxed} \end{aligned}$$

(6.38)

where zero flux boundary conditions are satisfied, Eq. (6.33). We highlight this important result showing the self-adjoint property of the Fokker-Planck equation subject to zero flux boundary conditions. The eigenvalues, $\lambda _n$, and eigenfunctions, $\psi _n(x)$, are determined with a numerical diagonalization of the matrix $\mathbf{L}^{(sp)}$ of dimension $N$. Although we have not demonstrated this explicitly here, the numerical results will show that the coefficients of the expansion $\psi _n(x)$ in the $S_n(x)$ basis set are linear variational parameters. Thus, the variational theorem is applicable and this spectral method will provide an upper bound to the eigenvalues for each $N$ and converge from above. The time dependent solution of the Fokker-Planck equation is given by Eq. (6.35).

We will use $L$ to denote several different Fokker-Planck operators in the sections that follow and each is defined at the outset. Otherwise the notation would become excessive.

6.2.2 Pseudospectral Methods with Nonclassical Quadratures

We introduce the basis set $R_n(x)$ orthonormal with respect to $w(x)$ defined by

$$\begin{aligned} R_n(x) = \sqrt{\frac{P_0(x)}{w(x)}}S_n(x), \end{aligned}$$

(6.39)

and evaluate the derivative $S'(x) \equiv dS(x)/dx$,

$$\begin{aligned} S'_n(x) = \sqrt{\frac{w}{P_0}}\left( \frac{w'}{2w} - \frac{P'_0}{2P_0}\right) R_n(x)+ \sqrt{\frac{w}{P_0}}R'_n(x). \end{aligned}$$

(6.40)

Thus, the matrix elements of the Fokker-Planck operator, Eq. (6.38), are given by

$$\begin{aligned} L^{(sp)}_{nm} = - \int \limits _0^\infty w(x)B(x)\left[ R^\prime _n(x)+h(x)R_n(x)\right] \left[ R^\prime _m(x)+h(x)R_m(x)\right] dx, \end{aligned}$$

(6.41)

where

$$\begin{aligned} \begin{boxed}{h(x) = \frac{w'(x)}{2w(x)} - \frac{P'_0(x)}{2P_0(x)},}\end{boxed} \end{aligned}$$

(6.42)

and is a measure of the departure of $w(x)$ from $P_0(x)$. This result is highlighted as it shows that an optimal choice of weight function might be $w(x) = P_0(x)$ giving $h(x) = 0$.

The matrix elements are evaluated with the quadrature based on the weight function $w(x)$, that is

$$\begin{aligned} L^{(sp)}_{nm} = -\sum _{k=1}^Nw_kB(x_k)[R'_n(x_k)+h(x_k)R_n(x_k)][R'_m(x_k)+h(x_k)R_m(x_k)], \end{aligned}$$

(6.43)

where $x_k$ and $w_k$ are the quadrature points and weights associated with the polynomials orthogonal with respect to the weight function, $w(x) = P_0(x)$. To express this spectral representation in the equivalent physical space representation, we transform

$$\begin{aligned} L^{(ps)}_{ij} = \sum _{m=0}^{N-1}\sum _{n=0}^{N-1}T_{in}L^{(sp)}_{nm}T_{jm}, \end{aligned}$$

(6.44)

where the superscript (ps) denotes the discrete pseudospectral representation and the transformation matrix between physical and spectral space is defined as in Chap. 3, namely

$$T_{in}=\sqrt{w_i}R_n(x_i).$$

We need only consider the first transformation with the sum over $n$ as the second over $m$ is similar. With the definition of $T_{in}$, the term in $R_n(x_k)$ is transformed as

$$\begin{aligned} h(x_k)\sum _{n=0}^{N-1}T_{in}R_n(x_k)&= h(x_k)\sum _{n=0}^{N-1}\sqrt{w_i}R_n(x_i)R_n(x_k),\nonumber \\&= \frac{h(x_k)}{\sqrt{w_k}}\delta _{ik}. \end{aligned}$$

(6.45)

The transformation of $R'_n(x_k)$ employs the derivative matrix operator giving

$$\begin{aligned} \sum _{n=0}^{N-1}T_{in}R'_n(x_k)&= \sum _{n=0}^{N-1}\sqrt{w_i}R_n(x_i)\sum _{\ell =1}^{N}D_{ k\ell }\sqrt{\frac{w_\ell }{w_k}}R_n(x_\ell ),\nonumber \\&= \sum _{\ell =1}^{N}D_{k\ell }\sqrt{\frac{w_iw_\ell }{w_k}}\sum _{n=0}^{N-1}R_n(x_i)R_n(x_\ell ),\nonumber \\&= \sum _{\ell =1}^{N}D_{k\ell }\sqrt{\frac{w_iw_\ell }{w_k}} \frac{\delta _{i\ell }}{w_i},\nonumber \\&= \frac{D_{ki}}{\sqrt{w_k}}. \end{aligned}$$

(6.46)

The transformation with $T_{jm}$ yields similar results and we find the discrete pseudospectral representation

$$\begin{aligned} \begin{boxed}{L^{(ps)}_{ij} = - \sum _{k=1}^N B(x_k)[D_{ki}+h(x_k)\delta _{ki}][D_{kj}+h(x_k)\delta _{kj}],\;\;\;(i,j) = 1,2, \ldots ,N.}\end{boxed}\nonumber \\ \end{aligned}$$

(6.47)

This result is also highlighted because for the quadrature defined with the stationary distribution, $w(x) = P_0(x)$, for which $h(x_k) = 0$, the representation of $L_{ij}$ has the simpler form,

$$\begin{aligned} L^{(ps)}_{ij} = - \sum _{k=1}^N B(x_k)D_{ki}D_{kj}. \end{aligned}$$

(6.48)

The representation of the Fokker-Planck operator given by Eq. (6.48) is straightforward to program in MATLAB for different choices of the basis set and associated quadrature points and weights. The derivative matrix operator, $D_{ki}$, is calculated as discussed in Chap. 3, Eqs. (3.138) and (3.139).

The pseudospectral solution of the Ornstein–Uhlenbeck Fokker-Planck equation is equivalent to the solution of the Schrödinger equation for the quantum harmonic oscillator. The drift coefficient, $B(x) = 1$, (Eq. 6.16), and $P_0(x) = w(x) = e^{-x^2},\; x\in (-\infty ,\infty )$, so that the derivative matrix operator, $D_{ki}$, is defined in terms of the Hermite polynomials as discussed in Chap. 4. The eigenvalues of the self-adjoint representation of the Fokker-Planck operator $\mathbf{L}^{(ps)}$ (Eq. 6.48) are $\lambda _n = n$ and the eigenfunctions are the Hermite polynomials.

For the Rayleigh Fokker-Planck equation $B(y) = y$, and with the Gauss Laguerre quadrature weights and points defined with $P_0(y) = w(y) = \sqrt{y}e^{-y}, \; y\in [0,\infty )$, the eigenvalues of $\mathbf{L}^{(ps)}$ are $\lambda _n = n$ and the eigenfunctions are the Laguerre polynomials. This is consistent with the spectral solutions given by Eqs. (6.20) and (6.28).

6.2.3 The Chang-Cooper Finite Difference Method of Solution of the Fokker-Planck Equation

The finite difference algorithm by Chang and Cooper (1970) has found numerous applications for the solution of the Fokker-Planck equation in many different applications (Larsen et al. 1985; Park and Petrosian 1996; Buet and Dellacherie 2010; Abolhassani and Matte 2012).

The self-adjoint form of the Fokker-Planck equation, Eq. (6.32), was shown to be consistent with zero flux at the boundaries, Eq. (6.33). This boundary condition is also related to particle conservation

$$\begin{aligned} \frac{\partial }{\partial t} \int \limits _0^\infty P_0(x) g(x,t) dx = P_0(x)B(x)\frac{ \partial g(x,t)}{\partial x}\bigg |_{x=0}^\infty = 0. \end{aligned}$$

(6.49)

Any useful discretization would have to ensure particle conservation which yields, $\lambda _0 = 0$.

We discretize the speed variable according to $0 = x_1 < x_2 <x_3 \cdots < x_N = x_{max}$ with $x_{i+1} = x_i + \varDelta x$, and $\varDelta x = x_{max}/(N-1)$ where $x_{max}$ is the speed point chosen large enough so that the flux boundary condition is satisfied. We also introduce a shifted grid at the midpoint defined by $x_{i+1/2} = x_i + \varDelta x/2$. With a centered difference for the derivative,

$$\begin{aligned} \frac{ \partial g(x,t)}{\partial x}|_{x_{i+1/2}} \approx \frac{g(x_{i+1},t) - g(x_i,t)}{\varDelta x}, \end{aligned}$$

(6.50)

the finite difference representation of the eigenvalue problem is,

$$\begin{aligned} \sum _{j=1}^N L_{ij} \phi _n(x_j) = \lambda _n \phi _n(x_i), \end{aligned}$$

(6.51)

where

$$\begin{aligned} \L _{ii} = \frac{1}{(\varDelta x)^2} \frac{x_i^2P_0(x_i)B_i + x^2_{i+1}P_0(x_{i+1})B_{i+1}}{x^2_{i+1/2}P_0(x_{i+1/2})}, \;\;\; i=1, \ldots , N, \end{aligned}$$

(6.52)

$$\begin{aligned} \L _{i,i-1} = - \frac{1}{(\varDelta x)^2} \frac{x_i^2P_0(x_i)B_i}{x^2_{i+1/2}P_0(x_{i+1/2})}, \;\;\; i=2, \ldots , N, \end{aligned}$$

(6.53)

$$\begin{aligned} \L _{i,i+1} = -\frac{1}{(\varDelta x)^2} \frac{x_{i+1}^2P_0(x_{i+1})B_{i+1}}{x^2_{i+1/2}P_0(x_{i+1/2})}, \;\;\; i=1, \ldots , N-1, \end{aligned}$$

(6.54)

with the understanding that the first term in the fraction on the right hand side of Eq. (6.52) vanishes for $i = 1$ and the second term vanishes for $i = N$ in order to enforce the boundary conditions.

We use a forward Euler difference algorithm for the time derivative, that is

$$\begin{aligned} \frac{\partial g(x_i,t)}{\partial t} = \frac{g_i^{(n+1)}-g_i^{(n)}}{\varDelta t}, \end{aligned}$$

(6.55)

where $t = n\varDelta t$ and $g_i^{(n)}=g(x_{i+1/2},n\varDelta t)$. The Chang-Cooper finite difference algorithm for the Fokker-Planck equation is

$$\begin{aligned} g_i^{(n+1)} = g_i^{(n)}+\varDelta t\frac{x_{i+1}^2P_0(x_{i+1})B_{i+1}[g_{i+1}^{(n+1)}-g_{i}^{(n+1)}] -x_i^2P_0(x_i)B_i[g_{i}^{(n+1)}-g_{i-1}^{(n+1)}]}{(\varDelta x)^2x_{i+1/2}^2P_0(x_{i+1/2})}.\nonumber \\ \end{aligned}$$

(6.56)

This result can be rewritten compactly as the matrix equation

$$\begin{aligned} \sum _{j=1}^NV_{ij}g_j^{(n+1)} = g_i^{(n)}, \end{aligned}$$

(6.57)

where the matrix $\mathbf{V}$ is tridiagonal with elements

$$\begin{aligned} V_{i,i}&= 1 + \varDelta t L_{i,i}, \;\;\;\; i=1, \ldots , N,\nonumber \\ V_{i,i-1}&= \varDelta t L_{i,i-1}, \;\;\;\; i=2, \ldots , N, \\ \nonumber V_{i,i+1}&= \varDelta t L_{i,i+1}, \;\;\;\; i=1, \ldots , N-1 \end{aligned}$$

(6.58)

At each time step, the updated values $g_i^{(n+1)}$ are obtained with the inversion of Eq. (6.57).

The Chang-Cooper finite difference scheme, as a discrete representation of $L$, does not give rapidly convergent eigenvalues and eigenfunctions (Leung et al. 1998). However, with the algorithm Eq. (6.56), the probability density function remains positive, entropy increases with time and a Maxwellian is recovered at equilibrium (Buet and Dellacherie 2010). We use this algorithm in Sect. 6.4.2 to solve the Fokker-Planck equation for a model that involves heating of a plasma by wave-particle interactions.

6.3 Electron Thermalization; The Lorentz Fokker-Planck Equation Revisited

The degradation or thermalization of energetic electrons in atomic and molecular moderators is an important aspect of radiation chemistry and physics (Mozumder 1999; Robson 2006), plasma processing of semiconductor devices (Petrović et al. 2009), the physics of the aurora (Stamnes 1980; Basu et al. 1993; Solomon 2001; Shematovich et al. 2008) fundamental aspects of the approach to equilibrium (Trunec et al. 2003; Sospedra-Alfonso and Shizgal 2011) and thermalization in condensed matter (Sakai 2007; White et al. 2010).

The electron anisotropic nonequilibrium distribution functions are often expanded in the direct product of the spherical harmonics in ($\theta , \phi$) and the Sonine–Laguerre polynomials for the reduced speed or reduced energy dependence (Kumar et al. 1980; Robson and Ness 1986; Robson 2006; White and Robson 2011; Abolhassani and Matte 2012). This methodology requires the matrix elements of the collision operator

in the Boltzmann equation as discussed in Chap. 5. There have also been solutions of the electron Boltzmann equation with a B spline representation (Pitchford and Phelps 1982) as well as with Monte Carlo methods (Koura 1983; Solomon 2001; Shematovich et al. 2008).

For molecular moderators such as N$_2$, O$_2$ and CH$_4$, inelastic collisions that involve changes in the rotational and/or vibrational states of the moderator (Pitchford and Phelps 1982; Kowari et al. 1992) and electron attachment to electronegative gases such as SF$_6$ and CCl$_4$ must be included (Kowari and Shizgal 1996; Kowari et al. 1998).

The relaxation of electrons of moderate energy in atomic moderators for which only elastic collisions need be included (Mozumder 1981; Knierim et al. 1982; Risken and Voigtlaender 1984; McMahon and Shizgal 1985; Shizgal and McMahon 1985) is presented in this section. In atomic moderators, there have been two notable phenomena exhibited, namely the transient negative mobility (Shizgal and McMahon 1985; Dyatko et al. 2001; Dyatko 2007) and the negative differential conductivity effect in gas mixtures (Shizgal 1990) previously thought to occur only for polyatomic gases with internal degrees of freedom.

Owing to the small electron mass $m_e$ to moderator mass $M$ ratio, the Boltzmann collision operator can be replaced by the Fokker-Planck operator corresponding to the Lorentz limit discussed in Sect. (6.1.3). The Boltzmann equation or the Fokker-Planck equation have been used in the study of electron thermalization in rare gases (Lin et al. 1979; Knierim et al. 1982; McMahon and Shizgal 1985; Shizgal and McMahon 1985). The physics of the problem is defined by the energy dependent momentum transfer cross section $\sigma _{mt}(v)$ for electron-atom collisions and the strength of the applied electric field, $E$. The electric field results in a drift of the electrons with a mobility determined by the electron-atom cross section.

To account for the electron drift in the applied electric field, there is a small anisotropy of the electron velocity distribution function which is expressed by the expansion in Legendre polynomials, that is,

$$\begin{aligned} f(\mathbf{v},t') = \sum _{\ell =0}^\infty f_\ell (v,t')P_\ell (\cos \theta ), \end{aligned}$$

(6.59)

where $\theta$ is the angle between $\mathbf{v}$ and the polar axis taken in the direction of the electric field. Owing to the small mass ratio, $m_e/M$, the anisotropy of the distribution remains small and only the terms in $\ell = 0$ and $\ell = 1$ in Eq. (6.59) need to be retained. This is referred to as the “two-term” approximation (Hagelaar and Pitchford 2005). The coupled equations for the first two terms, $f_0$ and $f_1$, are

$$\begin{aligned} \frac{\partial f_0}{\partial t'} + \frac{eE}{3m_e}\left( \frac{\partial }{\partial v}+\frac{2}{v}\right) f_1&= \frac{m_e}{Mv^2}\frac{\partial }{\partial v}\left[ v^3\gamma (v)\left( 1+\frac{k_B T_b}{m_ev}\frac{\partial }{\partial v}\right) \right] f_0,\nonumber \\ \frac{\partial f_1}{\partial t'} + \frac{eE}{m_e}\frac{\partial f_0}{\partial v}&= - \gamma (v) f_1, \end{aligned}$$

(6.60)

where $\gamma (v) = N_bv\sigma _{mt}(v)$ and $N_b$ is the density of the moderator. There is an initial fast transient that we ignore and thus we set $\partial f_1/\partial t' = 0$. This aspect of relaxation in the Lorentz limit was illustrated in Chap. 3; see Fig. 5.15.

We also define the reduced speed $x = v\sqrt{m_e/2k_BT_s}$ with an arbitrary scaling temperature $T_s$ and dimensionless time $t = t'/\tau$ where

$$\begin{aligned} \tau =\left[ \frac{nm_e\sigma _0}{2M}\sqrt{\frac{2k_BT_b}{m_e}} \right] ^{-1}. \end{aligned}$$

(6.61)

A representative hard sphere cross section is denoted by $\sigma _0$. We also consider a scaling of the speed variable in anticipation of the use of a quadrature based solution of the Fokker-Planck equation so that we define $s^2 =T_s/T_b$. With these definitions and the steady state value of $f_1$ given by

$$\begin{aligned} f_1 = -\frac{eE}{\gamma m_e}\frac{\partial f_0}{\partial v}, \end{aligned}$$

(6.62)

we have the Fokker-Planck equation for $f_0$, that is

$$\begin{aligned} \frac{\partial f_0}{\partial t} = \frac{s}{x^2}\frac{\partial }{\partial x}\left[ 2x^4\hat{\sigma }(x)f_0 +\frac{x^2}{s^2}B(x)\frac{\partial f_o}{\partial x}\right] , \end{aligned}$$

(6.63)

where

$$\begin{aligned} B(x) = x\hat{\sigma }x)+\frac{(\alpha /s)^2}{x\hat{\sigma }(x)}, \end{aligned}$$

(6.64)

and the field strength parameter is

$$\begin{aligned} \alpha ^2 = \frac{M}{6m_e}\left[ \frac{eE}{n\sigma _0k_BT_b}\right] ^2. \end{aligned}$$

(6.65)

In Eq. (6.63), $\hat{\sigma }(x) = \sigma _{mt}(x)/\sigma _0$ is a dimensionless cross section written as a function of reduced speed. The steady state distribution is from Eq. (6.63) given by

$$\begin{aligned} f_0(x,\infty ) = D(x) = C\exp \left[ -2s^2\int \limits _0^x\frac{(x')^2\hat{\sigma }(x')}{B(x')}dx'\right] . \end{aligned}$$

(6.66)

The steady solution, $D(x)$ given by Eq. (6.66), is precisely $P_0(x)$ in Sect. 6.2.1. The distribution, $D(x)$, is referred to as the Davydov distribution which reduces to a Maxwellian in the absence of an electric field. If we set $f(x,t)=D(x)g(x,t)$, the equation for $g(x,t)$ is given by

$$\begin{aligned} \frac{\partial g}{\partial t} = \frac{s}{D(x)x^2}\frac{\partial }{\partial x}\left[ \frac{x^2}{s}B(x)D(x)\frac{\partial g}{\partial x}\right] = - Lg,\;\;\;\; x\in [0,\infty ). \end{aligned}$$

(6.67)

The linear operator on the right hand side of Eq. (6.67) that we have denoted by $L$ is self adjoint with $D(x)$ as weight function with zero flux boundary conditions, that is,

$$\begin{aligned} x^2D(x)B(x)\frac{ \partial g(x,t)}{\partial x}\bigg |_{x=0}^{x=\infty } = 0. \end{aligned}$$

(6.68)

We are interested in the eigenvalue problem defined by

$$\begin{aligned} L\psi _n(x) = \lambda _n\psi _n(x). \end{aligned}$$

(6.69)

6.3.1 Hard Sphere Cross Section and Zero Electric Field, $E = 0$

If there is no external electric field and the cross section is a hard sphere $\sigma (x) = \sigma _0$, then $B(x) = x$ and the equilibrium distribution is the Maxwellian $P_0(x) = x^2\exp (-x^2)$. We consider a calculation of the eigenvalue spectrum of this Fokker-Planck operator with the Maxwell polynomials orthonormal with respect to the weight function $w(x) =x^2\exp (-x^2)$. The “traditional” method of solution of the Boltzmann equation involves the representation of the collision operator in the Sonine-Laguerre polynomials (Knierim et al. 1982) denoted by $L^{(SL)}_{nm}$.

The calculation of this matrix representation of the collision operator in the Boltzmann equation defined by $L_{nm}^{(SL)} = \langle L_n^{(\frac{1}{2})}|L|L_m^{(\frac{1}{2})}\rangle$ with weight function $w(y)=\sqrt{y}e^{-y}$ is straightforward but algebraically tedious (Shizgal and Fitzpatrick 1974) and the final expressions obtained can lead to considerable round-off errors in the numerical calculation of the matrix elements. In the Lorentz limit, the result is simpler and we have that

$$\begin{aligned} L^{(SL)}_{nm} = -2\sqrt{\pi }A\frac{m_e}{M}\sqrt{\frac{m!n!}{\varGamma (n+\frac{3}{2})\varGamma (m+\frac{3}{2})}} \sum _{r=1}^{min(n,m+1)}r(r+1)N(n-r)N(m-r) \end{aligned}$$

(6.70)

where $N(\ell )=\varGamma (\ell -\frac{1}{2})/(2\ell !\sqrt{\pi })$ and $A = 2n_b\pi d^2\sqrt{2k_BT_b/m_e}$. The matrix $\mathbf{L}^{(SL)}$ in the Sonine-Laguerre representation is a full matrix.

Our main objective is to compare the convergence of the eigenvalues of the Boltzmann collision operator as calculated with the representation given by Eq. (6.70) and a pseudospectral method of solution based on the non-classical polynomials orthogonal with respect to weight function $w(x) = P_0(x) = x^2 \exp (-x^2)$.

Table 6.1 Convergence of the eigenvalues of the hard sphere Lorentz Fokker-Planck equation with the Sonine-Laguerre polynomial basis set (left) in comparison with the Maxwell polynomial basis set (right)

The matrix representation of the Lorentz Fokker-Planck operator in the Maxwell polynomials, $M_n(x)$, $p = 2$, orthonormal with respect to a weight function $w(x) = x^2e^{-x^2}$ is defined by $L_{nm}^{(MP)}$ $=$ $\langle M_n|L|M_m\rangle$ where the scalar product is with respect to $w(x)$. Lo and Shizgal (2006) derived the explicit tridiagonal form of this spectral representation as

$$\begin{aligned} L^{(MP)}_{mn} = \left\{ \begin{array}{ll} (n-1)\alpha _{n-1} + \sum \nolimits _{k=0}^{n-2}\alpha _k, &{} \;\;\;\; m = n > 1, \\ 2(n-1)\sqrt{\beta _n}, &{} \;\;\;\; m = n + 1, \\ 2(m-1)\sqrt{\beta _m}, &{} \;\;\;\; m = n - 1, \\ 0, &{} \;\;\;\;\mathrm{otherwise}. \end{array}\right. \end{aligned}$$

(6.71)

where $\alpha _n$ and $\beta _n$ are the coefficients in the three term recurrence relation discussed in Chap. 2. It should be clear that the transformation of $L^{(MP)}_{nm}$ with the transformation $T_{in}=\sqrt{w_i}M_n(x_i)$, yields the discrete pseudospectral (ps) representation $L^{(ps)}_{ij}$ given by

$$\begin{aligned} L^{(ps)}_{ij} = -\sum _{k=1}^Nx_kD_{ki}D_{kj}, \end{aligned}$$

(6.72)

where in Eq. (6.48), $B(x_k) = x_k$. An equivalent pseudospectral representation can also be calculated for the Sonine-Laguerre quadratures.

A comparison of the convergence of the eigenvalues of the Fokker-Planck operator for a hard sphere cross section versus the size of the basis set $N$ is shown in Table 6.1. It is clear that the convergence of the eigenvalues is much faster with the Maxwell polynomials as basis functions. The first nonzero eigenvalue, $\lambda _1$, requires 150 Laguerre polynomials for convergence to five significant figures whereas only 6 Maxwell polynomials are required. The higher eigenvalues also converge very quickly with the Maxwell polynomial basis set.

Risken and Voigtlaender (1984) introduced the Maxwell polynomials in the transformation of the Fokker-Planck eigenvalue problem to a Schrödinger equation presented in the following section. They studied the relaxation of neutrons in a heavy gas moderator with the assumption that the Lorentz Fokker-Planck equation is applicable. They report eigenvalues in agreement with the results listed in Table 6.1 calculated with a continued fraction method (Risken and Till 1996).

The pseudospectral method of solution of the Lorentz Fokker-Planck equation is applicable also to realistic energy dependent momentum transfer cross sections (McMahon and Shizgal 1985) and also with a nonzero electric field (Shizgal and McMahon 1985). In these applications, the nonclassical polynomial basis sets that are used for the eigenvalue problem are orthonormal with respect to $P_0(x)$ parametrized by the momentum transfer cross section and the electric field strength, Eq. (6.66). The method was also applied to the relaxation of positrons in He and Ne (Shizgal and Ness 1987), for an oscillatory electric field (Viehland et al. 1988) and in systems with an admixture of a strongly electron attaching gas such as SF$_6$ and CCl$_4$ (Shizgal 1988). A review of this subject was presented by Shizgal et al. (1989).

6.3.2 Transformation of the Fokker-Planck Eigenvalue Problem to a Schrödinger Equation; Supersymmetric Quantum Mechanics

The eigenvalue problem of the Fokker-Planck equation is

$$\begin{aligned} A(x)\frac{d\psi _n}{dx} - B(x)\frac{d^2\psi _n}{dx^2} = \lambda _n\psi _n. \end{aligned}$$

(6.73)

We transform the independent variable $x$ to a new variable $z$ defined by

$$ z = \int \limits ^x\frac{1}{\sqrt{B(x')}}dx',$$

so that

$$\frac{dz}{dx} = \frac{1}{\sqrt{B}}\;\;\;\;\;\;\;\mathrm{and}\;\;\;\;\;\;\;\frac{d}{dx} = \frac{1}{\sqrt{B}}\frac{d}{dz},$$

giving

$$\frac{A}{\sqrt{B}}\frac{d\psi _n}{dz}- \sqrt{B}\left( -\frac{B'}{2B^{3/2}}\frac{d\psi _n}{dz}+ \frac{1}{\sqrt{B}}\frac{d^2\psi _n}{dz^2}\right) = \lambda _n\psi _n.$$

where the prime, ($B' \equiv dB[x(z)]/dz$), denotes differentiation with respect to $z$. The eigenvalue equation in the new variable $z$ is

$$\begin{aligned} - \frac{d^2\psi _n}{dz^2}+ \left( \frac{A}{\sqrt{B}} +\frac{B'}{2B}\right) \frac{d\psi _n}{dz}= \lambda _n\psi _n. \end{aligned}$$

(6.74)

A function $C(z)$ is defined by

$$\psi _n(z) = e^{C(z)}\phi _n(z),$$

where the functions $\phi _n(z)$ will be shown to satisfy a Schrödinger equation. With $\psi '_n(z) \equiv d\psi _n(z)/dz$, we have that

$$\psi _n' = C'e^C\phi _n +e^C\phi '_n,$$

and

$$\psi _n^{''} = C^{''}e^C\phi _n+(C')^2e^C\phi _n+2C'e^C\phi '_n+e^C\phi ''_n.$$

The Fokker-Planck eigenvalue equation, Eq. (6.74), is rewritten in terms of $\phi _n(z)$ as

$$\begin{aligned} -\left[ C^{''}\phi _n+(C')^2\phi _n+2C'\phi '_n+\phi _n^{''}\right] +\left( \frac{A}{\sqrt{B}} + \frac{B'}{2B}\right) (C'\phi _n+\phi '_n) = \lambda _n\phi _n. \end{aligned}$$

(6.75)

We set the coefficient of $\phi '_n$ to zero and get the defining equation for $C(z)$, that is,

$$\begin{aligned} \frac{dC(z)}{dz} = \frac{1}{2}\left( \frac{A}{\sqrt{B}} + \frac{B'}{2B}\right) , \end{aligned}$$

(6.76)

which when integrated gives

$$\begin{aligned} C(z) = \frac{1}{2}\int \limits ^z\frac{A(z')}{\sqrt{B(z')}}dz' +\frac{1}{4}\ln B(z). \end{aligned}$$

(6.77)

With these definitions, the partial differential equation, Eq. (6.75), for $\phi _n(z)$ is the Schrödinger equation

$$\begin{aligned} -\frac{d^2\phi _n}{dz^2} +V(z)\phi _n(z) = \lambda _n\phi _n(z), \end{aligned}$$

(6.78)

where Eq. (6.76) has been used and the coefficient of $\phi _n$ in Eq. (6.75) is the potential $V(z)$, given by

$$\begin{aligned} V(z) = (C'(z))^2 - C''(z). \end{aligned}$$

(6.79)

The orthonormality of the eigenfunctions $\psi _n(x)$ is defined in terms of $P_0(x)$ and given by

$$\begin{aligned} \int \limits _0^\infty P_0(x)\psi _n(x)\psi _m(x)dx&= \delta _{nm},\nonumber \\ \int \limits _0^\infty P_0[x(z)]e^{C(z)}\phi _n(z)e^{C(z)}\phi _m(z)\sqrt{B(x(z))}dz&= \delta _{nm}, \end{aligned}$$

(6.80)

which is consistent with the normalization

$$\begin{aligned} \int \limits _0^\infty \phi _n(z)\phi _m(z)dz = \delta _{nm}, \end{aligned}$$

(6.81)

so that

$$\begin{aligned} P_0[x(z)]e^{2C(z)}\sqrt{B[x(z)]} = 1, \end{aligned}$$

(6.82)

which is consistent with the definition of $C(z)$. Moreover, if we set $W(z) = 2C'(z)$ we have that

$$\begin{aligned} \begin{boxed}{V(z) = \frac{W(z)^2}{4}-\frac{W(z)'}{2},}\end{boxed} \end{aligned}$$

(6.83)

and

$$\begin{aligned} W(z) = \frac{A}{\sqrt{B}}+\frac{B'}{2B}. \end{aligned}$$

(6.84)

Thus the equilibrium solution of the Fokker-Planck equation can be expressed as

$$\begin{aligned} P_0(x) = \exp [-\frac{1}{2}\int \limits ^x W[z(x')] dx'], \end{aligned}$$

(6.85)

and is the ground state of the Fokker-Planck equation with $\lambda _0 = 0$ as can be easily verified by differentiating $P_0(x)$ twice.

The function $W(z)$ is the “superpotential” of supersymmetric quantum mechanics. We have derived the formal relationship between the Fokker-Planck equation and the Schrödinger equation (Comtet et al. 1985; Cooper et al. 1995; Risken and Till 1996; Feizi et al. 2011). This close relationship between these two large classes of problems has been exploited to advantage in the study of nucleation (Demeio and Shizgal 1993a), electron relaxation in molecular gases (Demeio and Shizgal 1993b), relaxation in plasmas (Shizgal 1992) and other applications (Gomez-Ullate et al. 2009).

6.3.3 Pseudospectral Representation of the Schrödinger Equation; Supersymmetric Quantum Mechanics

The spectral representation of the Hamiltonian in the Schrödinger equation, Eq. (6.78), for a basis set $\{S_n(y)\}$ orthonormal with unit weight function is

$$\begin{aligned} H^{(sp)}_{nm} = -\int \limits _0^\infty S_n(y)S^{''}_m(y)dy + \int \limits _0^\infty S_n(y)V(y)S_m(y)dy. \end{aligned}$$

(6.86)

We integrate the first integral by parts so that

$$\begin{aligned} H^{(sp)}_{nm} = \int \limits _0^\infty S_n^{'}(y)S^{'}_m(y)dy + V_{nm}, \end{aligned}$$

(6.87)

where the potential matrix element is $V_{nm} = \int S_n(y)V(y)S_m(y)dy$. Define a second polynomial set $\{F_n\}$ orthogonal with weight function $w(y)$, that is,

$$\begin{aligned} S_n(y) = \sqrt{w(y)}F_n(y), \end{aligned}$$

(6.88)

where the weight function is defined as $w(y) = \exp (-\int W(y')dy')$ analogous to the equilibrium distribution for the Fokker-Planck equation, Eq. (6.85). Equation (6.87) can then be rewritten as,

$$\begin{aligned} H^{(sp)}_{nm} = \int w [F'_m + \frac{w'}{2w}F_m][F'_n + \frac{w'}{2w}F_n]dy + V_{nm}. \end{aligned}$$

(6.89)

If one of the cross terms, $F_m^{'}F_n$, in the integrand above is integrated by parts we find that,

$$\begin{aligned} H^{(sp)}_{nm} = \int w F'_nF'_mdy + [V_{nm} - \tilde{V}_{nm}], \end{aligned}$$

(6.90)

where $\tilde{V}_{nm}$ are the matrix elements of the potential

$$\begin{aligned} \tilde{V}(y) = \frac{1}{4}W^2(y) - \frac{1}{2}W'(y), \end{aligned}$$

(6.91)

defined in terms of the “superpotential”, $W(y)$, in supersymmetric quantum mechanics. We transform the spectral representation $H^{(sp)}_{nm}$ to the discrete representation with the transformation $\mathbf{T}$, that is,

$$\begin{aligned} H^{(ps)}_{ij} = \sum _{n=0}^{N-1} \sum _{m=0}^{N-1} T_{in}H^{(sp)}_{nm}T_{jm}, \end{aligned}$$

(6.92)

to give the final desired result, namely

$$\begin{aligned} \begin{boxed}{H^{(ps)}_{ij} = \sum _{k=1}^N D_{ki}D_{kj} + [V(y_i) - \tilde{V}(y_i)]\delta _{ij},}\end{boxed} \end{aligned}$$

(6.93)

If the potential of interest can be written as in Eq. (6.91), then a useful weight function for the definition of the basis set is given by the equilibrium distribution function or the ground state wave function, Eq. (6.85). For this choice, $\tilde{V}(y) = V(y)$, and the pseudospectral representation of the Hamiltonian reduces to

$$\begin{aligned} H^{(ps)}_{ij} = \sum _{k=1}^N D_{ki}D_{kj}. \end{aligned}$$

(6.94)

This approach has been described in detail by Shizgal and Chen (1996) and Lo and Shizgal (2006).

6.4 Relaxation and Wave-Particle Heating in Space Plasmas

The Fokker-Planck equation plays a dominant role in plasma physics (Chandrasekhar 1942; Spitzer 1962; Hinton 1983; Shoub 1987) and stellar astrophysics (Spitzer and Härm 1958; Binney and Tremaine 2008; Lemou and Chavanis 2010). We consider as was done in Chap. 5, the kinetic theory of a test particle of mass $m$ and charge $Z$ dilutely dispersed in a large excess of a second species of mass $M$ and charge $Z_b$ at equilibrium with temperature $T_b$ and number density $N_b$. The Coulomb differential scattering cross section for collisions between the charged particles interacting via a Coulomb potential is given by

$$\begin{aligned} \sigma (g,\theta ) = \Bigg (\frac{Z_bZe^2}{2\mu g^2}\Bigg )^2\frac{1}{\sin ^4(\theta /2)}. \end{aligned}$$

(6.95)

The cross section varies inversely as the fourth power of the relative velocity, $g$, and diverges for small scattering angle, $\theta$.

The Coulomb Fokker-Planck equation finds numerous applications in space science and in particular for the modelling of the solar and polar winds. The solar wind consists primarily of protons and electrons that escape the solar gravitational field. The polar wind (Lie-Svendsen and Rees 1996; Pierrard and Lemaire 1998) is analogous to the solar wind (Parker 1965; Vocks 2002) and represents the escape of ions from the ionosphere along open magnetic field lines at high latitudes (Marsch 2006; Echim et al. 2011).

6.4.1 Pseudospectral Solution of the Coulomb Fokker-Planck and Associated Schrödinger Equations; The Approach to Equilibrium and the Continuous Spectrum

The Fokker-Planck equation for Coulomb collisions is derived from the Boltzmann equation with the differential cross section, Eq. (6.95), with the assumption of small energy transfers in individual binary collisions between charged particles. The small energy transfer collisions are those with large impact parameters. Consistent with this approximation, the small angle singularity in the momentum transfer differential cross section is eliminated by restricting the scattering angle such that $\theta > \theta _{min}$ where $\sin ^2(\theta _{min}/2) = [1+\varLambda ]^{-1}$, where $\varLambda = \lambda _D/b_0$; $\lambda _D$ is the Debye length and $b_0=ZZ_be^2/2k_BT_b$, the impact parameter that corresponds to the scattering angle $\theta = \pi /2$ (Spitzer 1962; Mitchner and Kruger 1973; Hinton 1983).

The Fokker-Planck equation derived from the Boltzmann equation as discussed in the previous paragraph is

$$\begin{aligned} \frac{\partial f(v,t')}{\partial t'} = \frac{A}{v^{2}}\frac{\partial }{\partial v}\left[ G(v) \left( 1+\frac{kT_{b}}{mv}\frac{\partial }{\partial v}\right) \right] f(v,t'), \end{aligned}$$

(6.96)

where $A = (4\pi N_be^{4}Z^{2}Z_{b}^{2}/mM)\ln \varLambda$ and the diffusion coefficient is

$$\begin{aligned} G(v) = \mathrm{erf}(\sqrt{\frac{Mv^{2}}{2kT_{b}}}) -\sqrt{\frac{2Mv^{2}}{\pi kT_{b}}} \mathrm{exp}(-\frac{Mv^{2}}{2kT_{b}}), \end{aligned}$$

(6.97)

as discussed elsewhere (Karney 1986; Shizgal 2004; Chavanis 2006).

The steady state distribution from Eq. (6.96) is a Maxwellian. A dimensionless time $t'=t/\tau$ is defined with $\tau = ([2A/3\sqrt{\pi }][M/2k_BT_b]^{3/2})^{-1}$ and the reduced speed $x = v\sqrt{m/2k_BT_s}$ where the temperature parameter $T_s = {s^{2}}T_b$. The parameter $s$ is the quadrature scaling parameter introduced in Chap. 3. With these definitions, the Fokker-Planck equation is

$$\begin{aligned} \frac{\partial f(x,t)}{\partial t} = \frac{2}{s^3x^2}\frac{\partial }{\partial x}\Bigg [G_1(sx)\bigg (1+\frac{1}{2xs^2}\frac{\partial }{\partial x}\bigg )\Bigg ]f(x,t), \end{aligned}$$

(6.98)

where $G_1(sx)=h(\gamma )G(v)$, $\gamma = \sqrt{M/m}$ and $h(\gamma )=3\sqrt{\pi }/4\gamma ^{3/2}$.

If we set $f(x,t)=e^{-s^2x^2}g(x,t)$, the Fokker-Planck equation is

$$\begin{aligned} \frac{\partial g}{\partial t}= \frac{1}{s^2}\Bigg [A(x)\frac{\partial g}{\partial x} - B(x)\frac{\partial ^2 g}{\partial x^2}\Bigg ] = Lg, \end{aligned}$$

(6.99)

with $B(x) = G_1(sx)/(sx)^3$. The drift coefficient in terms of $B(x)$ is

$$\begin{aligned} A(x) = 2s^2xB(x)-\frac{2B(x)}{x}-\frac{dB(x)}{dx}. \end{aligned}$$

(6.100)

We are concerned with the eigenvalue problem

$$L\psi _n(x) = -\lambda _n\psi _n(x),$$

with $L$ defined with Eq. (6.99). The physical space pseudospectral representation of this Fokker-Planck operator is

$$\begin{aligned} L^{(ps)}_{ij} = \frac{1}{s^2}\sum _{k=1}^NB(x_k)[D_{ki}+x_k(s^2-1)\delta _{ki}][D_{kj}+x_k(s^2-1)\delta _{kj}]. \end{aligned}$$

(6.101)

If the scaling parameter, $s = 1$, Eq. (6.101) reduces to Eq. (6.48).

The eigenvalues are determined with the numerical diagonalization of the matrix $\mathbf{L}^{(ps)}$ of dimension $N$. The convergence of the lower order eigenvalues is shown in Table 6.2 and the rapid convergence is clear.

Table 6.2 Convergence of the eigenvalues of the pseudospectral representation of the Coulomb Fokker-Planck operator, Eq. (6.101), for mass ratio, $\gamma = 0.3$

We transform the Fokker-Planck eigenvalue equation to a Schrödinger equation as discussed in Sect. 6.3.3 and derive, after some algebra, the potential in $x$, that is

$$\begin{aligned} V_{-}(x)=\frac{G_1(x)}{x}(1-\frac{9}{16x^4})-3\bigg [1+\frac{\gamma ^2}{2} - \frac{3}{8x^2}\bigg ]e^{-\gamma ^2x^2} -\frac{9x}{16G_1(x)}e^{-2\gamma ^2x^2}. \end{aligned}$$

(6.102)

The potentials are shown in Fig. 6.1 for two mass ratios, $M/m = 0.01$ and 0.04, and the eigenvalues (bound states) are indicated with the horizontal lines. There are a finite number of discrete eigenvalues and the number of states diminishes with increasing mass ratio. Since the potential barrier is finite, the eigenstates are not true bound states (Corngold 1981) and could be referred to as “quasi-bound” states. The only bound state is the ground state with $\lambda _0 = 0$. However, it is readily verified that in the Rayleigh limit, $\gamma = 0$, $V_{-}(x) = x^2-3$ and the eigenvalues are all discrete and given by $\lambda _n = 4n$, the harmonic oscillator eigenvalues.

Fig. 6.1

Potential, $V_{-}(y)$, in the Schrödinger equation corresponding to the Coulomb Fokker-Planck equation for mass ratios $M/m = 0.01$ and 0.04. There are a finite number of eigenstates which are strictly not discrete. Reproduced from Shizgal (1991) with permission from Beylich A.E.: Rarefied gas dynamics. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, Wiley-VCH Verlag GmbH and Co. KGaA. pp. 22–29, (1991)

The converged eigenvalues in the table are compared with the semiclassical Wentzel-Kramers-Brillouin (WKB) eigenvalues (Miller and Good 1953), namely

$$\begin{aligned} \int \limits _{x_1}^{x_2}\sqrt{\lambda _n - V_{-}(x)}dx = \bigg (n+\frac{1}{2}\bigg )\pi , \end{aligned}$$

(6.103)

and the corresponding supersymmetric, SWKB, eigenvalues (Fricke et al. 1988)

$$\begin{aligned} \int \limits _{x'_1}^{x'_2}\sqrt{\lambda _n - W^2(x)}dx = n\pi , \end{aligned}$$

(6.104)

where the integral limits are the classical turning points. The agreement with the SWKB and WKB approximations is very good.

We expand the solution in the eigenfunctions of $L$ and the time dependence of the average energy is

$$\begin{aligned} \frac{E(t')}{E_{th}} = \sum _{k=0}^\infty c_ke^{-\lambda _k t'}, \end{aligned}$$

(6.105)

where $E_{th} = 3k_BT_b/2$ is the thermal energy and the coefficients are

$$\begin{aligned} c_k=\frac{2}{3}s^5a_k\int \limits _0^\infty e^{-s^2x^2}\psi _k(x)x^4dx. \end{aligned}$$

(6.106)

The coefficients $a_k$ are the expansion coefficients of the intial condition

$$g(x,0) = \sum _{k=0}^\infty a_k\phi _k(x).$$

The 7 eigenvalues for $\gamma = 0.3$ are shown in Table 6.2 as the horizontal lines in the potential functions $V_{-}(x)$ and $W^2(x)$ in Fig. 6.2. The additional eigenvalue in $V_{-}(y)$ is $\lambda _0 = 0$. The eigenfunctions $\psi _4(x)$ and $\psi _6(x)$ are also shown in Fig. 6.2 with the WKB eigenfunctions denoted with the symbols. Two examples of continuum eiegenfunctions are shown in Fig. 6.3. The symbols that coincide with the solid curves are the results with the WKB approximation. These numerical eigenfunctions are $L^2$ square integrable with the discrete quadrature (Reinhardt 1979) defining the norm.

Fig. 6.2

(Left hand graphs) (A) The potential $V_{-}(y)$ in the Schrödinger potential; (B) The super potential $W^2(y)$ with a minimum value of 0. The bound states are shown with the dashed horizontal lines. (Right hand graphs) Eigenfunctions of the Fokker-Planck equation (A) $\psi _4(x)$ and (B) $\psi _6(x)$. The symbols are the WKB approximations (Miller and Good 1953). Reproduced from Shizgal (1992) with permission from Taylor and Francis

The relaxation of the temperature is given by $T(t')=2E(t')/3k_B$ with $E(t')$ as in Eq. (6.105). The time variation of $T(t')/T_b$ is shown in Fig. 6.4 for four mass ratios includes the sum over discrete and continuous eigenvalues and is convergent. For mass ratio $M/m = 0.4$, there are no bound states and the discrete sum is over the continuous spectrum.

In Chap. 5, we discussed the properties of the discrete and continuous portions of the eigenvalue spectra of the Boltzmann integral collision operators. It is these spectral properties that determine the time dependent approach to equilibrium (Sospedra-Alfonso and Shizgal 2013). For the Boltzmann equation, there is always a discrete spectrum whereas for the Coulomb Fokker-Planck equation we have demonstrated that the spectrum can be completely continuous, except for $\lambda _0 = 0$. The approach to equilibrium is an exponential if there is at least one discrete eigenvalue, that is “the spectral gap” as discussed in Chap. 5. For mass ratios for which there are no discrete “quasi-bound” states, the approach to equilibrium can be a complicated nonexponential function of time (Corngold 1981). This may be the case for curve d in Fig. 6.4.

Fig. 6.3

The continuum eigenfunction of the Fokker-Planck collision operator for Coulomb collisions (symbols) in comparison with the WKB approximation (solid curve) with $T_b = 300$ K, $s = 0.6042$ and $\gamma = 0.04$; (A) $\lambda = 5.855$ and (B) $\lambda = 7.603$. Reproduced from Shizgal (1992) with permission Beylich A.E.: Rarefied gas dynamics. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, Wiley-VCH Verlag GmbH and Co. KGaA. pp. 22–29, (1991)

Fig. 6.4

Temperature relaxation for $T(0)/T_b = 24$ and $\gamma =$ a 0.2, b 0.3, c 0.4 and d 0.5. Reproduced from Shizgal (1991) with permission Beylich A.E.: Rarefied gas dynamics. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, Wiley-VCH Verlag GmbH and Co. KGaA. pp. 22–29, (1991)

We consider the variation of the energy coefficients in the energy relaxation, Eq. (6.105), versus the numerical continuous eigenvalues. This variation of $c(\lambda )$ versus $\lambda$ is shown in Fig. 6.5 for different scaling parameters. The discrete values of $\lambda _k$ and $c_k$ (or $\lambda$ and $c(\lambda )$) in the continuum vary with a change in the scaling parameter, $s$, or with a change in the number of quadrature points, $N$, but the variation of $c(\lambda )$ versus $\lambda$ is on the same curve as shown in Fig. 6.5. As a consequence, the pseudospectral solution of the Fokker-Planck equation provides a converged solution even though the continuum has not been treated rigorously. However, the analytic form of the time variation of the average energy, equivalently the temperature, very close to equilibrium (Corngold 1981; Shizgal 1991) has not been confirmed.

Fig. 6.5

Variation of the coefficients, $c(\lambda )$ (in units of 10$^3$) for the temperature relaxation versus the continuous eigenvalue, $\lambda$. $T(0)/T_b = 24$. The symbols are the numerical results for 4 different values of the scaling parameter. Reproduced from Shizgal (1991) with permission Beylich A.E.: Rarefied gas dynamics. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, Wiley-VCH Verlag GmbH and Co. KGaA. pp. 22–29, (1991)

6.4.2 Fokker-Planck Equation for Wave Particle Heating of Ions; Kappa Distributions, and Tsallis Nonextensive Entropy

Tsallis (1995) derived the Kappa distribution

$$\begin{aligned} f_{\kappa }(x) = C_\kappa \left[ \frac{1}{1+\frac{x^2}{\kappa +1}}\right] ^{\kappa +1}, \end{aligned}$$

(6.107)

in the development of a new form of entropy functional for problems in statistical mechanics. In Eq. (6.107), $C_\kappa = 2\pi \varGamma (\kappa +1)/[\sqrt{\pi (\kappa + 1)}]^3\varGamma (\kappa -\frac{1}{2})$ is a normalization such that $4\pi \int _0^\infty f_\kappa (x)x^2dx = 1$. In the limit, $\kappa \rightarrow \infty$, the Kappa distribution tends to a Maxwellian. The Tsallis nonextensive entropy formalism is a controversial topic (Nauenberg 2003; Tsallis 2004; Lutsko and Boon 2011).

In Chap. 4, we discussed the expansion of the Kappa distribution function, in Laguerre polynomials, Eq. (4.60). We demonstrated that the expansion in Laguerre polynomials is a divergent asymptotic series as the decay of $f_\kappa (x)$ as $x \rightarrow \infty$ is slower than that of the Laguerre weight function, $w(x) = \sqrt{x}e^{-x}$; see Fig. 4.9 (Mintzer 1965; Leblanc and Hubert 1997). It is clear that the normalization $C_\kappa$ does not exist for $\kappa \rightarrow 1/2$. The average kinetic energy, that is the average of $mv^2/2$ with $f_\kappa (x)$, defines the nonequilibrium “temperature”

$$\begin{aligned} \frac{T_\kappa }{T_b} = \frac{\kappa +1}{\kappa - \frac{3}{2}}, \end{aligned}$$

(6.108)

which also diverges for $\kappa \rightarrow 3/2$. A nonphysical feature of the Kappa distribution is that the nth moment diverges for $\kappa \rightarrow (n+1)/2$ (Treumann et al. 2004; Shizgal 2007). For this reason, Magnus and Pierrard (2008) could not generate the Gaussian quadrature weights and points for the Kappa distribution and used instead modified weight functions.

In space physics, the Kappa distribution has been employed to explain the nature of energetic distributions in space physics (Meyer-Vernet 2001; Livadiotis and McComas 2009), the heating of the solar chromosphere (Scudder 1994), the escape of charged particles from the solar atmosphere and from the high latitude terrestrial ionosphere known as the solar and polar winds, respectively (Pierrard et al. 2004; Pierrard and Lazar 2010). There is an ongoing effort in space physics to better understand the complex mechanism for the energization of ions and electrons by plasma waves (Schulz and Lanzerotti 1974; Stix 1992; Gary 1993).

One approach is based on a Fokker-Planck equation where the wave-particle interactions are modelled with a second diffusion operator (Nicholson 1983) that is

$$\begin{aligned} \frac{\partial f(x,t)}{\partial t}&= \sqrt{\frac{m}{M}} \left( \frac{1}{x^2}\frac{\partial }{\partial x}\left[ D_1(v_{th}x) (1+\frac{1}{2x}\frac{\partial }{\partial x})\right] f(x,t)\right. \nonumber \\&\qquad \qquad + \left. \frac{\alpha v_{th}}{2}\frac{1}{x^{2}}\frac{\partial }{\partial x}\left[ x^2D_2(v_{th}x)\frac{\partial }{\partial x}f(x,t)\right] \right) , \end{aligned}$$

(6.109)

where in the second differential operator term the parameter $\alpha$ is an adjustable parameter that controls the strength of the wave-particle interactions relative to the Coulomb collision rate. It is clear that for $\alpha = 0$, the steady state distribution is a Maxwellian. Equation (6.109) has been written in dimensionless time, $t = t'/t_0$, where $t_0 = [2N\sigma _{eff}\sqrt{2kT_b/M}]^{-1}$ and $\sigma _{eff} = [4\pi NZ^2Z_b^2e^4\ln \varLambda ]/(2kT_b)^2$.

The steady distribution obtained by setting $\partial f/\partial t = 0$ in Eq. (6.109) is given by

$$\begin{aligned} \frac{df_{ss}(x)}{f_{ss}(x)} = -\left[ \frac{2x}{1+\alpha v_{th}x^3 \frac{D_2(v_{th}x)}{\hat{D}_1(z)}}\right] dx, \end{aligned}$$

(6.110)

where

$$\begin{aligned} \hat{D}_1(z) = \mathrm{erf}(z) - \frac{2z}{\sqrt{\pi }}e^{-z^2}, \end{aligned}$$

(6.111)

with $z = \sqrt{\gamma }x, \; \gamma = M/m$. As a consequence of the wave-particle interaction diffusion term, the steady state solution of Eq. (6.109), $f_{ss}(v)$, is no longer a Maxwellian and depends on the ratio of the strength of the wave-particle diffusion term relative to the strength of Coulomb collisional relaxation, that is on $\alpha$, as well as the mass ratio $M/m$. The velocity dependence of this steady-state distribution function depends on both $\hat{D}_1(z)$ and $D_2(v_{th}x)$.

The choice of the wave-particle diffusion coefficient has been discussed in the literature (Crew and Chang 1985; Stix 1992; Ma and Summers 1999; Vocks 2002; Shizgal 2007). There is at present no theoretical model for the occurrence of a Kappa distribution except for the works of Ma and Summers (1999) and Hasegawa et al. (1985). We consider the wave-particle diffusion coefficient to be of the form $D_2(v_{th}x) = 1/(v_{th}x)$ following on the work of Ma and Summers (1999). To reproduce the result obtained by them, one has to choose $D_2(v_{th}x) = 1/(v_{th}x)$ and $v \gg v_{th}$, that is $\gamma \rightarrow \infty$ in which case $\hat{D}_1(z) \rightarrow 1$. Thus the mass dependence and the behaviour $\hat{D}_1(z) \approx z^{-3}$ as $z \rightarrow 0$ are not retained by setting $\hat{D}_1(z) \equiv 1$.

The steady state distribution function that is obtained in this limit is

$$\begin{aligned} \frac{df_{\kappa }(x)}{f_{\kappa }(x)} = -\left[ \frac{2x}{1+\alpha x^2 }\right] dx, \end{aligned}$$

(6.112)

and when integrated leads to the Kappa distribution, Eq. (6.107). In this way, the adjustable $\kappa$ parameter in exospheric models is interpreted in terms of the strength of wave-particle interactions and $\kappa = (1-\alpha )/\alpha$.

The steady state distribution, $f_{ss}(x)$, the Kappa distribution, $f_\kappa (x)$ and the Maxwellian are compared in Fig. 6.6 for two mass ratios and the arbitrary choice $\alpha = 1/8$. The steady state distribution has a more extended high energy tail than either the Kappa distribution or the Maxwellian. The mass ratios chosen correspond to O$^+$ and Fe$^+$ in the solar atmosphere. For the larger mass ratio the tail of the steady distribution is more extended than for the smaller mass consistent with the observed heating of the heavy minor ions in the solar atmosphere.

Fig. 6.6

Comparison of the Maxwellian (dotted curve), Kappa (dashed curve) and steady, $f_{st}(x)$ (solid curve) distributions. The diffusion coefficient for wave-particle interactions is $\hat{D}_2(x) = 1/x$, the mass ratio $m/M$ and $\alpha$ are (A) 16, 1/8 (B) 55.845, 1/8. For the Kappa distribution, $\kappa = (1-\alpha )/\alpha$. Reprinted from Shizgal (2007) with permission from Springer

Fig. 6.7

Approach to a steady state distribution, $f_{ss}(x)$, with an initial Maxwellian and the inclusion of wave-particle energization; $\alpha = 0.25$ and $\hat{D}_2(x) = 1/x$. Successive distributions are at reduced times equal to 0.4, 2, 6, 10 and 20 for $m/M$ equal to (A) 16 and (B) 55.85. (C) The heating of the gas is shown as $T(t)/T_b$ and (D) the Kulback entropy increases with time. The mass ratio, $m/M$, is equal to (O) 16, (Mg) 24.3 and (Fe) 55.85. Reprinted from (Shizgal 2007) with permission from Springer

For this application, we use the Chang-Cooper finite difference algorithm described in Sect. 6.2.3 to integrate the Fokker-Planck equation given by Eq. (6.109) with an initial Maxwellian distribution. With this numerical method, there is no reference to the eigenvalue spectrum of the operator in Eq. (6.109) as we have done for all the other applications. The solutions converge provided that the grid spacing in the finite difference reduced speed discretization and the time step are sufficiently small. The evolution of the distribution functions showing the heating of the tail of the distributions is shown in Fig. 6.7a, b for $m/M = 16$ and 55.85, respectively. The increase in the temperature is shown in Fig. 6.7c for several heavy ions in the solar atmosphere. This heating is consistent with observations that the temperature of the minor ions in the solar atmosphere increases with mass (Pierrard et al. 2004).

In Fig. 6.7d, we show the monotonic increase in the Kullback–Leibler entropy functional defined by

$$\begin{aligned} \varSigma (t) =4\pi \int x^2f(x,t) \ln \frac{f(x,t)}{f_{ss}(x)} d{x}. \end{aligned}$$

(6.113)

This final result demonstrates that the usual notions of entropy rationalizes the generation of a non-equilibrium distribution which is neither a Maxwellian nor a Kappa distribution. The nonextensive entropy formalism of Tsallis (1995) is not required as previously suggested (Collier 2004; Leubner and Vörös 2005) and references therein. A pseudospectral solution of the Fokker–Planck operator equation, Eq. (6.109), is of considerable interest, especially with concern to the properties of the eigenvalue spectrum of the operator with wave-particle interactions ($\alpha \ne$ 0).

6.5 Fokker-Planck or Smoluchowski Equation for Bistable Potentials

Potentials $U(y)$, where $y$ denotes a coordinate for internal rotation in a molecule about some symmetry axis, are known for many molecules including for example butane, C$_4$H$_{10}$, (Ryckaert and Bellemans 1978; Montgomery et al. 1979; Blackmore and Shizgal 1985b; Pastor and Karplus 1989; Travis and Searles 2006), hydrogen peroxide, HOOH, (Koput et al. 2001; Lin and Guo 2003; Lynch et al. 2004; Le et al. 2009) and chlorine peroxide, ClOOCl (Gomes and Pacios 1996). The cis-trans isomerization kinetics for such molecules can be modelled with a Fokker-Planck or Smoluchowski⁵ equation of the form

$$\begin{aligned} \frac{\partial P(y,t)}{\partial t} = \frac{\partial }{\partial y}\left[ U(y)P(y,t) + \frac{\partial B(y)P(y,t)}{\partial y} \right] = LP(y,t),\;\;\;y \in (-\infty ,\infty ),\nonumber \\ \end{aligned}$$

(6.114)

where $y$ is a reaction coordinate.

A simple model for the isomerization kinetics used by many researchers (Larson and Kostin 1978; Bernstein and Brown 1984; Voigtlaender and Risken 1985; Blackmore and Shizgal 1985a, b; Cartling 1987; Drozdov 1999; Drozdov and Tucker 2001; Felderhof 2008) is defined with the drift and diffusion coefficients given by

$$\begin{aligned} U(y) = y^3 -y,\;\;\;\;\; B(y) = \epsilon . \end{aligned}$$

(6.115)

This model potential is bimodal with two minima at $y = \pm 1$. The steady distribution is

$$\begin{aligned} P_0(y) = C\exp [-\frac{y^4}{4\epsilon } + \frac{y^2}{2\epsilon }], \end{aligned}$$

(6.116)

and has two sharp maxima at $y = \pm 1$, especially for $\epsilon$ small as shown in Fig. 6.8. The constant $C$ is a normalization. The model is also referred to as the quartic potential because of the form of $P_0(y)$. This type of Fokker-Planck equation with two stable states also has application to climate models (Nicolis and Nicolis 1981; Nicolis 1982; Shizgal and Chen 1997) and laser physics (Blackmore et al. 1986; Shizgal and Chen 1997). The recent work by (Blaise et al. 2012) provides an extensive bibliography on diffusion in a double well potential.

Fig. 6.8

Steady state bimodal distribution $P_0(x) = C\exp [-\frac{1}{\epsilon }(x^4/4 - x^2/2)]$ for $\epsilon = 0.02$ on the left and $\epsilon = 0.005$ on the right. The distribution of quadrature points with $N = 20$ is also shown

We study the time evolution of this system in terms of the eigenfunction expansion discussed in Sect. 6.2. The eigenvalues and eigenfunctions can be calculated numerically with the diagonalization of the spectral matrix representation of the linear Fokker-Planck operator, $L$, defined by Eq. (6.114), that is

$$\begin{aligned} L^{(sp)}_{mn} = \int \limits _{-\infty }^\infty w(y)B_n(y)LB_m(y)dy, \end{aligned}$$

(6.117)

where the nonclassical polynomials, $B_n(y)$, are orthonormal with respect to the weight function, $w(y) = P_0(y)$, that is

$$\begin{aligned} \int \limits _{-\infty }^\infty \exp \left[ -\frac{1}{\epsilon }\left( \frac{y^4}{4}-\frac{y^2}{2}\right) \right] B_n(y)B_m(y)dx = \delta _{nm}, \end{aligned}$$

(6.118)

as discussed in Chap. 2, Sect. 2.5.2.

Since $w(y)$ is even and the integrals are evaluated over $(-\infty ,\infty )$, the recurrence coefficients, $\alpha _n = 0$. The polynomials $B_n(y)$ are even when $n$ is an even number and odd when $n$ is an odd number. After some detailed algebraic manipulations presented in Appendix B of Lo and Shizgal (2006), the symmetric matrix representation of the Fokker-Planck operator is given by

$$\begin{aligned} L^{(sp)}_{mn} = \left\{ \begin{array}{ll} -(\beta _2 + \beta _1 - 1), &{}\;\; m = n = 2, \\ (n-1)(\beta _n + \beta _{n-1} - 1) + 2\sum \nolimits _{k=1}^{n-2} \beta _k, &{}\;\; m = n > 2, \\ (n-1)\sqrt{\beta _{n+1}\beta _n}, &{}\;\; m = n + 2, \\ (m-1)\sqrt{\beta _{m+1}\beta _m}, &{}\;\; m = n - 2, \\ 0 &{}\mathrm {\;\; otherwise}, \end{array}\right. \end{aligned}$$

(6.119)

where $\beta _n$ are the recurrence coefficients in the three term recurrence relation discussed in Chap. 2. The eigenvalue problem is $L\psi _n(y) = -\lambda _n\psi _n(y)$, with $L$ as defined by Eq. (6.114).

This matrix representation of the Fokker-Planck operator for the bimodal model is pentadiagonal where the off-diagonal elements for $m \ne n\pm 2$ are zero. As a consequence, the eigenfunctions are odd or even and there is no coupling between them. Thus, the matrix representation $L^{(sp)}_{nm}$ can be split into two separate matrices, $\mathbf{L}^{even}$ with even $n$ and $m$ and $\mathbf{L}^{odd}$ with odd $n$ and $m$. The spectrum of the Fokker-Planck operator is composed of a set of singlet and nearly degenerate triplets. With $B(y_i) = \epsilon$, the pseudospectral discrete matrix representation of the Fokker-Planck operator is given by Eq. (6.48)

$$\begin{aligned} L^{(ps)}_{ij}=-\epsilon \sum _{k=1}^ND_{ki}D_{kj}, \end{aligned}$$

(6.120)

and yields the same eigenvalues and eigenfunctions as obtained with the spectral representation. However, $L^{(ps)}_{ij}$, is a full matrix and the symmetry properties of the problem as determined from the structure of $L^{(sp)}_{nm}$ are not apparent. As demonstrated in Chap. 2, the quadrature points are distributed nonuniformly within the domain, as shown in Fig. 6.8 for two choices of $\epsilon$. Therefore, the pseudospectral approach is more flexible than the polynomial based spectral method as different weight functions can be easily used to improve the convergence. The matrix elements for such nonclassical basis functions may be difficult to calculate analytically as given by Eq. (6.119).

The convergence of the lower order eigenvalues for $\epsilon = 0.01$ is shown in Table 6.3 versus the number of quadrature grid points, $N$, for three different grids as defined by the weight functions shown in the table. As can be seen, the first nonzero eigenvalue $\lambda _1$ is extremely small relative to the other eigenvalues. The reciprocal of this eigenvalue represents the isomerization rate as discussed later. The three eigenvalues $\lambda _3 - \lambda _5$ are nearly degenerate and converge at different rates. The first set of results are obtained with the grid defined by $w_a(y) = P_0(y),\; y \in (-\infty ,\infty )$.

The convergence can be improved by taking advantage of the symmetry of the eigenstates and calculating the even and odd eigenvalues with different weight functions, $w_b^{(e)}(y)$ and $w_b^{(o)}(y)$, $y \in [0,\infty )$, as shown in the middle of the table. The convergence requires about half the number of quadrature points as with $w_a(y) = P_0(y)$ over the whole interval. The bottom portion of the table shows the convergence with the uniform grid for the Sinc collocation method (SCM) (Wei 1999; Amore 2006) as well as a comparison with the limited results by Dekker and van Kampen (1979).

Table 6.3 Convergence of the eigenvalues of the Fokker-Planck operator with the bistable potential, $U(y) = y^3 - y$; $\epsilon = 0.01$

Table 6.4 Convergence of the eigenvalues of the Fokker-Planck operator with the bistable potential, $U(x) = x^3 - x$; $\epsilon = 0.001$

Fig. 6.9

The bimodal potential, $V(y)$, for $\epsilon$ equal to (A) 0.1, (B) 0.01 and (C) 0.001. The quadrature points with weight functions $w_a(y) = \exp [-(\frac{y^4}{4\epsilon }-\frac{y^2}{2\epsilon })] / \exp (1/4\epsilon )$; $y \in (-\infty ,\infty )$, $w_b^{(e)}(y) = w_a(y)\;\; y \in [0,\infty )$; $w^{(o)}_b(y) = y^2w_a(y)$; $y \in [0,\infty )$ and $w_c(y) = \exp [-(\frac{y^4}{4\epsilon }-\frac{y^2}{2\epsilon }] \exp (\frac{1}{4\epsilon }) + \exp (-\frac{y^2}{2\epsilon })$ are shown. Reprinted from Lo and Shizgal (2006) with permission of the American Institute of Physics

Fig. 6.10

Eigenvalue spectrum of the Fokker-Planck operator for the bistable potential; (Top graph) Variation of the smallest nonzero $\lambda _1$ eigenvalue versus $\epsilon$. (Bottom graph) Variation of the higher eigenvalues showing the splitting into singlet and triplet states. Reprinted from Blackmore and Shizgal (1985a); Copyright 1985 by the American Physical Society

The convergence of the eigenvalues for $\epsilon = 0$.001 is shown in Table 6.4 and a third weight function is chosen in order to accelerate the convergence. The weight function is $w_c(y) = P_0(y)+ \exp (-\frac{y^2}{2\epsilon })$ where the added exponential term yields quadrature points in the middle of the interval near the origin. The convergence of the eigenvalues is extremely rapid relative to the Sinc collocation method with a uniform grid.

The distribution of quadrature points relative to the bimodal potential is shown in Fig. 6.9 for three values of $\epsilon$. The distribution of grid points is uniform for all three values with the SCM. The quadrature points labelled QDM (a) are densely distributed over the region of the outermost potential wells. The acronym QDM is for the Quadrature Discretization Method which is the pseudspectral representation given by Eq. (6.120). The quadrature points for QDM (b) are defined over the interval $x \in [0,\infty )$ and are more densely distributed near the origin. The points for QDM (c) are distributed densely about the origin. The rate of convergence of the eigenvalues in Tables 6.3 and 6.4 is consistent with the distribution of the quadrature points for these different weight functions.

The structure of the eigenvalue spectrum is made clearer by considering the transformation to a Schrödinger equation and one finds the potential

$$\begin{aligned} V(y)=\frac{y^2(y^2-1)^2}{4\epsilon }-\frac{1}{2}(3y^2-1), \end{aligned}$$

(6.121)

which has three minima at

$$\begin{aligned} y^0 = 0,\;\;\;\; y^\pm =\pm \sqrt{\left[ \frac{2}{3}+\sqrt{\frac{1}{9}+2\epsilon }\right] }. \end{aligned}$$

(6.122)

In the limit $\epsilon \rightarrow 0$, the potential barriers between the two minima become larger and the potentials near the minima are quadratic, that is,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}V^\pm (y)&\rightarrow \frac{(y-y^\pm )^2}{\epsilon }-1,\;\;\;\;y\approx y^\pm , \\ \nonumber \lim _{\epsilon \rightarrow 0}V^0(y)&\rightarrow \frac{y^2}{4\epsilon }+\frac{1}{2} \;\;\;\;y\approx 0. \end{aligned}$$

(6.123)

and the corresponding eigenvalues are

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\lambda _k^\pm&\rightarrow 2k,\;\;\; k = 0,1,2, \ldots \\ \nonumber \lim _{\epsilon \rightarrow 0}\lambda _k^0&\rightarrow k+1,\;\;\; k =0,1,2, \ldots \end{aligned}$$

(6.124)

Thus in the very small $\epsilon$ limit the eigenvalues approach integer values, the zero eigenvalue is doubly degenerate and the remaining even eigenvalues are triply degenerate.

The importance of the distribution of grid points is illustrated in Fig. 6.9 where the grid points are shown in relation to the potentials in the Schrödinger equation. For $\epsilon = 0.1$ in Fig. 6.9a, the grid points are well distributed in the two wells of the potential. For $\epsilon = 0.01$, the potential has a minimum near the origin and $w_a(y) = P_0(y)$ does not properly capture the eigenfunction in this region whereas the quadrature over the positive interval defined with $w_b(y),\; y\in [0,\infty )$ has more quadrature points close to the origin than does $w_a(y),\; y \in (-\infty ,\infty )$. For $\epsilon = 0$.001, we use a weight function centred about the origin together with $w_a(y)$. These results illustrate the flexibility of a pseudospectral method based on nonclassical weight functions that accelerate the convergence.

The variation of the eigenvalues versus $\epsilon$ is shown in Fig. 6.10. The top graph illustrates the very rapid decrease of $\lambda _1$ with decreasing $\epsilon$. This eigenvalue represents the slowest mode and the reciprocal can be identified with the long time isomerization rate coefficient. The division of the other eigenvalues into singlet and triplet states is shown in the bottom graph with the triplet states converging to integer values for $\epsilon \rightarrow 0$. The objective of such modelling is to determine the nonequilibrium isomerization rate coefficient.

The cis-trans isomerization of n-butane has been studied by numerous researchers (Ryckaert and Bellemans 1978; Montgomery et al. 1979; Pastor and Karplus 1989; Shizgal et al. 1991; Travis and Searles 2006) with a particular potential reported by Montgomery et al. (1979) and also used by Marechal and Moreau (1984).

The solution of the time dependent Fokker-Planck equation for an initial delta function, $\delta (y-y_0)$, with all of the particles in one well at $y_0$ is given by

$$\begin{aligned} P(y,t) = \sum _{n=0}^\infty \psi _n(y_0)\psi _n(y)e^{-\lambda _nt}, \end{aligned}$$

(6.125)

where $\lambda _n$ are the eigenvalues of the Smoluchowski operator, $L$, in Eq. (6.114). The eigenvalues are calculated with the pseudospectral method with quadrature points and weights defined with the equilibrium density, $P_0(y)$ and the associated discrete derivative operator in physical space (Blackmore and Shizgal 1985a).

The number density of isomers in the potential well on the right for $y \in [0,\infty )$ is denoted by $N_A(t)$ and the isomerization rate equation is defined by

$$\begin{aligned} \frac{dN_A(t)}{dt} = -k(t)\bigg [ N_A^{eq}-N_A(t) \bigg ]. \end{aligned}$$

(6.126)

where the nonequilibrium time dependent rate coefficient is determined from the time dependent reactive flux over the barrier. From a correlation function formalism (Montgomery et al. 1979; Marechal and Moreau 1984; Blackmore and Shizgal 1985a), the time dependent nonequilibrium rate coefficient can be written in the form

$$\begin{aligned} k(t) = \sum _{n=0}^\infty A_n e^{-\lambda _n t}, \end{aligned}$$

(6.127)

with the $A_n$ coefficients given by

$$\begin{aligned} A_n = \lambda _n \bigg [\int \limits _0^\infty \psi _n(y)dy\bigg ]^2, \end{aligned}$$

(6.128)

and determined numerically with the nonclassical quadrature points. The details of this calculation were provided by Blackmore and Shizgal (1985a).

The nonequilibrium rate coefficient is compared with the transition state theory estimate given by

$$\begin{aligned} k_{tst}=S(0)\sqrt{\frac{k_BT_b}{2\pi m}}, \end{aligned}$$

(6.129)

where

$$S(y)=\frac{e^{-U(y)/k_BT_b}}{\int _{-\infty }^\infty e^{-U(y)/k_BT_b}dy}.$$

The time dependent rate coefficient for butane isomerization in the potential reported by Montgomery et al. (1979) is summarized in Fig. 6.11. The time dependent rate coefficient given by Eq. (6.127) relative to the equilibrium transition state theory (tst) rate coefficient, Eq. (6.129), is shown in the figure as the solid curves (a) and (b). The curve labelled (b) is for the potential as reported and the one labelled (a) is for an harmonic fit to this potential. The curve denoted by MM and MCB are the results by Marechal and Moreau (1984) and the simulations by Montgomery et al. (1979), respectively. There are many reactive systems and diffusion processes that are modelled with the Smoluchowski equation (Szabo et al. 1980; Bagchi et al. 1983; Chavanis 2006; Felderhof 2008) including protein folding (Bicout and Szabo 2000), dielectric relaxation (Coffey et al. 2009) and a Smoluchowski equation with a capture term (Spendier et al. 2013) that overlaps in some respects the studies of the nonequilibrium reactive system in Sect. 5.4.4.

Fig. 6.11

The time dependent nonequilibrium rate coefficient for butane isomerization with the potential reported by Montgomery et al. (1979) (b) and with an harmonic fit to the potential (a). The dashed curve (MM) is the result by Marechal and Moreau (1984) and the dotted curve (MCB) is the result by Montgomery et al. (1979). The collision frequency $\nu = 3\times 10^{12}$ sec$^{-1}$. Reprinted with permission from Blackmore and Shizgal (1985a); Copyright 1985 by the American Institute of Physics

6.6 Kramers Equation and Nonequilibrium Chemical Kinetics; A Spectral Solution

It has been long recognized that reactive processes for gaseous systems proceed with the perturbation of the species velocity distribution functions from Maxwellian (Ross and Mazur 1961; Shizgal and Karplus 1970; Shizgal and Napier 1996; Kustova and Giordano 2011; Dziekan et al. 2012). These analyses of the departure from Maxwellian are based on the Boltzmann equation. The fundamental quantities that define the Boltzmann equation are the cross sections for elastic and reactive collisional processes. We considered spectral methods of solution of the chemical kinetic Boltzmann equation in Chap. 5, Sect. 5.4.4. The theoretical description of the kinetics of isomerization reactions presented in Sect. 6.5 based on the Fokker-Planck or equivalently the Smoluchowski equation in position space assumed a Maxwellian distribution function of the particles in velocity space.

The Kramers equation (Kramers 1940) for the distribution function, $f(r,v,t)$, of a test particle at position $r$ and velocity $v$, at time $t$ in the potential $U(r)$ is given by

$$\begin{aligned} \frac{\partial f}{\partial t} -v\frac{\partial f}{\partial r} - \frac{F}{m}\frac{\partial f}{\partial v} = \nu \frac{\partial }{\partial v}\left( v + \frac{k_BT_b}{m}\frac{\partial }{\partial v}\right) f. \end{aligned}$$

(6.130)

The equation is comparable to the Boltzmann equation in Chap. 5, Eq. (5.30) with a drift term on the left hand side and a collision term on the right hand side. The force $F = -\partial U(r)/\partial r$ is derivable from an internal potential $U(r)$. The collision term is a particular choice which can be recognized as the Ornstein-Uhlenbeck Fokker-Planck operator in Sect. 6.1.2. The strength of the collision operator is denoted by the collision frequency $\nu$. Kramers equation is generally used to model isomerization reactions in liquids for which the potential is the internal molecular torsional potential about a symmetry axis in a molecule with cis-trans isomers. The frequency $\nu$ is related to the viscosity of the background liquid in which the isomerization rate is measured. There exists experimental data of the isomerization rates versus the viscosity of the liquid. The magnitude of $\nu$ reflects the strength of the coupling of the nonequilibrium system with the surroundings at equilibrium. When $\nu$ is large, the coupling is strong and the velocity distribution function approaches the equilibrium Maxwellian distribution at the heat bath temperature, $T_b$.

We continue the analogy of Kramers equation with the Boltzmann equation and relate the problem to rarefied gas dynamical problems where instead of $\nu$ we have the Knudsen number, 1/Kn, playing a similar role. In the small Kn collision dominated regime, departures from equilibrium are small and pertubation type methods such as the Chapman-Enskog method work remarkably well. The other extreme is the almost fully collisionless situation when Kn is very large and the Boltzmann equation can be solved with Liouville’s theorem. It is the intermediate situation when Kn $\approx$ 1 that is the most difficult to treat theoretically.

A review of the many different applications of Kramers equation in chemistry and physics is beyond the scope of this book. The reader is directed to the excellent review by Hänggi et al. (1990) that provides a large bibliography up to about 1990. Another good overview is the book by Risken and Till (1996). A more recent summary was provided by Pollak and Talkner (2005). References 1–40 in the introduction section of the paper by Voigtlaender and Risken (1985) refers to a large number of applications of the Kramers equation. Numerical methods for the efficient solution of the Kramers equation are of considerable importance (Berezhkovskii et al. 1996; Bicout et al. 2001; Schindler et al. 2005; Bi and Chakraborty 2009; Coffey et al. 2009; Müller et al. 2012).

In this section we describe a particular pseudospectral method of solution of the Kramers equation with the symmetric bistable double Morse potential introduced by Garrity and Skinner (1983) given by

$$\begin{aligned} U(r)=\frac{U_{m}}{[1-e^{-br_{0}}]^4}\bigg [1-e^{-b(r_{0}+r)}\bigg ]^2\bigg [1-e^{-b(r_{0}-r)}\bigg ]^2, \end{aligned}$$

(6.131)

where $r$ is an internal spatial coordinate, $r_{0}$ and $-r_{0}$ are the positions of the minima and $br_{0}$ is a parameter which controls the width of the barrier of height $U_m$ that separates cis-trans isomers, as in the previous section. The potential is shown for two values of $br_0$ in Fig. 6.12.

Fig. 6.12

The model bimodal double Morse potential (Garrity and Skinner 1983)

The “collision” operator on the right hand side of Eq. (6.130) that describes the coupling of the system with the surrounding heat bath is the Ornstein-Uhlenbeck Brownian motion Fokker-Planck operator discussed in Sect. 6.1.2. This is not the only choice for the operator that couples the reactive system with the surrounding heat bath. Garrity and Skinner (1983) used the Bhatnagar–Gross–Krook (Bhatnagar et al. 1954) model of the Boltzmann collision operator. The correspondence between the Boltzmann equation and the Kramers equation was made in the paper by Skinner and Wolynes (1980) and they proposed alternate collision operators in the Kramers equation that couple the reactive system with the heat bath.

In the high collision frequency limit, $\nu \rightarrow \infty$, the Kramers equation can be transformed to the Smoluchowski equation (Risken 1996; Gardiner 2003) for the distribution in $r$. This transformation involves the integration over the Maxwellian velocity distribution for the probability density in position (Blackmore 1985)

$$\begin{aligned} P(r,t)=\int \limits _{-\infty }^{\infty }f(r,v,t)dv, \end{aligned}$$

(6.132)

and the result is the Smoluchowski equation

$$\begin{aligned} \frac{\partial P}{\partial t} = L_{S}P, \end{aligned}$$

(6.133)

where $L_{S}$ is defined by,

$$\begin{aligned} L_{S}P = \frac{1}{\nu m}\frac{\partial U^{'}(r)P}{\partial r} + D\frac{\partial ^{2}P}{\partial r^{2}}, \end{aligned}$$

(6.134)

and $D = k_BT_b/m\nu$ is the diffusion coefficient as treated in the previous section.

The theoretical maximum reaction rate is the transition state theory (tst) estimate. The extent of the nonequilibrium reactive effects is determined by the magnitude of the coupling of the reactive system with the heat bath. If the coupling is strong, the nonequilibrium effects are small and conversely if the coupling is weak, the nonequilibrium effects can be large. This is similar to the treatment in Chap. 5 with the chemical kinetic Boltzmann equation where the elastic cross section controls the coupling with the heat bath.

The collision frequency $\nu$ plays the same role in Kramers equation as does 1/Kn in gas kinetic theory. We are interested in the solution of Kramers equation over the whole range of friction coefficient. In the large collision frequency limit, the Kramers equation is approximated by the Smoluchowski equation.

The Kramers equation in dimensionless variables, $x=v\sqrt{m/2kT}$, $\rho =r/r_{0}$, and $t'=t\nu /2$, is

$$\begin{aligned} \frac{\partial P(x,\rho ,t)}{\partial t'}&= \frac{\partial }{\partial x}\left[ \frac{\partial }{\partial x} +2x\right] P(x,\rho ,t) - \frac{1}{\nu }\left[ 2\nu \frac{\partial }{\partial \rho } - V(\rho )\frac{\partial }{\partial x}\right] P(x,\rho ,t) \nonumber \\&= L_{K}P(x,\rho ,t), \end{aligned}$$

(6.135)

where $\gamma = \sqrt{mr_{0}^{2}\nu ^{2}/2kT}$ is the friction coefficient and

$$\begin{aligned} V[\rho (r)] = -\frac{1}{kT}\frac{dU(r)}{dr}. \end{aligned}$$

(6.136)

We expand the probability density in the eigenfunctions of the Kramers operator, $L_{K}$. The eigenfunctions and eigenvalues can be complex as the Kramers operator is not Hermitian. We expand the eigenfunctions in Hermite polynomials in velocity $v$ and in the eigenfunctions of the Smoluchowski operator in the spatial coordinate $\rho$. The Hermite polynomials are the eigenfunctions of the Ornstein-Uhlenbeck “collision” operator as discussed in Sect. 6.1.2.

The system is initially prepared to be entirely in one of the potential wells, that is

$$\begin{aligned} P(x,\rho ,0) = \left\{ \begin{array}{ll} P_{0}(x,\rho ),\;\;\; &{} \rho > 0, \\ \\ 0, &{} \mathrm{otherwise}, \end{array} \right. \end{aligned}$$

(6.137)

where the equilibrium distribution is

$$\begin{aligned} P_{0}(x,\rho ) = Ne^{-x^{2}}\exp (\int \limits _{}^{\rho }V(\rho ')d\rho '), \end{aligned}$$

(6.138)

and $N$ is a normalization constant.

We expand $P(x,\rho ,t)$ in the eigenfunctions of $L_{K}$,

$$\begin{aligned} P(x,\rho ,t') = \sum _{n=0}^{\infty }a_{n}e^{-\lambda _{n}t'}\varPsi _{n}(x,\rho ), \end{aligned}$$

(6.139)

where

$$\begin{aligned} L_{K}\varPsi _{n}(x,\rho ) = - \lambda _{n}\varPsi _{n}(x,\rho ). \end{aligned}$$

(6.140)

The expansion coefficients, $a_{n}$, are determined by the initial condition,

$$\begin{aligned} a_{n}&= \int \limits _{-\infty }^{\infty }\int \limits _{-\infty }^{\infty }P(x,\rho ,0)\varPsi _{n}^{*}(x,\rho ) /P_{0}(x,\rho )dxd\rho , \nonumber \\&= \int \limits _0^{\infty }\int \limits _{-\infty }^{\infty }\varPsi _{n}^{*}(x,\rho )dxd\rho . \end{aligned}$$

(6.141)

With Eq. (6.139), it can be shown (Shizgal et al. 1991) that the time dependent relaxation time is given by

$$\begin{aligned} \tau ^{-1}(t') = \sum _{n=0}^{\infty }A_{n}\exp (-\lambda _{n}^{r}t')\left[ \lambda _{n}^{r}\cos ( \lambda _{n}^{i}t')+|\lambda _{n}^{i}|\sin ( |\lambda _{n}^{i}|t')\right] , \end{aligned}$$

(6.142)

where $\lambda _{n} = \lambda _{n}^{r} + i\lambda _{n}^{i}$ are the complex eigenvalues and

$$A_{n} = a_{n}\int \limits _{-\infty }^{\infty }\int \limits _{0}^{\infty }P_{n}(x,\rho )dxd\rho .$$

For a sufficiently large barrier separating the two minima, $\lambda _{1}$ will be much less than the higher eigenvalues and the relaxation time will tend to the limiting value,

$$\begin{aligned} \frac{1}{\tau } \rightarrow \lambda _{1}A_{1}, \end{aligned}$$

(6.143)

as $t \rightarrow \infty$. This result also requires that $A_{1}$ is of the order of unity and the remaining coefficients $A_{n}$ are very much smaller (Blackmore and Shizgal 1985b).

We compare the results obtained with Eq. (6.143) with the transition state theory (tst) value of the relaxation time

$$\begin{aligned} \frac{1}{\tau _{tst}} = S(0)\sqrt{kT/2\pi m}, \end{aligned}$$

(6.144)

where

$$\begin{aligned} S(r)=e^{-U(r)/kT}/\int \limits _{-\infty }^{\infty }\exp [-U(r')/kT]dr'. \end{aligned}$$

(6.145)

It is important to note that this result is independent of $\nu$.

The eigenvalue equation for the Kramers operator, $L_{K}\varPsi _{n} = -\lambda _{n}\varPsi _{n}$, is written in terms of the eigenfunctions, $\phi _{n}(x,\rho ) = \varPsi _{n}/P_{0}$ which satisfy the eigenvalue problem,

$$\begin{aligned} \tilde{L}_{K}\phi _{n} = [\frac{\partial }{\partial x} - 2x]\frac{\partial }{\partial x}\phi _{n} - \frac{1}{\gamma }[2x\frac{\partial }{\partial \rho } - V(\rho )\frac{\partial }{\partial x}]\phi _{n} = -\lambda _{n}\phi _{n}. \end{aligned}$$

(6.146)

Since the first operator in $x$ alone is diagonal in the Hermite polynomials, $H_{j}(x)$ (normalized to unity), we consider the expansion

$$\begin{aligned} \phi _{n}(x,\rho )=\sum _{j=0}^{\infty }c_{j}(\rho )H_{j}(x), \end{aligned}$$

(6.147)

in Eq. (6.146) and find that the coefficients satisfy the set of operator equations (Brinkmann 1956; Risken and Till 1996; Blackmore and Shizgal 1985b; Shizgal et al. 1991),

$$\begin{aligned} \sum _{j=0}^{\infty }\left( 2j\delta _{jk}+\frac{\sqrt{2j}}{\gamma } \left[ \frac{\partial }{\partial x}\delta _{j,k-1} + (\frac{\partial }{\partial \rho } - V(\rho ))\delta _{j,k+1}\right] \right) c_{j}(\rho ) = \lambda _n c_{k}(x).\nonumber \\ \end{aligned}$$

(6.148)

Equation (6.148) is a tri-diagonal system of coupled differential operator equations in the spatial variable (Brinkmann 1956; Risken and Till 1996).

The eigenfunctions, $S_{\ell }(x)$, of the Smoluchowski operator, $\tilde{L}_{S}$, defined by,

$$\begin{aligned} \tilde{L}_{S}S_{\ell }(\rho ) = -\frac{1}{\gamma ^{2}}\left[ \phi (\rho ) - \frac{\partial }{\partial \rho }\right] \frac{\partial }{\partial \rho } = -\lambda _{\ell }^{S}S_{\ell }(\rho ), \end{aligned}$$

(6.149)

are used as nonclassical basis functions to expand

$$\begin{aligned} c_{j}(\rho ) =\sum _{\ell =0}^{\infty }d_{j\ell }S_{\ell }(\rho ). \end{aligned}$$

(6.150)

The set of eigenvalue differential operator equations, Eq. (6.148), reduces to the matrix eigenvalue equation

$$\begin{aligned}&\sum _{k'=0}^{\infty }\sum _{\ell '=0}^{\infty }\left( 2k\delta _{k,k'}\delta _{\ell , \ell '} -\sqrt{2(k+1)}\delta _{k',k+1}G_{\ell ',\ell }/\gamma + \sqrt{2k}\delta _{k',k-1}G_{\ell ,\ell '}/\gamma \right) d_{k',\ell '}\nonumber \\&\qquad = \lambda d_{k,\ell }. \end{aligned}$$

(6.151)

The quantities

$$\begin{aligned} G_{\ell ,\ell '} = \int \limits _{-\infty }^{\infty }v(\rho )S_{\ell }(\rho )\frac{dS_{\ell '}(\rho )}{d\rho }d\rho , \end{aligned}$$

(6.152)

are the matrix elements of the derivative operator over the spatial variable with the eigenfunctions of the Smoluchowski operator, where the weight function is $v(\rho )= \exp (-\int _{}^{\rho }V(\rho ')d\rho ')$. These eigenfunctions form the nonclassical basis set for the solution of the Kramers equation.

The eigenfunctions of the Smoluchowsi operator which define the $G_{\ell , \ell '}$ matrix elements are determined with the pseudospectral solution of the Smoluchowski equation as discussed in Sect. 6.5. This particular numerical approach permits the efficient numerical evaluation of the derivative of the eigenfunctions with the derivative matrix operator, $\mathbf{D}$, and the evaluation of the matrix elements with the associated quadrature. These matrix elements are evaluated with the derivative operator and quadrature as

$$\begin{aligned} \begin{boxed}{G_{\ell \ell '} = \sum _{m=1}^Mw_m\frac{v(x_m)}{w(x_m)}S_\ell (x_m)\sum _{m'=1}^MD_{mm'}S_{\ell '}(x_m),}\end{boxed} \end{aligned}$$

(6.153)

where

$$w(x) = \exp [-U_m(x^4-2x^2)/(k_BT_b)],$$

defines the quadrature points, $x_m$ and weights, $w_m$. Thus we make use of both the pseudospectral derivative evaluation and the Gaussian quadrature, both for the nonclassical Smoluchoski eigenfunctions as basis functions. It is useful to mention that the matrix elements $G_{\ell \ell '}$ are the representations of the derivative operator in the basis set of Smoluchowski eigenfunctions.

A detailed consideration of the form of Eq. (6.151) shows that the characteristic polynomial which determines the eigenvalues factors into two polynomials, with eigenvalues that correspond to even and odd eigenfunctions (Voigtlaender and Risken 1985), and the dimensionality of the eigenvalue problem is reduced. The reaction rate, which is given by $\lambda _{1}$, is of primary interest. Consequently, the eigenfunction corresponding to this eigenvalue must satisfy the conditions, $P_{1}(x,-\rho )=-P_{1}(x,\rho )$ and $P_{1}(-x,\rho ) = P_{1}(x,\rho )$. Since the overall parity is odd, the eigenvalue $\lambda _{1}$ can be determined by restricting the calculation to the space of odd eigenfunctions.

Table 6.5 Convergence of $\lambda _1$ of the Kramers operator in units of 10$^{-2}$ ($\sqrt{k_BT_b/2m}/r_0$); $br_0 = 4$; $U_m/k_BT_b = 5$

Table 6.5 shows the convergence of $\lambda _{1}$ versus the number of Hermite polynomials $M$ and the number of Smoluchowski eigenfunctions $N$ for $br_{0} = 4$. The convergence of this eigenvalue is very rapid for large $\gamma$ as expected in view of the use of the Smoluchowski eigenfunctions as basis functions. We show the results only for an even number of Smoluchoswki basis functions, $N$ even. The results for $N$ odd are nonphysical as $\lambda _{1}$ increases with $N$. This feature of the matrix representation of the Kramers operator has been interpreted and discussed in detail by Shizgal et al. (1991).

Fig. 6.13

The ratio $\tau ^{-1}/\tau _{TST}^{-1}$ versus the friction coefficient $\gamma$ demonstrates the classic “turnover problem”. The value of $br_0$ is equal to (a) 0, (b) 2, (c) 3 and (d) 4. Reproduced from (Shizgal et al. 1991) with permission Beylich A.E.: Rarefied gas dynamics. In: Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, Wiley-VCH Verlag GmbH and Co. KGaA. pp. 85–92, (1991)

The variation of the relaxation rate relative to the transition state estimate is shown in Fig. 6.13 versus the friction coefficient. The present method of calculation of these isomerization rates gives reliable estimates down to $\gamma = 0.05-0.1$. This variation of the reaction rate with $\gamma$ with a maximum at some intermediate $\gamma$ is referred to as the “turnover” problem that has been investigated experimentally and theoretically since the publication of Kramers paper (Kramers 1940). It is the turnover of the graph in the Figure that has been difficult to calculate. With increasing $\gamma$, the convergence becomes more rapid.

There have been and continue to be many experimental studies of the isomerization rates in different solvents with different viscosities. The viscosity is related to the friction coefficient, $\gamma$, and thus the turnover in Fig. 6.13 has been verified experimentally (Pollak et al. 1989; Anna and Kubarych 2010; Pollak and Ianconescu 2014).

6.7 Sturm-Liouville Problems and the Schrödinger Equation

The Sturm-Liouville problem (Pryce 1993; Al-Gwaiz 2008) refers to the solution, either analytically or numerically, of the eigenvalue problem

$$\begin{aligned} L\psi _n(x) = \lambda _nw(x) \psi _n(x), \end{aligned}$$

(6.154)

where $w(x) > 0$ is a weight function and $L$ is the second order differential operator, defined by

$$\begin{aligned} Lf(x) = \frac{d}{dx}\left[ p(x)\frac{df(x)}{dx}\right] + q(x)f(x). \end{aligned}$$

(6.155)

It is useful to notice that this operator is in the form of a diffusion equation where $p(x)$ is a diffusion coefficient in a Fokker-Planck equation and $q(x)$ is a gain or loss term. We assume that $p(x) > 0$, $dp(x)/dx$, $q(x)$ and $w(x) > 0$ are real valued and piecewise continuous. Any linear second order differential equation can be written in this form. The eigenfunction, $\psi _n(x)$, defined on the interval $[a,b]$ is subject to two homogeneous boundary conditions which are linear combinations of the value of the function and derivative at the two interval end points and are of the form

$$\begin{aligned} A_1\psi _n(a) + B_1\psi '_n(a)&= 0,\nonumber \\ A_2\psi _n(b) + B_2\psi '_n(b)&= 0, \end{aligned}$$

(6.156)

where for $A_k = 0$ we have a Neumann boundary condition and if $B_k = 0$ we have a Dirichlet boundary condition.

In Sect. 6.2.1 we demonstrated the self-adjoint property of the Fokker-Planck operators subject to zero flux boundary conditions. The linear operator defined by the Sturm-Liouville problem is self-adjoint with respect to the boundary conditions, Eq. (6.156).

With the transformation of the independent variable from $x$ to $y$,

$$\begin{aligned} y=\int \sqrt{w(x)/p(x)}dx, \end{aligned}$$

(6.157)

and the transformation of the dependent variable $\psi _n(x)$ to $\phi _n(y)$ of the form

$$\begin{aligned} \psi _n(x)=m(x)\phi _n[y(x)], \end{aligned}$$

(6.158)

where $m(x)=[p(x)w(x)]^{-1/4}$, the Sturm-Liouville equation can be written in so-called Liouville normal form which is identical to a Schrödinger equation of the form

$$\begin{aligned} -\frac{d^2\phi _n(y)}{dy^2} + V(y) \phi _n (y) = \lambda _n \phi _n(y), \end{aligned}$$

(6.159)

where the potential function $V(y)$ is

$$\begin{aligned} V(y)=\frac{q[x(y)]}{w[x(y)]} + m[x(y)]\frac{d^2}{dy^2}(\frac{1}{m[x(y)]}). \end{aligned}$$

(6.160)

as derived by Pryce (1993).

6.7.1 Classical Polynomials as Eigenfunctions of the Sturm-Liouville and Schrödinger Equations

The classical polynomials discussed in this chapter (and other orthogonal polynomials) satisfy a Sturm-Liouville eigenvalue problem related to an associated Schrödinger equation. Many of the details of these relationships can be found in standard textbooks so we here outline the main results and the reader is referred to other references for a complete development. We presented a preliminary discussion in Sect. 3.9.3.

6.7.2 Legendre Polynomials; Quantized Rotational States of a Rigid Rotor

The rigid rotor model for a diatomic molecule has a fixed internuclear distance at $r_e$ and it is only the orientation of $\mathbf{r} = (r_e, \theta , \phi )$ in terms of the spherical coordinates that is of concern. The dependence on the azimuthal angle $\phi$ does not influence the rotational energy and thus only the dependence on $\theta$ is considered. The Schrödinger equation is

$$\begin{aligned} -\frac{\hbar ^2}{2I}\left[ \frac{1}{\sin \theta }\frac{d}{d\theta }(\sin \theta \frac{d\psi (\theta )}{d\theta })\right] = E\psi (\theta ), \end{aligned}$$

(6.161)

where $E$ is quantized, $I$ is the moment of inertia and the differential operator in $\theta$ is from the form of $\nabla ^2$ in spherical polar coordinates. With the substitution $x = \cos \theta$, Eq. (6.161) can be expressed as

$$\begin{aligned} H\psi _\ell (x) = -\frac{d}{dx}\left[ (1-x^2)\frac{d\psi _\ell (x)}{dx}\right] = \lambda _\ell \psi _\ell (x), \end{aligned}$$

(6.162)

where $H$ is the dimensionless Hamiltonian, $E_\ell = \lambda _\ell \frac{\hbar ^2}{2I}$ is the energy eigenvalue and

$$\begin{aligned} \lambda _\ell = \ell (\ell +1). \end{aligned}$$

(6.163)

These rigid rotor energy eigenvalues are precisely the eigenvalues of the total angular momentum operator $L^2$. There are two aspects that are important to note. The differential operator in Eq. (6.163) is of the Sturm-Liouville type and the differential operator on the left hand side is self-adjoint on the interval $x \in [-1,1]$. The eigenvalue equation, Eq. (6.163), is the defining equation for the Legendre polynomials, that is

$$\begin{aligned} \frac{d}{dx}\left[ (1-x^2)\frac{dP_\ell (x)}{dx}\right] = -\ell (\ell +1)P_\ell (x). \end{aligned}$$

(6.164)

Thus, the solution of this problem is $\psi _\ell (x) \equiv P_\ell (x)$. We have found the basis for which the Hamiltonian is diagonal $H_{\ell ,\ell '}= \ell (\ell +1)\delta _{\ell ,\ell '}$. This is the physical space representation. The discrete space representation can be obtained with the transformation $T_{\ell j}$ defined in terms of the Legendre polynomials $P_\ell (x)$, that is

$$\begin{aligned} H^{(ps)}_{ij} = \sum _{\ell =0}^{N-1} \sum _{\ell '=0}^{N-1} T_{i\ell }H_{\ell \ell '}T_{\ell 'i}. \end{aligned}$$

(6.165)

With the transformation, $T_{i\ell } = \sqrt{w_i}P_\ell (x_i)$, one can show that

$$\begin{aligned} H^{(ps)}_{ij} = \sum _{k=1}^N(1-x_k^2)D_{ki}D_{kj}, \end{aligned}$$

(6.166)

where $\mathbf{D}$ is the derivative matrix operator. The numerical diagonalization of this discrete matrix representation of the Hamiltonian of order $N$ gives $N$ eigenvalues $\lambda _\ell = \ell (\ell +1)$ exactly. This formalism was introduced in Chap. 3, Sect. 3.9.3, and Fig. 3.26 shows the exact eigenfunction obtained with Eq. (6.166). For Legendre polynomials, $P_\ell (x)$, defined in $x \in [-1,1]$ with $w(x) = 1$, $p(x) = (1-x^2)$ and $q(x) = 0$, we find easily from Eq. (6.155) that

$$\begin{aligned} -(1-x^2)P_\ell ^{''}(x) + 2xP_\ell ^{'} = \ell (\ell +1)P_\ell (x) \end{aligned}$$

(6.167)

which is the defining differential equation for Legendre polynomials.

6.7.3 Hermite Polynomials; Quantum Harmonic Oscillator

The Hermite polynomials $H_n(x)$ on $x \in (-\infty ,\infty )$, satisfy a Sturm-Liouville problem defined by $w(x) = p(x) = e^{-x^2}$ and $q(x) = 0$ in the general form Eq. (6.155). With these definitions, Eq. (6.155) gives the differential equation

$$\begin{aligned} H^{''}_n(x) - 2xH_n^{'} = -2nH_n(x). \end{aligned}$$

(6.168)

This differential equation can be written as a Schrödinger equation in terms of $h_n(x) = e^{-x^2/2}H_n(x)$. Notice that where $H_n(x)$ polynomials are orthogonal with respect to $w(x) = e^{-x^2}$, the basis functions $h_n(x)$ are orthogonal with unit weight function. The defining Schrödinger differential equation for these functions from Eq. (6.168) is

$$\begin{aligned} -h^{''}_n(x) + x^2h_n = (2n+1)h_n(x), \end{aligned}$$

(6.169)

where the term in $h^{'}_n(x)$ does not appear. This is precisely the dimensionless Schrödinger equation for a quantum harmonic oscillator as a simple model for the vibrational states of a non-rotating diatomic molecule.

If the interaction potential between the nuclei of a diatom is $V(r)$ where $r$ is the internuclear separation, the harmonic oscillator model involves the quadratic approximation of the potential at the minimum of the potential, that is,

$$\begin{aligned} V(r) \approx V(r_e) + \frac{1}{2}\frac{d^2V}{dr^2}\bigg |_{r=r_e}(r-r_e)^2. \end{aligned}$$

(6.170)

If we define the force constant as $k =\frac{dV}{dr}|_{r=r_e}$ and the displacement from $r_e$ as $x = r-r_e$, the one-dimensional Schrödinger equation is given by

$$\begin{aligned} -\frac{\hbar ^2}{2\mu }\psi ^{''}_n(x) +\frac{kx^2}{2}\psi _n(x) = E_n\psi _n(x), \end{aligned}$$

(6.171)

where $\hbar$ is the Planck constant and $\mu = m_1m_2/(m_1+m_2)$ is the reduced mass with $m_1$ and $m_2$ the masses of the two nuclei. Comparing Eqs. (6.171) and (6.169), we get the quantized vibrational states given by

$$\begin{aligned} E_n = (n+\frac{1}{2})\hbar \omega , \end{aligned}$$

(6.172)

where $\omega = \sqrt{k/\mu }$ is the fundamental frequency of the oscillator.

Fig. 6.14

(A) Relative accuracy $= 1-\lambda _n/(n+1/2)$ for eigenvalues of the quantum harmonic oscillator versus the vibrational quantum number $n$ obtained with the diagonalization of the $H_{ij}$ matrix (Eq. (6.174)) of dimension $N$; $x \in [-7,7]$. (B) The eigenfunction, $\psi _{11}(x)$, with the 10 Hermite quadrature points shown as the solid circles; $N = 112$

The harmonic oscillator is a typical benchmark problem for which a variety of different discretizations are compared. One such method is based on the representation of the second derivative operator on $x \in (-\infty ,\infty )$ with a uniform grid with spacing $\varDelta x$. The grid points are thus $x_i = i\varDelta x, i = 0, \pm 1, \pm 2, \ldots ,\pm N/2$, on the finite interval $[-N\varDelta x/2, N\varDelta x/2]$. This representation of the second derivative operator

$$\begin{aligned} D^2_{ij} = \left\{ \begin{array}{ll} \pi ^3/3, &{}\;\;\;\;\; i = j, \\ 2(-1)^{(i-j)}/(i-j)^2, &{}\;\;\;\;\; i\ne j, \end{array} \right. \end{aligned}$$

(6.173)

has been reported by Schwartz (1985); Colbert and Miller (1992); Mazziotti (1999); Amore (2006); Baye (2006) and others. The Hamiltonian matrix for the dimensionless quantum harmonic oscillator (Colbert and Miller 1992) is approximated by

$$\begin{aligned} H_{ij} = \frac{1}{2(\varDelta x)^2} \left\{ \begin{array}{ll} -\pi ^3/3, &{}\;\;\;\;\; i = j, \\ -2(-1)^{(i-j)}/(i-j)^2, &{}\;\;\;\;\; i\ne j. \end{array} \right\} + \frac{x_i^2}{2}\delta _{ij}. \end{aligned}$$

(6.174)

The relative accuracy of the approximate harmonic oscillator eigenvalues determined with the diagonalization of the matrix $H_{ij}$ of dimension $N \times N$ given by Eq. (6.174) is shown in Fig. 6.14(A) versus the quantum number $n$. The lower order eigenstates are well approximated but the error increases with increasing $n$. The size of the matrix is increased by halving the step size and keeping the interval fixed as shown in the figure and there is not much change for $N = 56$ to $N = 224$. Of interest is the variation of $\psi _{11}(x)$ in Fig. 6.14(B) with $N = 112$. The filled circles are the 10 Hermite quadrature points which coincide rather well with the nodes of the wave function. In order to get a good result for the higher eigenstates the interval has to be made larger.

The optimal basis functions are the Hermite polynomials which are the eigenfunctions of the harmonic oscillator Hamiltonian and the matrix representation of the Hamiltonian in this basis set is diagonal, $\langle n|H|m \rangle = (n+1/2)\delta _{nm}$, and provides the exact result. This result can be derived with the recurrence relations for the Hermite polynomials. This is the spectral solution of this elementary problem.

Baye and Heenen (1986) use a pseudospectral method (a Lagrange mesh method) based on the discrete physical space representation of the second derivative matrix operator in the Hermite polynomial basis

$$\begin{aligned} H_{ij} = \left\{ \begin{array}{ll} (4N-1-2x_i^2)/12 , \;\;\;\;\;\;\;\;\;\;\; i = j, \\ (-1)^{(i-j)}\bigg [\frac{1}{(x_i-x_j)^2}-\frac{1}{4}\bigg ]\;\;\;\;\;\; i \ne j.\end{array} \right\} +\frac{x_i^2}{2}\delta _{ij}. \end{aligned}$$

(6.175)

The diagonalization of this discrete matrix representation gives the eigenvalues $\lambda _n = n+1/2$ to machine accuracy for all but one eigenvalue even though the basis functions used are the exact eigenfunctions of the Hamiltonian. The results of this calculation are summarized in Table 6.6. The four eigenvalues, $\lambda _0$ to $\lambda _3$, for $N = 6$ to 9 are determined to machine accuracy, although only shown to three significant figures. For each $N$ there is a nonphysical eigenvalue referred to as a“ghost” level (Wei 1997; Willner et al. 2004; Kallush and Kosloff 2006) that are framed in the table. For the harmonic oscillator problem, $\lambda _{ghost} = (3N-2)/4$, and for $N = 8$ this coincides with an eigenvalue so there are two degenerate eigenvalues. This pattern repeats for $N = 10$ to 13, 14 to 18, etc.

Table 6.6 Ghost levels of the quantum harmonic oscillator determined with Eq. (6.175)

The reason for the appearance of nonphysical eigenvalues is often attributed to the inexact calculation of the matrix elements of the harmonic potential with the quadrature of order $N$ represented by the diagonal physical space matrix $V(x_i) = (x_i^2/2)\delta _{ij}$. The potential energy matrix elements

$$\begin{aligned} V_{nm} = \frac{1}{2}\int \limits _{-\infty }^\infty e^{-x^2}H_n(x)x^2H_m(x)dx, \end{aligned}$$

(6.176)

evaluated with an $N\mathrm{th}$ order quadrature

$$\begin{aligned} V_{nm}^{(N)} \approx \frac{1}{2}\sum _{i=1}^N w_iH_n(x_i)x_i^2H_m(x_i), \end{aligned}$$

(6.177)

is not exact. For the element $n = m = N$, the integrand is a polynomial of degree $2N+2$. The quadrature of order $N$ is exact only for polynomials of order up to $2N+1$. We have considered this calculation in detail in Chap. 3, Sect. 3.7.2. The transformation to spectral space of the physical space representation of the multiplicative potential operator with the diagonal matrix $V(x_i)\delta _{ij}$ gives an inaccurate result for the $V_{N,N}$ matrix element. The result with a quadrature of order $N$ in Eq. (6.177) is not correct.

The physical space pseudospectral representation of the harmonic oscillator Hamiltonian based on the Hermite polynomials is

$$\begin{aligned} H^{(ps)}_{ij}= \frac{1}{2}\sum _{k=1}^N D_{ki}D_{kj}. \end{aligned}$$

(6.178)

where $D_{kj}$ is given by Eq. (3.139). This representation does not have explicit reference to the harmonic potential as does Eq. (6.175). The usual concerns about the accuracy of the quadrature evaluated matrix elements of the potential related to the physical space representation as $V(x_i)\delta _{ij}$ do not play a role (Harris et al. 1965; Dickinson and Certain 1968).

A short MATLAB code constructs the derivative matrix operator $\mathbf{D}$ for the Hermite polynomials. The diagonalization of physical space matrix representation $\frac{1}{2}\mathbf{D}^t \cdot \mathbf{D}$ gives exactly $\lambda _n \equiv n$ relative to the ground state and $\psi _n(x_i) \equiv H_n(x_i)e^{-x^2/2}$. The solid curve in Fig. 6.15 shows the exact $H_{12}(x)e^{-x^2/2}$. The corresponding eigenfunction, $\psi _{12}(x_i)$, from the diagonalization of $\frac{1}{2}\mathbf{D}^t \cdot \mathbf{D}$ with $N = 12$ is shown with the symbols evaluated at the quadrature points. There is exact agreement between the numerical and analytical result.

Fig. 6.15

The solid curve is the exact Hermite polynomial whereas the symbols represent the values of the 12th eigenfunction of $\mathbf{D}^t\cdot \mathbf{D}$ calculated at 12 quadrature points defined by the Hermite weight function, $w(x)=e^{-x^2}$

6.7.4 The Schrödinger Equation for the Electron Relaxation Problem

We return to the electron relaxation problem discussed in Sect. 6.3 for the hard sphere cross section, ${\hat{\sigma }} = 1$, and zero electrostaic field, $\alpha = 0$. The Fokker-Planck equation, Eq. (6.63), leads to the eigenvalue problem Eq. (6.73) with $B(x) = x$ and $A(x)= 2x^2-3$ (Shizgal 1979). The transformation to the new variable $z$ in Sect. 6.3.2 which is $y$ in this section is defined by

$$y = \int \limits ^x \frac{1}{\sqrt{B(x')}}dx' = 2\sqrt{x}.$$

The coefficients in the Fokker-Planck eigenvalue problem are in terms of $y$,

$$A(y) = \frac{y^4}{2} - 3,$$

and

$$ B(y) = \frac{y^2}{4}. $$

The superpotential given by Eq. (6.84) is

$$ W(y) = \frac{y^3}{4} - \frac{5}{y}.$$

The potential $V_{-}(y)$ in the Schrödinger equation

$$\begin{aligned} -\frac{d^2\psi _n(y)}{dy^2}+V_{-}(y)\psi _n(y)=\lambda _n\psi _n(y), \end{aligned}$$

(6.179)

is defined in terms of $W(y)$ in Eq. (6.83) and

$$\begin{aligned} V_{-}(y)=\frac{y^6}{64}-y^2+\frac{15}{4y^2},\;\;\;\;\;\;\;\; y\in [0,\infty ). \end{aligned}$$

(6.180)

The notation $V_{-}(y)$ refers to one of the partner potentials in supersymmetric quantum mechanics, the other being $V_{+}(y)$. We refer the reader to references (Comtet et al. 1985; Cooper et al. 1995, 1987) for further details.

Fig. 6.16

Supersymmetric potential $V_{-}(y)$, Eq. (6.180), in the Schrödinger equation corresponding to the hard sphere Lorentz Fokker-Planck equation. The horizontal lines show the ordering of the eigenvalues in the potential

Table 6.7 Convergence of the eigenvalues of the Schrödinger equation with the potential, Eq. (6.180), with the quadrature defined with the weight function $w(y)=y^5e^{-y^4/16}$ and the pseudospectral representation $\mathbf{H} = \mathbf{D}^t\cdot \mathbf{D}$, Eq. (6.94)

The potential in Eq. (6.180) is shown in Fig. 6.16. The horizontal lines indicate the positions of the eigenvalues calculated as discussed below. At first glance one might consider the basis set of associated Laguerre polynomials or the discrete representation based on the Laguerre quadrature points. However, the optimal polynomial basis set is defined with the weight function equal to the known ground state wave function, that is

$$\begin{aligned} w(y) =\left\{ \begin{array}{ll} &{} \exp {[-\int \limits ^y \sqrt{W(y')}}dy'], \\ \\ &{}y^5e^{-y^4/16}, \end{array} \right. \end{aligned}$$

(6.181)

which gives $V(y) = {\tilde{V}}(y)$ in Eq. (6.93) and the pseudospectral matrix representation is as in Eq. (6.94) with the physical space derivative operator defined by the weight function Eq. (6.181).

A MATLAB code provides the recurrence coefficients for the polynomials orthogonal with respect to this weight function and the physical space derivative operator. The representation of the Hamitonian for this potential is $\mathbf{H} = \mathbf{D}^t\cdot \mathbf{D}$ given by Eq. (6.94). With the use of this nonclassical basis set and associated quadrature the matrix elements of the potential are not required as with other pseudospectral methods (Harris et al. 1965; Dickinson and Certain 1968).

The convergence of the eigenvalues calculated in this way versus the number of quadrature points is shown in Table 6.7. The convergence is rapid and from above so that the calculation is variational. At each order $N$ an upper bound to the eigenvalue is obtained. There is no occurrence of ghost levels. The bottom row of the table shows the eigenvalues obtained with the solution of the Fokker-Planck eigenvalue problem in complete agreement with the calculations based on the Schrödinger equation. The eigenfunctions corresponding to four eigenvalues are shown in Fig. 6.17. With $N = 80$, the oscillations of these converged eigenfunctions are well resolved.

Fig. 6.17

Eigenfunctions of the Schrödinger equation for the potential, $V(y)=\frac{y^6}{64}-y^2+\frac{15}{y^2}$ that arises from the transformation of the hard sphere Lorentz Fokker-Planck equation to the Schrödinger equation. Eigenfunctions calculated with the diagonalization $\mathbf{H} = \mathbf{D}^t\cdot \mathbf{D}$ with the pseudospectral derivative matrix operator based on the quadrature defined with the weight function $w(y)=y^5e^{-y^4/16}$, $N = 80$

6.7.5 Quantum Mechanics for the Vibrational States of a Diatomic Molecule; Morse Potential

The Morse interatomic potential for a diatomic molecule (Morse 1929) is given by

$$\begin{aligned} V(r) = D_e\bigg [1-e^{-\alpha (y-y_e)}\bigg ]^2, \end{aligned}$$

(6.182)

where $D_e$ is the dissociation energy and $\alpha$ determines the spatial variation relative to the equilibrium position $y_e$. The exact vibrational eigenvalues with $\hbar = 1$ and reduced mass $\mu = 1$ are

$$\begin{aligned} \epsilon _n = \bigg [2\alpha \sqrt{D_e}-\alpha ^2\bigg ]n -\alpha ^2n^2,\;\;\; n = 1,2, \ldots , n_{max}, \end{aligned}$$

(6.183)

relative to the ground state. There are a finite number of bound states denoted by $n_{max}$. Table 6.8 lists several diatomic molecules that have been studied by researchers to benchmark numerical methods of solution of the Schrödinger equation. For most of these studies, the interatomic potential is approximated with a Morse potential (Morse 1929). The numerical methods include finite difference methods, pseudospectral methods, methods based on B splines, the Discrete Variable Representation, the Lagrange mesh method, the Fourier grid method, the Sinc interpolation, and the Quadrature Discretization Method. Each method is based on the physical space representation of the derivative operator as determined with an interpolation. The methods are all variants of a pseudospectral method (Gottlieb and Orszag 1977; Francisco 1995; Fornberg 1996; Boyd 2001; Canuto et al. 2006b).

Table 6.8 References to numerical solutions of the Schrödinger equation for the vibrational states of diatomic molecules

Table 6.9 Convergence of the eigenvalues in cm$^{-1}$ for the Morse oscillator for HF with $D_e = 49383.407073$ cm$^{-1}$, $\beta = 1$.1741$a_0^{-1}$, $x_e = 1$.7329$a_0$ and $\mu = 1$744.4453572532$m_e$

The numerical methods differ primarily with regards to the choice of the basis functions and the application of boundary conditions. Fourier methods are applied on a uniform grid and the number of grid points per wavelength of the eigenfunction is an important parameter (Colbert and Miller 1992). Associated with some of the methods is a variational theorem so that the $N\mathrm{th}$ approximation represents an upper bound. For some methods there are nonphysical eigenvalues calculated that are referred to as “ghost” levels (Wei 1997; Kokoouline et al. 1999; Willner et al. 2004).

The Morse potential belongs to the class of potentials in supersymmetric quantum mechanics (Dutt et al. 1988; Cooper et al. 1995). The basis set defined with the weight function

$$\begin{aligned} w(x) = \exp \bigg [-2\sqrt{D_e}(x+\frac{e^{-\alpha x}}{\alpha }) + \alpha x\bigg ], \end{aligned}$$

(6.184)

for which the ground state wavefunction is $\psi _0(x) =\sqrt{w(x)}$ and $V(x) = \tilde{V}(x)$. The pseudospectral representation of the Hamiltonian, Eq. (6.93), reduces to Eq. (6.94). The diagonalization of $\mathbf{D}^t\cdot \mathbf{D}$ gives the eigenvalues and eigenfunctions.

The convergence of the vibrational energies $\epsilon _2$, $\epsilon _8$ and $\epsilon _{14}$ for the Morse potential for HF calculated with the quadrature defined with $w(x)$, Eq. (6.184), are shown in Table 6.9 and the rapid convergence of the eigenvalues is evident. It is clear that there is a variational theorem inherent in the method as the convergence of the eigenvalues is from above. For each $N$, an upper bound to the vibrational state is obtained.

The convergence demonstrated here is faster than reported by other researchers (Balint-Kurti and Pulay 1995; Braun et al. 1996; Hoffman et al. 1998; Baye and Vincke 1999) with different numerical methods. The spectral convergence of the eigenvalues is shown in Fig. 6.18 and several eigenfunctions are shown in Fig. 6.19. Unlike Fourier methods, this high order pseudospectral method with the particular basis set constructed with $w(x) = \psi ^2_0(x)$, does not require a particularly dense distribution of quadrature points to accurately calculate the higher order oscillatory vibrational eigenfunctions (Gottlieb and Orszag 1977; Francisco 1995; Fornberg 1996; Boyd 2001; Canuto et al. 2006b).

Fig. 6.18

The convergence of the lower order eigenvalues, $\lambda _n$, of the Morse potential with diagonalization of $\mathbf{D}^t\cdot \mathbf{D}$; Morse potential for HF with $D_e = 49383.407073$ cm$^{-1}$, $\beta = 1$.1741$a_0^{-1}$, $x_e = 1$.7329$a_0$ and $\mu = 1$744.4453572532$m_e$; Accuracy $= \log _{10}|1-\lambda _n^{(N)}/\lambda _n^{(exact)}|$

Fig. 6.19

Eigenfunctions determined from the diagonalization of $\mathbf{D}^t\cdot \mathbf{D}$ for selected vibrational states of HF versus $x$ in angstroms

An important aspect of these benchmark calculations is the total number of bound states for the potential chosen. If the potential supports $n_{max}$ states, the calculation of the lower states up to vibrational quantum number $n \approx 3n_{max}/4$ are relatively easy to calculate in spite of the oscillatory form of the eigenfunctions. It is the vibrational states close to the dissociation limit that are the most difficult to calculate.

We illustrate this feature with the model Morse potential employed by Pryce (1993) and Weideman (1999)

$$V(x) = 9(1-e^{-x})^2 -9,$$

which supports only three bound states,

$$\lambda _n = -n^2+5n-\frac{25}{4},\;\;\; n = 0, 1\; \mathrm{and} \; 2.$$

The vibrational states are determined with the quadrature based on the weight function

$$w(x) = \exp (-5x+6e^{-x})$$

for which $V(x) = \tilde{V}(x)$ and pseudospectral representation of the Hamiltonian is $\mathbf{D}^t\cdot \mathbf{D}$. The convergence of the 3 states to 14 significant figures requires 2, 20 and 55 quadrature points (Chen and Shizgal 2001). The numerical difficulty in the accurate calculation of the highest state with only two nodes is the diffuse nature of the eigenfunction just below the dissociation limit. This is illustrated in Fig. 6.20 for a higher order eigenfunction for H$_2^+$ determined with 100 B splines (ONeil and Reinhardt 1978). The numerical challenge is to capture both the oscillatory behaviour at small internuclear distance as well as the variation on a much larger scale for larger ($r>$20 $a_0$) distances. This behaviour was also demonstrated in Fig. 1 of Meshkov et al. (2008) for the eigenfunctions for a Lennard–Jones potential.

Fig. 6.20

Variational approximation to the $n = 19$ vibrational eigenfunction of H$_2^+$ with the potential from Wind (1965) with 100 B splines (Shore 1973). Reproduced from ONeil and Reinhardt (1978) with permission of the American Physical Society

The diatoms, He$_2$ and Ne$_2$, with 1 and 3 bound states, respectively, illustrate the same difficulty in the accurate representation of the highest bound state. The potentials for He$_2$ and Ne$_2$ were reported by Aziz and Slaman (1991) and Tang and Toennies (2003), respectively. A Morse potential that approximates the true potential (Lo and Shizgal 2008a) is used to define a quadrature based on the weight function, Eq. (6.184). For this realistic potential, $V(y) \ne \tilde{V}(y)$ and the pseudospectral representation of the Hamiltonian given by Eq. (6.93) is diagonalized.

In addition, a mapping, $u = \rho (x)$, is used to redistribute the points so as to best capture the variation of the wave function. Two such mappings are

$$\begin{aligned} \rho (x) = s_1\ln \bigg (\frac{x-b_2}{s_2}\bigg ) \end{aligned}$$

(6.185)

$$\begin{aligned} \rho (x)=s_1\sinh ^{-1}\bigg (\frac{x-b_2}{s_2}\bigg )+b_1, \end{aligned}$$

(6.186)

where $s_1$, $s_2$, $b_1$ and $b_2$ are adjustable parameters chosen empirically.

The single ground vibrational state for He$_2$ and the second excited vibrational state for Ne$_2$ are shown in Fig. 6.21. The variation of the wave function of the one bound state for He$_2$ occurs on two different spatial scales. There is a rapid variation near the origin and a very slow decay over a very large distance. The collocation points shown in the figure are distributed nonuniformly on the large interval of interest. The curves labelled QDM are calculated with the quadrature discretization method (Lo and Shizgal 2008a) and the two mappings above. The curves labelled by “Lag” refer to the Lagrange mesh method (Baye 2006).

Fig. 6.21

(Left hand panel) Single bound state eigenfunction, $\psi _0(x)$, for He$_2$ with the Aziz and Slaman (1991) interatomic potential. QDM is the quadrature discretization method with a nonclassical weight function with the mapping, Eq. (6.186); Lag refers to the Lagrange mesh method with the same map. (Right hand panel) The second excited state eigenfunction, $\psi _2(x)$, for Ne$_2$ with the Tang and Toennies (2003) potential determined with the QDM, with and without the mapping, Eq. (6.185). Reproduced from Lo and Shizgal (2008a) with permission of the American Institute of Physics

Pseudospectral methods applied to the entire interval or in subdomains of interest in which case it is referred to as a spectral element method (Deville et al. 2002; Pasquetti and Rapetti 2004) belong to that class of spectral and higher order numerical methods (Azaez et al. 2012). In every application in chemistry, physics and engineering, there are important applications to three and multidimensional problems. The extension from one-dimension to several dimensions generally involves a direct product of several one-dimensional polynomial basis sets. The size of the matrices for such problems increases dramatically with an increase with the number of degrees of freedom especially for the calculation of the rotational–vibrational states of polyatomic molecules (Friesner et al. 1993; Littlejohn et al. 2002; Dawes and Carrington 2004, 2005; Cassam-Chenaï and Liévin 2012). The numerics is then a problem in linear algebra to find the eigenvalues of a very large matrix.

A reduction in the dimensionality of the problem can be achieved by making use of available symmetries and with other techniques. Also, with a judicious choice of basis functions and/or grid points the number for each vibrational mode can be significantly decreased so as to achieve computational economy. In their case study of the vibrational states of methane, Mielke et al. (2013) introduce the use of optimized vibrational quadratures for the efficient computation of one-dimensional matrix elements. Any reduction in the number of grid points for each degree of freedom could dramatically decrease the dimension of the matrices resulting from the direct product of the different spaces for multidimensional problems. There are ongoing efforts to develop more efficient schemes for the development of sparse grids with algorithms related to cubatures discussed in Chap. 2, Sect. 2.8 (Avila and Carrington 2013; Lauvergnat and Nauts 2014).

6.7.6 Pseudospectral Solution of the Two Dimensional Schrödinger Equation for the Henon-Heles Potential; Nonclassical Basis Sets

Quantum problems in two and higher dimensions are often solved with a direct product of the basis sets for each one dimensional variable (Parrish and Hohenstein 2013) (and references therein). The resulting matrix representation of the Hamiltonian for a multidimensional system in either the spectral space or the physical space is the product of matrix representations for each dimension. Consequently the size of the matrices involved can increase very quickly if many basis functions or grid points are required in each dimension. This becomes a computationally challenging problem in order to reduce the dimensionality of the matrices by applying symmetries or particular numerical algorithms to reduce memory requirements and computational speed. The Milne problem (Lindenfeld and Shizgal 1983) and the associated planetary escape problem (Shizgal and Blackmore 1986) discussed in Chap. 5 are examples of problems in kinetic theory in three dimensions.

In this section, we consider the calculation of the eigenvalues of the two dimensional Hamiltonian

$$\begin{aligned} -\frac{1}{2}\bigg [\frac{\partial ^2 \psi _{nm}(x,y)}{\partial x^2} + \frac{\partial ^2 \psi _{nm}(x,y)}{\partial y^2}\bigg ] + V(x,y)\psi _{nm}(x,y) =\lambda _{nm}\psi _{nm}(x,y), \end{aligned}$$

(6.187)

where the potential is the Henon-Heles potential

$$\begin{aligned} V(x,y) = \frac{x^2+y^2}{2} -\lambda x(\frac{x^3}{3}-y^2). \end{aligned}$$

(6.188)

This potential was introduced by Henon and Heiles (1964) in their study of the motion of a star in the potential of the galaxy and the determination of a third conserved integral of the classical two dimensional motion with this Hamiltonian. The potential, Eq. (6.188), was chosen for its analytic simplicity so as to make the trajectory computations easy and to obtain interesting dynamical results. This system has received considerable attention as a model for classical and quantum chaotic behaviour. For $\lambda = 0$, the problem reduces to two uncoupled harmonic oscillators.

For the two dimensional Schrödinger equation, we choose basis functions

$$\begin{aligned} X_n(x)&= \sqrt{u(x)}G_n(x),\nonumber \\ Y_n(y)&= \sqrt{v(y)}H_m(y), \end{aligned}$$

(6.189)

where $u(x)$ and $v(y)$ are the weight functions and we denote the logarithmic derivatives of the weight functions by

$$\begin{aligned} U(x)&= -\frac{u'(x)}{u(x)}, \nonumber \\ V(x)&= -\frac{v'(x)}{v(x)}, \end{aligned}$$

(6.190)

We extend the pseudospectral analysis of the one-dimensional applications presented in Sect. 6.3.3 to two dimensions by defining the spectral space representation of the Hamiltonian as

$$\begin{aligned} H_{n^\prime m^\prime ,nm}&=\delta _{m^\prime m}\int u(x)G^\prime _{n^\prime }(x)G_n^\prime (x)dx + \delta _{n^\prime n} \nonumber \\&\quad \int v(y)H^\prime _{m^\prime }(y)H_m^\prime (y)dy +(V_{n^\prime m^\prime ,nm} - \tilde{V}_{n^\prime m^\prime ,nm}). \end{aligned}$$

(6.191)

The potential matrix elements are $V_{n^\prime m^\prime ,nm} = \langle X_{n^\prime } Y_{m^\prime }|V(x,y)|X_n Y_m\rangle$ and the matrix elements of the reference potential are

$$\begin{aligned} \tilde{V}_{n^\prime m^\prime ,nm}&= \delta _{m^\prime m}\int \bigg (\frac{U^2(x)}{4}-\frac{U^\prime (x)}{2}\bigg )u(x)G_{n^\prime }(x)G_n(x)dx \nonumber \\&\quad + \delta _{n^\prime n}\int \bigg (\frac{V^2(y)}{4}-\frac{V^\prime (y)}{2}\bigg )v(y)H_{m^\prime }(y)H_m(y)dy. \end{aligned}$$

(6.192)

We transform this spectral space representation with the appropriate transformation matrices, Eq. (1.24), and obtain the discrete physical space representation

$$\begin{aligned} H_{ij,k\ell } = \delta _{k\ell }\sum _{k'=0}^{N_x} D_{k'i}D_{k'j} + \delta _{ij}\sum _{k'=0}^{N_y} D_{k'k}D_{k'\ell } +\bigg [V(x_i,y_j)-{\tilde{V}}(x_i,y_k)\bigg ], \end{aligned}$$

(6.193)

where

$$\begin{aligned} {\tilde{V}}(x,y) = \bigg (\frac{U^2(x)}{4}-\frac{U^\prime (x)}{2}\bigg ) + \bigg (\frac{V^2(y)}{4}-\frac{V^\prime (y)}{2}\bigg ). \end{aligned}$$

(6.194)

is the usual reference potential in two-dimensions. The pseudospectral representation, Eq. (6.193) is the two-dimensional analog of the one-dimensional representation, Eq. (6.93). The details are similar to the transformations for the Fokker-Planck operator in Sect. 6.2.2.

Table 6.10 Eigenvalues of the Henon-Heles potential with $\lambda = \sqrt{0.0125}$ with $u(x) = \exp [-x^2+2\lambda x^3/9]$ and $v(y) = \exp (-y^2)$; $N_x = N_y = 32$

Two sets of quadratures were used; (1) Hermite quadratures for both dimensions and (2) a quadrature in $x$ based on a nonclassical weight function, $u(x) = \exp [-x^2+2\lambda x^3/9]$, chosen empirically and Hermite quadratures in $y$. With the Hermite quadrature in both dimensions, the lowest order eigenvalues required as few as 8 quadrature points in each variable to get convergence to 5 significant figures and up to 50 quadrature points for the higher states to the same accuracy. The nonclassical weight function provides faster convergence giving 8 significant figure accuracy with 32 quadrature points in each dimension.

The eigenvalues calculated in this way are listed in Table 6.10 in comparison with the previous calculations. Echave and Clary (1992) used Fourier basis functions to solve two one-dimensional reference problems and used the eigenfunctions of these hamiltonians as basis functions for the two-dimensional Henon-Heiles potential. They refer to this method that follows on the earlier work by Hamilton and Light (1986) as the potential optimized discrete variable representation. The results listed in the table are also in agreement with the results by Wei (1999) who used a collocation method referred to as a discrete singular convolution analogous to a Sinc interpolation as used by Amore (2006) and Amore et al. (2009). We discussed the Sinc interpolation in Chap. 2. We list the eigenvalues in the same manner as done by Noid and Marcus (1977). The results are in agreement to the accuracy in the table except for the (12,12) state for which Wei (1999) reports the value of 12.748431. The reason for this discrepancy is not known.

The contour plots of several eigenfunctions are shown in Fig. 6.22 and converged with 50 Hermite basis functions in each variable. The C$_{3v}$ symmetry is evident from the figure and several fine details of the eigenfunctions are recovered. It should be mentioned that this model system with the small value of $\lambda = \sqrt{0.0125}$ is very close to two uncoupled harmonic oscillators in each variable. The convergence of the Hermite polynomial basis set for each dimension for the lower states works as well as it does owing to the small value of $\lambda$. It would be of interest to consider larger values and experiment with nonclassical basis sets that might provide faster convergence.

Fig. 6.22

Contour plots of the eigenfunctions of the Schrödinger equation for the Henon-Heles potential with $n$ and $\ell$ equal to (A) 2, 0 (B) 6, 0 (C) 9, $-$9 and (D) 10, 6; Hermite quadratures were used, $N_x = N_y = 32$; the vertical scale is $y$ and the horizontal scale is $x$, both in the interval [$-$7, 7]. Reproduced from Shizgal and Chen (1997) with permission of the American Institute of Physics