Abstract
The Langevin equation is connected to the Brownian motion formulated by Einstein and Smoluchowski. The Langevin equation for a free particle with mass m is given by Langevin (CR Acad Sci Paris 146:530, 1908)
where x(t) is the particle displacement, v(t) is its velocity, γ is the friction coefficient and ξ(t) is a Gaussian random noise with zero mean 〈ξ(t)〉 = 0 (so-called white noise).
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The Langevin equation is connected to the Brownian motion formulated by Einstein and Smoluchowski. The Langevin equation for a free particle with mass m is given by Langevin [35] (for details, see Ref. [11])
where x(t) is the particle displacement, v(t) is its velocity, γ is the friction coefficient, and ξ(t) is a Gaussian random noise with zero mean 〈ξ(t)〉 = 0 (so-called white noise). Its correlation has the form
where k B is the Boltzmann constant, T is the absolute temperature of the environment in which the particle is immersed, and 2γk BT is the so-called spectral density. The notation 〈⋅〉 means ensemble averaging, i.e., statistical averaging over an ensemble of particles at a given moment of time t. Relation (6.2) represents the second fluctuation-dissipation theorem , which is valid only in case of internal noise ξ(t). The Langevin equation (6.1) actually is obtained from the second Newton law of motion of a particle in presence of viscous dynamic friction force \(-\gamma \dot {x}(t)\) and an internal random noise ξ(t), which is a residual force due to the interaction of the surrounding molecules on the particle. For a free particle, the MSD at long times reads
which is Einstein relation for the Brownian motion . From the MSD, one concludes that the Langevin equation (6.1) describes normal diffusion process, with diffusion coefficient given by
For the same process, the VACF has exponential decay in respect of time (for details, see next section)
For a particle in a given potential V (x), the corresponding Langevin equation becomes
where
is an additional force which acts on the particle due to the potential V (x). For harmonic potential
the Langevin equation (6.3) turns to
where ω is the frequency of the oscillator, and m is its mass.
For an internal noise whose correlation is not of the form (6.2), then the Langevin equation (6.3) becomes a GLE [34],
where we set m = 1, and γ(t) is the generalized friction memory kernel. The internal noise ξ(t) is of a zero mean (〈ξ(t)〉 = 0), whose correlation is given by
When the system reaches an equilibrium state, i.e., the noise is internal, the correlation is related to the friction memory kernel via the second fluctuation-dissipation theorem [34, 42, 72] in the following way:
This means that fluctuation and dissipation come from the same source. The friction memory kernel satisfies [12]
where \(\hat {\gamma }(s)=\mathcal {L}[\gamma (t)](s)\) is the Laplace transform of γ(t). If the fluctuation and dissipation do not come from the same source (in case of external noise ), then the second fluctuation-dissipation theorem (6.7) is not satisfied, and the system does not reach a unique equilibrium state. The GLE (6.5) for a free particle (V (x) = 0) in case of a stationary Gaussian random force ξ(t), in case when the second fluctuation-dissipation theorem holds, describes a stationary, Gaussian, non-Markovian process [19, 20].
GLE has been used to describe anomalous diffusion processes. In the pioneer work of Mainardi and Pironi [42], the authors introduced fractional Langevin equation and showed that it is a special case of a GLE. The M-L function appears in the analysis of the MSD and VACF for a given GLE. Thus, Mainardi and Pironi [42] for the first time in the literature represented the velocity and displacement correlation functions in terms of the M-L functions, and generalized the results for the standard Brownian motion (see also Ref. [40]).
6.1 Free Particle: Generalized M-L Friction
In this section we consider anomalous diffusion of a free particle with mass m = 1, driven by stationary random force ξ(t) [34, 42, 72]:
where the noise ξ(t) is internal noise. Therefore, the second fluctuation-dissipation theorem (6.7) holds.
The anomalous diffusion process can be modeled by GLE with internal noise, which correlation is of power-law form [5, 6, 12, 39, 68]
where C λ is a proportionality coefficient independent on time and which dependents on the exponent λ (0 < λ < 1 or 1 < λ < 2). In some investigations [40, 42] the friction memory kernel is represented as a superposition of Dirac delta and power-law function.
Generalization of the power law correlation function is the one parameter M-L correlation function [7, 66, 67]
where τ is the characteristic memory time, 0 < λ < 2, and E λ(z) is the one parameter M-L function (1.1). Furthermore, more generalized friction memory kernel of the form
was introduced [8, 16], where E λ,ν(z) is the two parameter M-L function (1.4).
We have introduced the three parameter M-L friction memory kernel [59]
where τ is the characteristic memory time, C α,β,δ may depend on α, β, and δ (α > 0, β > 0, δ > 0), and \(E_{\alpha ,\beta }^{\delta }(z)\) is the three parameter M-L function (1.14). This noise (6.9) contains several parameters and a number of limiting cases, which means that the obtained results can be used for better description and fits of experimental data. Note that, from relation (1.29), for the generalized M-L noise (6.9) one has
For fulfillment of the condition the friction memory kernel γ(t) goes to zero for t →∞ [12],
where \(\hat {\gamma }(s)=\mathcal {L}[\gamma (t)](s)\), one should consider such values of parameters for which β < 1 + αδ is satisfied.
The three parameter M-L noise (6.9) is a generalization of the two parameter M-L noise, which is obtained for δ = 1. For β = δ = 1 it yields the one parameter M-L noise. From the asymptotic behavior of three parameter M-L noise (6.9), for τ → 0, β = δ = 1 and α ≠ 1, one recovers the power-law correlation function. Setting α = δ = 1, the correlation function corresponds to the one for the standard Ornstein-Uhlenbeck process
which for τ → 0 turns to the correlation function for the standard Brownian motion.
6.1.1 Relaxation Functions
In order to find the MSD and VACF we use the Laplace transform method [40, 42], and the so-called relaxation functions . Thus, from Eq. (6.8) it follows
From relation (6.12) for the displacement x(t) and velocity \(v(t)=\dot {x}(t)\) one obtains
where
and
The function g(t) represents inverse Laplace transform of \(\hat {g}(s)\),
where
is obtained from Laplace transform formula (1.17) for the three parameter M-L function. The function
is also of interest in the analysis of the velocity and displacement correlation functions as we will see later. Therefore,
and
These functions I(t), G(t), and g(t) are known as relaxation function, and by analysis of their behavior one can show the existence of anomalous diffusion.
From relation (6.18) it follows
where \(\gamma _{\alpha ,\beta ,\delta }=\frac {C_{\alpha ,\beta ,\delta }}{k_BT\tau ^{\alpha \delta }}\). Relaxation function g(t) can be obtained by applying relation (1.18) with \(\alpha \rightarrow \frac {1+\beta }{2}\), ρ → α, γ → δ, λ →−γ α,β,δ, ν →−τ −α, μ → 1 to (6.22). Thus, we obtain [59]
By using relation (1.19) in (6.19) and (6.17), one finds
The mean velocity (6.16) and mean particle displacement (6.15) then become
Note that for τ → 0, by using relation (1.28), for the relaxation functions we have
from where for β = δ = 1, which corresponds to the power-law correlation function, we obtain the well-known results (see, for example, [39])
Remark 6.1 ([59])
The function g(t) given by (6.23) is uniformly convergent series with argument t∕τ for all \(t \in \mathbb {R}\). This can be shown in the following way. The function g(t) is a double series of form
where \(b_k=(-1)^k\gamma _{\alpha ,\beta ,\delta }^k\), and
To show that the series (6.32) converges uniformly, we have to demonstrate that both series with respect to columns (keeping k fixed and summing m) and the series with respect to the rows (summing k for fixed m) lead to uniformly convergent series. In that case the resulting function g(t) is continuous within the radius of convergence and can be integrated within the interval of convergence. As the three parameter M-L function (1.14) defines an absolutely converging function, which is easily demonstrated by a ratio test, we only need to verify the summation over k with fixed m. Let us use
By using [31]
we find that
which goes to zero if k →∞. Thus we prove that the series is uniformly convergent. The convergence of series in M-L functions has been extensively studied by Paneva-Konovska in a series of works [49,50,51].
6.1.2 Velocity and Displacement Correlation Functions
From the general expressions for the velocity and displacement correlation functions [12, 52]
we obtain the following exact results for the generalized M-L memory kernel [59]
For t = t′ it eventually leads to
Thus, for the time-dependent diffusion coefficient [42, 53]
by using relation (1.19) we obtain
Here we consider thermal initial conditions x 0 = 0 and v 0 = k BT. From the general expressions of the velocity and displacement correlation functions (6.35a) and (6.35b), one finds that the relaxation functions, under the assumption (6.11), are connected to the MSD, time dependent diffusion coefficient and VACF in the following way [12, 42, 66], respectively,
Furthermore, these relaxation functions can be used to find variances [12, 17, 66, 68]
Therefore, for thermal initial conditions, x 0 = 0 and \(v_{0}^{2}=k_{B}T\), for the MSD (6.39), D(t) (6.41) and VACF (6.36), we obtain [59]
respectively. Graphical representation of the MSD (6.46) and VACF (6.48), in case of thermal initial conditions is given in Figs. 6.1, 6.2 and 6.3.
Graphical representation of the MSD (6.46) for τ = 1, C α,β,δ = 1, k BT = 1; (a) β = δ = 1: α = 1 (solid line), α = 1∕2 (dashed line), α = 3∕2 (dot-dashed line); (b) α = δ = 1: β = 1 (solid line), β = 1∕2 (dashed line), β = 3∕2 (dot-dashed line); (c) α = 3∕2, β = 1, δ = 1∕2 (solid line), α = β = 1∕2, δ = 3∕4 (dashed line), α = 3∕4, β = 1∕2, δ = 1 (dot-dashed line). Reprinted from Physica A, 390, T. Sandev, Z. Tomovski, and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise, 3627–3636, Copyright (2011), with permission from Elsevier
Graphical representation of the VACF (6.48) for τ = 1, C α,β,δ = 1, k BT = 1; (a) β = δ = 1: α = 1 (solid line), α = 1∕2 (dashed line), α = 3∕2 (dot-dashed line); (b) α = δ = 1: β = 1 (solid line), β = 1∕2 (dashed line), β = 3∕2 (dot-dashed line); (c) α = 3∕2, β = 1, δ = 1∕2 (solid line), α = β = 1∕2, δ = 3∕4 (dashed line), α = 3∕4, β = 1∕2, δ = 1 (dot-dashed line). Reprinted from Physica A, 390, T. Sandev, Z. Tomovski, and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise, 3627–3636, Copyright (2011), with permission from Elsevier
Graphical representation of exact results (6.46) and (6.48), respectively, for τ = 10, C α,β,δ = 1, k BT = 1, α = 1∕2, and β = δ = 1 (solid line), β = 3∕4, δ = 1 (dashed line), β = 3∕4, δ = 1∕2 (dot-dashed line); (a) MSD (6.46); (b) VACF (6.48). Reprinted from Physica A, 390, T. Sandev, Z. Tomovski, and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise, 3627–3636, Copyright (2011), with permission from Elsevier
6.1.3 Anomalous Diffusive Behavior
The anomalous diffusive behavior of the particle can be obtained either from the exact results by using properties of the three parameter M-L function or by using the Tauberian theorems [18] (see Appendix B), as it was done by Gorenflo and Mainardi in Ref. [24]. From relation (1.28) it follows that
for long times (αδ ≠ β), so from the Tauberian theorems, one obtains [59]
From (6.18), (6.17), (6.19), (6.112), and (1.17) it follows
From the asymptotic expansion formula (1.28) of the three parameter M-L function, one finds
Thus, the time-dependent diffusion coefficient gets the form [59]
From (6.57) we conclude that for β − 1 < αδ < β in the long time limit the particle motion is subdiffusive , and for β < αδ—superdiffusive [59]. Note that for β = 1 the obtained results are same as those in Ref. [57] (where β = 1, ω = 0). For β = δ = 1, the results obtained in Refs. [39, 53, 59] are recovered. The case with α = β = δ = 1 corresponds to the one considered in Refs. [42, 53, 59]. For δ = 1 one derives the relaxation functions obtained in Ref. [8] (ω = 0, α = 2, β = 1). Comparison of the asymptotic and exact results for the MSD and VACF for thermal initial conditions is given in Fig. 6.4. In Fig. 6.5 comparison with the results for the Brownian motion is given.
Graphical representation of asymptotic and exact results for τ = 1, C α,β,δ = 1, k BT = 1, α = 1∕2, β = δ = 1; (a) MSD; asymptotic solution (6.56) (solid line), exact solution (6.25) (dashed line); (b) VACF, asymptotic solution (6.54) (solid line), exact solution (6.23) (dashed line). Reprinted from Physica A, 390, T. Sandev, Z. Tomovski, and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise, 3627–3636, Copyright (2011), with permission from Elsevier
Graphical representation of MSD and VACF, respectively, for k BT = 1; (a) standard Brownian motion \( \frac {\langle x^2(t)\rangle }{2k_{B}T}=t\) (solid line) and exact result (6.25) for α = β = δ = 1, τ = 1, C α,β,δ = 1 (dashed line); (b) standard Brownian motion C V(t) = e −t (solid line) and exact result (6.23) for α = β = δ = 1, τ = 1, C α,β,δ = 1 (dashed line). Reprinted from Physica A, 390, T. Sandev, Z. Tomovski, and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag-Leffler noise, 3627–3636, Copyright (2011), with permission from Elsevier
In the short time limit, the relaxation functions behave as [59]
These results can be obtained either by using Tauberian theorems or from the exact results by using the first two terms in the corresponding series. For β = 1 the results from Ref. [57] are obtained (β = 1, ω = 0), and for δ = 1 those given in Ref. [8] (ω = 0, α = 2, β = 1).
6.2 Mixture of Internal Noises
6.2.1 Second Fluctuation-Dissipation Theorem
Let us now consider a stationary Gaussian internal noise ξ(t) with a zero mean \(\left \langle \xi (t)\right \rangle =0\), represented as a mixture of N independent noises [58]
for which \(\left \langle \xi _{i}(t)\xi _{j}(t')\right \rangle =0\) (i ≠ j), each of zero mean \(\left \langle \xi _{i}(t)\right \rangle =0\), with correlation functions of the form
Thus, for the correlation function C(t) we have
Therefore, the second fluctuation-dissipation theorem (6.7) gives
Two (N = 2) distinct independent noises (white noise and an arbitrary noise) were analyzed in Ref. [65], and various diffusive regimes are observed. Such situations with two types of noises have been shown to govern the motion of the tracked particles in several experimental works by Weigel et al. [69], Tabei et al. [64], and Jeon et al. [27]. Therefore, our investigation of GLE (6.8) for a particle driven by mixture of noises is justified with such experimental observations.
6.2.2 Relaxation Functions
Here we use the known relations for the relaxation function (6.18), (6.20), and (6.21) in order to analyze the diffusive behavior of the particle. The Laplace transformation to relation (6.63) yields
In what follows we consider different forms of the noise that are of importance in the anomalous diffusion theory.
6.2.3 White Noises
First, let us consider the motion of a free particle driven by N internal white noises , i.e., ζ i(t) = δ(t) (\(\hat {\zeta }_{i}(s)=1\)). Relation (6.63) then becomes
The inverse Laplace transform for the relaxation function G(t) gives
from where the MSD (6.42) and VACF (6.44) read
From relation (6.66), one concludes that in the long time limit (t →∞), the MSD has a linear dependence on time
i.e., normal diffusive behavior of the particle, as it was expected, with diffusion coefficient
and exponential relaxation of the VACF. Graphical representation of the MSD and VACF for different values of N is given in Fig. 6.6.
Graphical representation of: (a) MSD (6.66), (b) VACF (6.67) for \(\alpha _{i}^{2}=2\), in case of thermal initial conditions v 0 = k BT = 1, x 0 = 0, and a mixture of Dirac delta noises N = 1 (solid line), N = 2 (dashed line); N = 3 (dot-dashed line); N = 4 (dotted line). Reprinted from Phys. Lett. A, 378, T. Sandev and Z. Tomovski, Langevin equation for a free particle driven by power law type of noises, 1–9, Copyright (2014), with permission from Elsevier
6.2.4 Power Law Noises
Next we analyze the case of N independent noises with power-law correlation functions
i = 1, 2, …, N, 0 < λ 1 < … < λ N < 1, λ i ≠ 1. From relation (6.63) one gets
Here we note that we can extend the analysis for the case of \(\hat {\gamma }(s)\) with 0 < λ 1 < … < λ N < 2, but in such a case the memory kernel γ(t) is defined only in the sense of distributions [9, 13, 22, 42, 43, 70]. For the relaxation function G(t), by using the approach given in Refs. [26, 37], we obtain
where \(A_{i}=\frac {\alpha _{i}^{2}}{k_{B}T}\),
are the so-called multinomial coefficients, and \(E_{\left (a_{1},a_{2},\ldots ,a_{N}\right ),b}\left (z_{1},z_{2},\ldots ,z_{N}\right )\) is the multinomial M-L function (1.35).
Let us analyze the case with 0 < λ 1 < λ 2 < 2, λ 1, λ 2 ≠ 1. From (6.68) we obtain
and thus
where \(E_{\alpha ,\beta }^{\delta }(z)\) is the three parameter M-L function (1.14) [54].
For the long time limit behavior, from (1.28), we obtain
From the MSD
we conclude that the particle shows anomalous diffusive behavior with the lower diffusion exponent λ 1 (0 < λ 1 < λ 2 < 2). Therefore, subdiffusion appears for 0 < λ 1 < 1 and superdiffusion for 1 < λ 1 < 2. VACF becomes
For the short time one finds
Thus, we conclude that the noise with the greater exponent λ 2 has dominant contribution to the particle behavior in the short time limit. For the variance in the short time limit we have
Graphical representation of the MSD and VACF for different values of parameters λ 1 and λ 2 is given in Fig. 6.7. The anomalous diffusive behavior of the particle is evident.
Graphical representation of: (a) MSD (6.70), (b) VACF (6.71) for A 1 = A 2 = 1, in case of thermal initial conditions v 0 = k BT = 1, x 0 = 0, and a mixture of two power law noises, for λ 1 = 1∕2, λ 2 = 3∕4 (solid line), λ 1 = 1∕2, λ 2 = 3∕2 (dashed line); λ 1 = 5∕4, λ 2 = 3∕2 (dot-dashed line). Reprinted from Phys. Lett. A, 378, T. Sandev and Z. Tomovski, Langevin equation for a free particle driven by power law type of noises, 1–9, Copyright (2014), with permission from Elsevier
6.2.5 Distributed Order Noise
Furthermore, let us instead of mixture of noises consider an internal noise of distributed order , i.e.,
Such memory kernel was used by Kochubei [33] in the theory of evolution equations with distributed order derivative, which is a useful tool for modeling ultraslow relaxation and diffusion processes. The Laplace transform of the memory kernel then becomes
We note that the assumption (4.32) is satisfied for this memory kernel since
Thus, we have
from where, by inverse Laplace transform, the relaxation function G(t) becomes
Here
whose Laplace transform reads
\(\Re (\alpha )>-1\), \(\Re (s)>-1\) [14]. For detailed properties and relations of these and related Volterra functions, we refer to the literature [2,3,4, 21, 46]. Thus, the relaxation functions are represented in terms of series in special functions \(\mu \left (t,\beta ,\alpha \right )\), and their representation in a closed form is an open problem. Here we use Tauberian theorems (see Appendix B and Refs. [24, 41] for details) to find the asymptotic behavior of the relaxation functions. In the long time limit (t →∞, i.e., s → 0 according to the Tauberian theorems) we obtain
where \(A=\frac {\alpha ^{2}}{k_{B}T}\), γ = 0.577216 is the Euler-Mascheroni (or Euler’s) constant ,
is the exponential integral function [14], and
From the asymptotic expansion formula
for t →∞ [41], which has error of order \(O\left (n!t^{-n}\right )\), for the relaxation function we obtain \(I(t)\simeq \frac {\gamma }{A}+\frac {1}{A}\log t\). The MSD has logarithmic dependence on time
and therefore the particle shows ultraslow diffusive behavior. In the same way, for the VACF in the long time limit (t →∞) we find
Similar relaxation functions were obtained by Mainardi [41] in analysis of fractional relaxation equation of distributed order. The short time limit (t → 0, i.e., s →∞) becomes
In the same way, for the VACF in the short time limit we obtained
Here we note that the same result can be obtained directly from the series expression (6.75).
A more general distributed order internal noise is of form
where p(λ) is the weight function. The case with p(λ) = α 2 yields the already considered uniformly distributed noise
For \(p(\lambda )=\sum _{i=1}^{N}\alpha _{i}^{2}\delta \left (\lambda -\lambda _{i}\right )\), where δ(λ) is the Dirac delta, 0 < λ i < 1, i = 1, 2, …, N, the mixture of N internal power-law noises
is recovered.
6.2.6 Mixture of White and Power Law Noises
As an addition, we analyze the GLE with mixture of P white noises , and Q power-law noises , where P + Q = N,
whose Laplace transform pair is given by
Here we also note that we can extend our analysis to exponents between 1 and 2, but in such a case the memory kernel is defined only in the sense of distributions [9, 13, 22, 42, 43, 70]. In the same way as previously described, we obtain
where \(A_{i}=\frac {\alpha _{i}^{2}}{k_{B}T}\) and \(B_{j}=\frac {\beta _{j}^{2}}{k_{B}T}\). We rewrite relation (6.79) in the following way
where \(0<\tilde {\lambda }_{1}<\ldots <\tilde {\lambda }_{Q+1}<2\). \(\tilde {\lambda }_{i}\) actually have the values of λ j and 1. For a given value r = i ∈{1, 2, …, Q + 1},
Note that if 0 < λ j < 1 then r = Q + 1, and if 1 < λ j < 2 then r = 1. Therefore, the relaxation function G(t) is represented through the multinomial M-L function (1.35),
Mixture of white and power law noises of the form
was considered in Ref. [65], for 0 < λ < 1. From (6.81) we obtain
where \(A=\frac {\alpha ^{2}}{k_{B}T}\) and \(B=\frac {\beta ^{2}}{k_{B}T}\). This relation yields
By using the asymptotic expansion formula (1.28), in the long time limit we obtain
Thus, the MSD becomes
which means that the particle shows subdiffusive behavior. From (6.85a) we note that the power law noise has dominant contribution to the particle behavior in the long time limit. These results are in agreement with those obtained in Ref. [65]. The short time limit yields
from where we conclude that both noises contribute to the particle behavior. The contribution of the white noise to the particle behavior in the short time limit is dominant. For variance (6.45a) we recovered the result obtained in Ref. [65],
for t → 0. Graphical representation of the MSD and VACF is given in Fig. 6.8.
Graphical representation of: (a) MSD (6.83), (b) VACF (6.84), in case of thermal initial conditions v 0 = k BT = 1, x 0 = 0, and a mixture of Dirac delta (A = 2) and power law (B = 1) noises, for λ = 1∕4 (solid line), λ = 1∕2 (dashed line); λ = 3∕4 (dot-dashed line). Reprinted from Phys. Lett. A, 378, T. Sandev and Z. Tomovski, Langevin equation for a free particle driven by power law type of noises, 1–9, Copyright (2014), with permission from Elsevier
In the same way, from (6.81), the case 1 < λ < 2 yields
and thus
From the asymptotic expansion formula we obtain the asymptotic behavior of relaxation functions
so the MSD is
It means that the particle shows normal diffusive behavior. Therefore, the white noise has dominant contribution to the particle behavior in the long time limit. This result is obtained by Mainardi et al. [42, 43] in case of friction memory kernel represented as superposition of white and power law noises. For the short time limit it follows
so the power-law noise has dominant contribution to the particle behavior in the short time limit. Here we note that the friction memory kernel, which represents superposition of white and power law noises in sense of distributions, was considered by Mainardi et al. [42, 43] for λ = 3∕2 and it was shown that the VACF behaves as C V ≃ t −3∕2. This result can be obtained from asymptotic expansion of relation (6.90),
λ = 3∕2, and represents a proof of the computer simulations of the VACF observed by Alder and Wainwright [1]. Graphical representation of the MSD and VACF is given in Fig. 6.9.
Graphical representation of: (a) MSD (6.88), (b) VACF (6.89), in case of thermal initial conditions v 0 = k BT = 1, x 0 = 0, and a mixture of Dirac delta (A = 2) and power law (B = 1) noises, for λ = 5∕4 (solid line), λ = 3∕2 (dashed line); λ = 7∕4 (dot-dashed line). Reprinted from Phys. Lett. A, 378, T. Sandev and Z. Tomovski, Langevin equation for a free particle driven by power law type of noises, 1–9, Copyright (2014), with permission from Elsevier
Let us now consider mixture of three noises, one of which is the white noise,
where 0 < λ 1 < 1 and 1 < λ 2 < 2. From relation (6.81) we obtain
i.e.,
and
where \(A=\frac {\alpha ^{2}}{k_{B}T}\), \(B_{1}=\frac {\beta _{1}^{2}}{k_{B}T}\) and \(B_{2}=\frac {\beta _{2}^{2}}{k_{B}T}\). The long time limit yields
which means that dominant contribution to the particle behavior in the long time limit has the noise with the exponent 0 < λ 1 < 1. Thus, the particle shows a subdiffusive behavior. The short time limit, again, yields ballistic motion \(I(t)\simeq \frac {t^{2}}{2}\). Graphical representation of the MSD and VACF is given in Fig. 6.10.
Graphical representation of: (a) MSD (6.94), (b) VACF g(t) = G′(t), in case of thermal initial conditions v 0 = k BT = 1, x 0 = 0, and a mixture of Dirac delta (A = 2) and power law (B 1 = B 2 = 1) noises, for λ 1 = 1∕4, λ 2 = 5∕4 (solid line); λ 1 = 1∕2, λ 2 = 5∕4 (dashed line); λ 1 = 3∕4, λ 2 = 5∕4 (dot-dashed line). Reprinted from Phys. Lett. A, 378, T. Sandev and Z. Tomovski, Langevin equation for a free particle driven by power law type of noises, 1–9, Copyright (2014), with permission from Elsevier
Here we note that combinations of white noise and anomalous diffusion were studied by Eule and Friedrich [15] and Jeon et al. [29].
6.2.7 More Generalized Noise
The mixture of white and two parameter M-L noise of form
for which
is further generalization of the previous cases of white and power-law noises. For ν = 1 we obtain the one parameter M-L noise, and for τ → 0—the power law noise. The case μ = ν = 1 gives the exponential noise, and the case μ = ν = 1 with τ → 0 recovers the Dirac delta noise. Similar M-L noises have been introduced in the literature to describe complex data related to anomalous diffusion [7, 12, 55, 57, 59, 60]. In case of the Dirac delta and the two parameter M-L noise,
the relaxation function G(t) becomes
where \(A=\frac {\alpha ^{2}}{k_{B}T}\), \(B=\frac {\beta ^{2}}{k_{B}T}\), \(C_{i}\in \left \{\tau ^{-\mu },A,A\tau ^{-\mu },B\tau ^{-\mu }\right \}\), and 
. Same approach can be performed in case of combination of the power-law and M-L noises,
since in this case for the relaxation function one finds
which can be represented in terms of the multinomial M-L functions (1.35).
6.3 Harmonic Oscillator
In this section we analyze the behavior of a harmonic oscillator driven by generalized M-L internal noise (6.9). The corresponding GLE for the harmonic oscillator with mass m = 1 and frequency ω driven by stationary random force ξ(t) is given by:
The GLE describes the particle dynamics bounded in the harmonic potential well and immersed in complex or viscoelastic media. The internal noise ξ(t) is of a zero mean (〈ξ(t)〉 = 0). Again we apply the second fluctuation-dissipation theorem since the considered noise is internal.
GLE (6.97) represents a suitable model for description of anomalous dynamics within proteins. Within given protein, the movements are bounded in small domains, thus the potential energy can be well approximated by the harmonic potential. Furthermore, the movements of the proteins are in a given complex liquid environment and its influence on the particle movement can be described by appropriate friction memory kernel. The high viscous damping, which is characteristic for the proteins in a liquid environment, will be described by neglecting the inertial term in Eq. (6.97). Information for the behavior of the oscillator will be obtained from the MSD, time dependent diffusion coefficient, and VACF. The normalized displacement correlation function, which is an experimental measured quantity, will be analyzed as well.
6.3.1 Harmonic Oscillator Driven by an Arbitrary Noise
Let us formally solve the GLE (6.8). From the initial condition x(0) = x 0 and \(\dot {x}(0)=v(0)=v_{0}\), one obtains
where \(\hat {F}(s)=\mathcal {L}[\xi (t)](s)\) and \(\hat {\gamma }(s)=\mathcal {L}[\gamma (t)](s)\). From Eq. (6.98) for x(t) and \(v(t)=\dot {x}(t)\) one finds
where
are the average displacement and average velocity, respectively. The function G(t) is the Laplace pair of
The same relations for the relaxation functions are valid, \(I(t)=\int _{0}^{t}G(t')\,\mathrm {d}t'\) and \(g(t)=\frac {\mathrm {d}G(t)}{\mathrm {d}t}\), as previously.
The MSD, time dependent diffusion coefficient, and VACF are related with the relaxation functions as previously, i.e., \(\left \langle x^{2}(t)\right \rangle =2k_{B}TI(t)\), D(t) = k BTG(t) and C V(t) = g(t), respectively [12]. These relations are valid for friction memory kernels which satisfy the assumption (6.11).
6.3.2 Overdamped Motion
From relation (6.103) we note that for the M-L noise (6.9) very complex expressions for the relaxation functions are obtained, and exact results are very difficult to be obtained. For simpler friction memory kernels of the Dirac delta type (standard Langevin equation), power-law type (fractional Langevin equation), one and two parameter M-L types the corresponding relaxation functions can be found exactly. In case of the three parameter M-L noise (6.9) the calculations become very complex, and thus one analyzes the asymptotic behavior of the oscillator in the short and long time limit. Therefore, instead of that, we analyze the overdamped motion, which means that there is high viscous damping, i.e., the inertial term \(\ddot {x}(t)\) vanishes. This case of high friction leads to same asymptotic behavior in the long time limit as the one for the GLE, so the overdamped motion can be used to analyze the anomalous diffusive behavior of the oscillator in the long time limit. This case of high viscous damping appears in the analysis of conformational dynamics of proteins, due to the liquid environment in which the proteins are immersed [10]. Thus, the relaxation functions \(\hat {g}(s)\), \(\hat {G}(s)\) and \(\hat {I}(s)\) become
By substitution of the friction memory kernel (6.9) in (6.104), by applying the Laplace transform formula (1.18), for \(\hat {I}(t)\) we obtain [55]
For the long time limit (s → 0), one finds the asymptotic behavior
Therefore, the MSD reads
and the VACF becomes
At long times t →∞, the MSD reaches the equilibrium value
For a free particle (ω = 0) from (6.107) one obtains
which is identical to (6.56) for the GLE for a free particle. In a similar way, for the relaxation functions G(t) and g(t), which are directly related to the time dependent diffusion coefficient and VACF, follow results (6.55) and (6.54), respectively.
Remark 6.2
Previous studies [57] showed that analytical treatment of the GLE with internal three parameter M-L noise with correlation function of the form
where C α,β,δ does not depend on time, and can depend on α, β and δ, where α > 0, β > 0, δ > 0, 0 < αδ < 2, is very complex. The difficulty of analytical treatment of the GLE with an internal noise with correlation (6.110) is due to the Laplace transform of three parameter M-L function, see relation (1.61) for κ = 1. Therefore, we only analyze the asymptotic behavior of relaxation functions by using Tauberian theorems (see Appendix B). For the Laplace pair of γ(t), from Eq. (1.61), we have
For the long time limit (t →∞) the frictional memory kernel has the following behavior
so the Tauberian theorem yields
where
Here we use that β ≠ αδ, β ≠ αδ − 1 and αδ ≠ 1. By substitution of (6.112) in the relaxation function
for 0 < αδ < 2, one obtains
From the Laplace transform formula (1.3) for the one parameter M-L function, it follows [57]
The MSD and VACF then read [57]
respectively. The case with β = δ = 1 corresponds to the results obtained in [12, 66, 67]. For a free particle (ω = 0) we obtain [57]
where we apply the Laplace transform formula (1.6). The MSD then becomes
and the time dependent diffusion coefficient and VACF turn to
respectively. Same result can be obtained from the L’Hôpital’s rule, i.e., [57]
Thus, the particle shows anomalous diffusive behavior. The well-known result for β = δ = 1 was obtained in Ref. [39, 52]. For α = β = δ = 1 one obtains ρ(t) ≃ t and \(C_{V}(t)\simeq E_{1}\left (-\gamma _{1,1,1}t\right )=e^{-\gamma _{1,1,1}t}\), which in fact is the result for Brownian motion [42, 52]. The case with αδ = 1∕2 gives \(C_{V}(t)\simeq t^{-\frac {3}{2}}\), which is theoretically obtained in Ref. [42] for superposition of the Dirac delta and power-law memory kernel, and previously confirmed by computer simulations for the VACF [1]. We can show, as well, that in case of a friction memory kernel which is a sum of the generalized M-L noise (6.110) and Dirac delta noise, the VACF has a form C V(t) ≃ t −αδ, so for \(\alpha \delta =\frac {3}{2}\) and β = 1 again we obtain the same result \(C_{V}(t)\simeq t^{-\frac {3}{2}}\) [57].
Remark 6.3
Let us now consider the following thermal initial conditions \(\left \langle x_{0}^{2}\right \rangle =\frac {k_{B}T}{\omega ^{2}}\), \(\left \langle x_{0}v_{0}\right \rangle =0\), and \(\left \langle \xi (t)x_{0}\right \rangle =0\) for the GLE for a harmonic oscillator. For the normalized displacement correlation function , which is an experimental measured quantity, and which is defined by Burov and Barkai [5, 6]
one obtains
For the friction memory kernel of form (6.9) in the limit τ → 0, for C X(t) we find
The graphical representation of the normalized displacement correlation function (6.124) is given in Fig. 6.11. Note that for ω = 0.3, C X(t) is a decreasing monotone function and C X(t) > 0. For ω = 3 and ω = 1, C X(t) has an oscillation-like behavior passing the zero line, and goes asymptotically to zero. For ω = 0.74, C X(t) > 0, but it is a non-monotone function. It approaches the zero line asymptotically. These results are different than those obtained for the Langevin equation for a harmonic oscillator, for which the oscillator has only two different behaviors; either overdamped motion with 〈x(t)〉 > 0 for all t under the condition 〈x 0〉 > 0, for which C X(t) is monotone function, or underdamped motion when 〈x(t)〉 has oscillation-like behavior passing the zero line [5, 6]. The frequency at which the oscillator turns from overdamped to underdamped motion is the so-called critical frequency. For the GLE for a harmonic oscillator there is a need for definition to additional critical frequencies on which C X(t) changes its behavior, and their computation is a non-trivial problem [5, 6]. Such behaviors of C X(t) were observed in the molecular dynamic simulations of fluctuation of the donor-acceptor distance within proteins [38]. Moreover, such oscillation-like behavior and power law decay of the fluorescein-tyrosine distance within a protein are experimentally observed in Ref. [47].
6.4 GLE with Prabhakar-Like Friction
As we showed before, the regularized Prabhakar derivative (2.88) is a special case of the generalized derivative (2.89), therefore we conclude that the GLE with regularized Prabhakar friction memory kernel of the form
has the form [56]
Here \({{ }_{C}}\mathcal {D}_{\rho ,-\nu ,t\,}^{\delta ,\mu }\) is the regularized Prabhakar derivative (2.88), 0 < μ, δ < 1, 0 < μ∕δ < 1, 0 < μ∕δ − ρ < 1, ν = τ −μ, τ is a time parameter, and γ μ,ρ,δ is the generalized friction coefficient. This equation is a generalization of the fractional Langevin equation considered by Lutz [39], which is recovered by setting δ = 0. The Laplace transform of the friction memory kernel (6.125) reads
By asymptotic expansion of the three parameter M-L function (1.28) and the Laplace transform of the friction memory kernel (6.127), we show that the assumption (4.32) is satisfied for μ > ρδ. We consider that the noise ξ(t) is internal, i.e., the second fluctuation-dissipation theorem of the form [56]
is satisfied.
6.4.1 Free Particle
From the general formulas for the relaxation functions, the MSD, D(t), and VACF become [56]
respectively.
The asymptotic expansion of the three parameter M-L function (1.29) for the long time limit yields
where \(\bar {\gamma }=\gamma _{\mu ,\rho ,\delta }\tau ^{-\rho \delta }\). Therefore, one concludes that in the system exists subdiffusion \(\left \langle x^{2}(t)\right \rangle \simeq t^{\alpha }\) with anomalous diffusion exponent α = μ − ρδ, where 0 < α < δ < 1.
Graphical representation of the MSD (6.129) is given in Fig. 6.12. From the figure we see that the MSD shows oscillation-like behavior for intermediate times which can be explained as a result of the cage effect of the environment represented by the M-L memory kernel [5].
6.4.2 High Friction
The high viscous damping, corresponding to vanishing of the inertial term \(\ddot {x}(t)=0\), yields [56]
Therefore, we conclude that decelerating subdiffusion exists in the system, since the anomalous diffusion exponent from μ for the short time limit turns to μ − ρδ in the long time limit.
6.4.3 Tempered Friction
We further consider the GLE with a friction term represented through the tempered regularized Prabhakar derivative (2.92), i.e.,
where b > 0, and all the parameters are the same as in Eq. (6.126). From definition (2.92) one concludes that the friction memory kernel is given by [56]
The second fluctuation-dissipation theorem then reads
For the MSD, we find [56]
where \(I_{0+}^{\alpha }\) is the R-L integral (2.2). In absence of truncation (b = 0), from (6.139), by using that [30]
we recover the result (6.129).
For high viscous damping, \(\ddot {x}(t)=0\), the following result for the MSD is obtained
Therefore, the short time limit yields subdiffusion
and the long time limit normal diffusion \(\left \langle x^{2}(t)\right \rangle \simeq t\). This means that accelerating diffusion —from subdiffusion to normal diffusion—exists in the system. Such crossover from subdiffusion to normal diffusion has been observed, for example, in complex viscoelastic systems [28].
Graphical representation of the MSD (6.139) is given in Fig. 6.13. From the figure, one observes the influence of the truncation parameter b on the MSD behavior. The case with no truncation (b = 0) shows subdiffusive behavior (blue line), and the case with truncation (red and green lines) normal diffusion in the long time limit.
6.4.4 Harmonic Oscillator
We further consider the GLE (6.141) for a harmonic oscillator with tempered regularized Prabhakar friction [56]
where ω is the frequency of the oscillator. From the Laplace transform method we find exact result for the MSD
where \(\left ({\mathbf {E}}_{\alpha ,\beta ,-\omega ^2,0+}^{\delta }f\right )(t)\) is the Prabhakar integral (2.46). For ω = 0, the Prabhakar integral corresponds to the R-L integral (2.2), therefore, from (6.142) one finds the previously obtained result for a free particle (6.129).
We are particularly interested in the normalized displacement correlation function
under the conditions \(x_{0}^{2}=\frac {k_{B}T}{\omega ^{2}}\), \(\left \langle x_{0}v_{0}\right \rangle =0\), and \(\left \langle \xi (t)x_{0}\right \rangle =0\) [5]. C X(t) then becomes [56]
and the case with no truncation (b = ) yields
Graphical representation of the C X(t) (6.145) and (6.144) is given in Figs. 6.14 and 6.15, respectively. In Fig. 6.14 different behaviors of C X(t) are observed, such as monotonic or non-monotonic decay without zero crossings (for ω < 1.44), critical behavior between the situations with and without zero crossings (at critical frequency ω ≈ 1.44), and oscillation-like behavior with zero crossings (for ω > 1.44), which appear due to the cage effect of the environment [5]. The friction, depending on the memory kernel parameters, forces either diffusion or oscillations. In Fig. 6.15 we note that with increasing of tempering, oscillation behavior with zero crossings appears. Thus, by tuning the values of friction parameters contained in the tempered Prabhakar derivative, we increase the versatility to fit complex experimental data.
6.5 Tempered GLE
Here we consider truncated three parameter M-L memory kernel of the form [36]
where γ > 0 is a constant, b ≥ 0, δ ≥ 0, τ > 0 is a time parameter, and \(E_{\alpha ,\beta }^{\delta }(z)\) is the three parameter M-L function (1.14) [54]. Tempered diffusion with memory kernel of the form (6.146) with δ = 1 was obtained within the CTRW theory in Ref. [61]. Similar kernels were considered in Refs. [48, 62, 63] in the context of tempered subdiffusion.
The Laplace transform of the kernel is given by
where we use the shift rule \(\mathcal {L}\left [f(t)e^{-at}\right ]=\hat {F} (s+a)\), \(\mathcal {L}\left [f(t)\right ]=\hat {F}(s)\), and the Laplace transform of the three parameter M-L function. It is obvious that the tempered memory kernel (6.146) satisfies the assumption (4.32). The tempered memory kernel is quite general and contains a number of limiting cases. For example, for τ → 0 (τ −1 →∞) it becomes truncated power-law memory kernel
such that
For δ = 1 and δ = β = 1, one finds the truncated two parameter and one parameter M-L kernel, respectively. In absence of truncation (b = 0), it yields the three parameter M-L memory kernel [59]
which for α = β = 1 corresponds to the Kummer’s confluent hypergeometric memory function
considered in Ref. [32].
6.5.1 Free Particle: Relaxation Functions
The relaxation functions for the truncated three parameter M-L memory kernel (6.146) becomes
where \(I_{0+}^{\alpha }f(t)\) is the R-L integral (2.2) of order α > 0. Respectively, the other relaxation functions read
In absence of truncation (b = 0), one finds the results obtained in Ref. [59]. We note that the relaxation functions (6.148)–(6.150) can also be written without the R-L integral in terms of the confluent hypergeometric function 1F 1(a;b;z) [14], as follows [36]
Graphical representation of the relaxation function (6.148) for different values of parameters is given in Fig. 6.16. From the figure one concludes that in the case of truncation the relaxation function, which is proportional to the MSD, has a linear dependence on time in the long time limit. In absence of truncation for the chosen parameters the MSD shows subdiffusive behavior of the form t 0.7. Due to the complex form of the memory kernel in the intermediate times the MSD has an oscillation-like behavior. Such behavior can be explained due to the cage effects [5], which appear as a result of influence of the environment (represented by the friction memory kernel) on the particle motion.
Graphical representation of: (a) relaxation function I(t) (6.148), (b) relaxation function g(t) (6.150), for α = 1.5, β = 1.2, δ = 0.6, τ = 1, γ = 1, b = 0 (black line), b = 0.2 (blue line), b = 0.5 (violet line). Reprinted from Physica A, 466, A. Liemert, T. Sandev and H. Kantz, Generalized Langevin equation with tempered memory kernel, 356–369, Copyright (2017), with permission from Elsevier
From the exact result for the relaxation functions, we analyze the MSD and VACF. For the short time limit one finds
while the VACF becomes
The long time limit yields normal diffusion
Therefore, characteristic crossover dynamics from ballistic motion to normal diffusion is observed.
6.5.2 High Viscous Damping Regime
Let us now consider high viscous damping, which means that \(\dot {v}(t)=0\). The GLE (6.8) then reads
The relaxation functions become
For the truncated memory kernel (6.146), we find exact result for the MSD
Here we note that the MSD can also be written in terms of the regularized hypergeometric function [14], i.e.,
where
For δ = 1 we obtain the same result as the one obtained in Ref. [61] within the CTRW theory. Therefore, two different diffusion models which describe different stochastic processes may give same results for the MSD. For the short time limit subdiffusive behavior is observed,
while for the long time limit—normal diffusive behavior
which is same as (6.156).
The case with b = 0, for the short time limit gives subdiffusive behavior
while for the long time limit diffusive behavior of the form
Therefore, the MSD has subdiffusive behavior for αδ < β, normal for αδ = β, and superdiffusive for αδ > β. This means that the particle shows accelerating diffusion, from subdiffusion it turns either to subdiffusion with greater anomalous diffusion exponent, normal diffusion or superdiffusion. Note that in the long time limit in both cases, with and without inertial term, same behavior for the MSD is obtained.
6.5.3 Harmonic Oscillator
For a particle bounded in a harmonic potential we use the previously presented general expressions for the relaxation functions, see (6.103). For the tempered memory kernel (6.146) one finds exact result for the relaxation function,
For the special case β = δ = 1, and τ → 0, we obtain the result for tempered power-law memory kernel \(\gamma (t)=e^{-bt}\frac {t^{-\alpha }}{\varGamma (1-\alpha )}\), 0 < α < 1,
The normalized displacement correlation function is represented through I(t), as C X(t) = 1 − ω 2I(t). Therefore, we have [55]
For tempered power-law memory kernel \(\gamma (t)=e^{-bt}\frac {t^{-\alpha }}{\varGamma (1-\alpha )}\), 0 < α < 1, the normalized displacement correlation function C X(t) reduces to
Graphical representation of the normalized displacement correlation function (6.165) for different values of parameters is given in Fig. 6.17. From the figures one concludes that the normalized displacement correlation function shows different behaviors: monotonic decay, non-monotonic decay without zero crossings, critical behavior which distinguishes the cases with and without zero crossings, and oscillation-like behavior with zero crossings. These behaviors are based on the cage effects of the environment as shown by Burov and Barkai [5]. This means that, depending on the values of the friction memory kernel parameters, the friction caused by the complex environment may force either diffusion or oscillations. These effects are observed in the analysis of the relaxation functions as well (Fig. 6.16). From Fig. 6.17 one concludes that the critical frequencies in case of truncated power-law memory kernel are different than those in case of no truncation. Thus, for example, for α = 1∕2 the critical frequency in case of no truncation is 1.053 [5], while in case of truncation b = 1∕2 it is equal to 0.903. The truncation decreases the critical frequency for α = 3∕4 from 0.965 [5] to 0.825, while for α = 1∕5 from 1.035 [5] to 0.889. Note that in case of classical harmonic oscillator two types of motion are observed, monotonic decay of C X(t) without zero crossings, and oscillation-like behavior with zero crossings. These two types of motions are separated at a critical frequency equal to γ∕2.
Graphical representation of the normalized displacement correlation function (6.165) for truncated power-law memory kernel with b = 1∕2 and different frequencies ω; (a) α = 1∕2, (b) α = 3∕4, (c) α = 1∕5. Reprinted from Physica A, 466, A. Liemert, T. Sandev and H. Kantz, Generalized Langevin equation with tempered memory kernel, 356–369, Copyright (2017), with permission from Elsevier
6.5.4 Response to an External Periodic Force
It has been shown that the stochastic force either in classical oscillator [23], fractional oscillator [71], or in the GLE [5, 44, 45] yields some interesting behaviors in the system, such as stochastic resonance , and the double-peak phenomenon . Similar phenomena are observed if one considers the GLE with tempered memory kernel [36]. The external periodic force is of the form \(A_{0}\cos {}(\varOmega t)\), where A 0 and Ω are the amplitude and frequency of the periodic driving force, respectively. Therefore, we consider the following GLE
By using the Laplace transform method, for the mean displacement one finds
where
From here, for the long time limit (s → 0, t →∞) it follows [5]
where the response R(Ω) and the phase shift θ(Ω) will be defined below. Here we consider the complex susceptibility
where
and
The real and imaginary parts of the complex susceptibility are experimental measured quantities. From the complex susceptibility, one finds the response
and the space shift
Graphical representation of the (a) response R(Ω), (b) loss χ″(Ω), for a free particle with tempered power-law memory kernel for α = 0.1, γ = 1, and different values of b, b = 0 (blue line), b = 0.5 (brown line), b = 1.0 (green line), b = 1.5 (red line). Reprinted from Physica A, 466, A. Liemert, T. Sandev, and H. Kantz, Generalized Langevin equation with tempered memory kernel, 356–369, Copyright (2017), with permission from Elsevier
Graphical representation of: (a) response R(Ω), (b) loss χ″(Ω), for tempered power-law memory kernel with α = 0.1, ω = 0.3, γ = 1, and different values of b, b = 0 (blue line), b = 0.2 (brown line), b = 0.4 (green line), b = 0.6 (red line). Reprinted from Physica A, 466, A. Liemert, T. Sandev and H. Kantz, Generalized Langevin equation with tempered memory kernel, 356–369, Copyright (2017), with permission from Elsevier
Particularly, we consider the special case of tempered power-law memory kernel \(\gamma (t)=e^{-bt}\frac {t^{-\alpha }}{\varGamma (1-\alpha )}\), which for b = 0 corresponds to the case considered in Ref. [5]. Therefore, for the complex susceptibility we find
which for b = 0 reduces to [5]
From Fig. 6.18 one concludes that resonance appears even for a free particle driven by truncated power-law noise, and that the resonant behavior depends on the truncation parameter b. We observe that the resonant peak which exists for b = 0 becomes smaller for b = 0.5, and disappear for b = 1.0 and b = 1.5. Here we note that the response function for the Brownian motion is a monotonic decaying function and resonance does not appear. In Fig. 6.19 same situation is observed for the harmonic oscillator driven by truncated power-law noise. The imaginary part of the complex susceptibility, or the so-called loss, shows double peak phenomenon. In Fig. 6.20 we observe similar behavior for the harmonic oscillator driven by truncated M-L noise. We find that by increasing the truncation parameter the resonance frequency is increasing. The dependence of the response and the loss on parameter γ for fixed values of α, β, δ, b is given in Fig. 6.21. By increasing parameter γ, the resonant frequency is increasing. One also concludes that by increasing parameter γ, from one peak the loss exhibits double-peak phenomena. Such double-peak phenomena have been observed in the investigation of relaxation processes in supercooled liquids [25].
Graphical representation of: (a) response R(Ω), (b) loss χ″(Ω), for tempered Mittag-Leffler memory kernel with α = 0.1, β = 1, δ = 3∕4, τ = 1, γ = 1, ω = 0.1, and different values of b, b = 0 (blue line), b = 0.3 (brown line), b = 1 (green line), b = 3 (red line). Reprinted from Physica A, 466, A. Liemert, T. Sandev and H. Kantz, Generalized Langevin equation with tempered memory kernel, 356–369, Copyright (2017), with permission from Elsevier
Graphical representation of: (a) response R(Ω), (b) loss χ″(Ω), for the tempered Mittag-Leffler memory kernel with α = 0.1, β = 1, δ = 3∕4, τ = 1, b = 1∕2, ω = 0.1, and different values of γ, γ = 0.1 (blue line), γ = 0.3 (brown line), γ = 1 (green line), γ = 3 (red line). Reprinted from Physica A, 466, A. Liemert, T. Sandev and H. Kantz, Generalized Langevin equation with tempered memory kernel, 356–369, Copyright (2017), with permission from Elsevier
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Sandev, T., Tomovski, Ž. (2019). Generalized Langevin Equation. In: Fractional Equations and Models. Developments in Mathematics, vol 61. Springer, Cham. https://doi.org/10.1007/978-3-030-29614-8_6
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