Kinetic Exchange Models as D Dimensional Systems: A Comparison of Different Approaches

Abstract

The Kinetic Exchange Models represent a charming topic in both interdisciplinary physics, e.g. in the study of economy models and opinion dynamics, as well as in condensed matter physics, where they represent a simple but effective kinetic model of perfect gas, with the peculiar feature that the dimension D of the system is a real variable which can be tuned continuously. Here we study kinetic models of energy exchange between particles of a perfect gas in D dimensions and discuss their relaxation toward the canonical equilibrium characterized by the energy distribution in D dimensions ($D \in \mathbb {R}$), comparing various theoretical approaches with results from numerical simulations.

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1 Introduction

Kinetic Exchange Models (KEMs) have attracted considerable attention not only in the interdisciplinary physics, whether opinion dynamics or the studies of wealth exchange models, but also in condensed matter physics as in the case of prototypical and general systems of units exchanging energy (Patriarca and Chakraborti 2013). A noteworthy feature of KEMs is that a suitable tuning of some parameters regulating the energy exchange in the basic homogeneous versions leads to a situation where the system relaxes toward a Boltzmann canonical energy distribution characterized by an arbitrary dimension D. It is not that D can assume only a positive integer value. In fact, it is a real variable that can assume any value greater than or equal to 1. In this contribution we discuss a basic version of KEM in D dimensions using different theoretical approaches, including numerical simulations of a perfect gas in D dimensions. This provides a historical overview of KEMs from the statistical mechanical point of view, a snapshot of the current status of research, as well as an educational presentation of the different ways to look at the same problem.

2 KEMs with No Saving: A Micro-canonical Ensemble Approach (Exponential Distribution)

In the basic versions of KEMs, N agents exchange a quantity x which represents the wealth. The state of the system is characterized by the set of variables $\{x_i\}$, ($i = 1, 2, \ldots , N$). The total wealth is conserved (here set conventionally equal to one),

$$\begin{aligned} X = x_1 + x_2 + \cdots + x_{N-1} + x_N = 1 , \end{aligned}$$

(11.1)

The evolution of the system is carried out according to a prescription, which defines the trading rule between agents. Dragulescu and Yakovenko introduced the following simple model (Dragulescu and Yakovenko 2000): at every time step two agents i and j, with wealths $x_i$ and $x_j$ respectively, are extracted randomly and a random redistribution of the sum of the wealths of the two agents takes place, according to

$$\begin{aligned} x_i'= & {} \varepsilon (x_i + x_j), \nonumber \\ x_j'= & {} (1-\varepsilon ) (x_i + x_j), \end{aligned}$$

(11.2)

where $\varepsilon$ is a uniform random number $\varepsilon \in (0,1)$, while $x_i'$ and $x_j'$ are the agent wealths after the “transaction”. This rule is equivalent to a random reshuffling of the total wealth of the two interacting agents. After a large number of iterations, the system relaxes toward an equilibrium state characterized by a wealth distribution f(x) which numerical experiments show to be perfectly fitted by an exponential function,

$$\begin{aligned} f(x) = \frac{1}{\langle x \rangle } \exp (-x/\langle x \rangle ) \, , \end{aligned}$$

(11.3)

where $\langle x \rangle$ is the average wealth of the system. The exponential function (see below for a demonstration) represents the distribution of kinetic energy in a two-dimensional gas, $D=2$, with an effective temperature T defined by $T = 2 \langle x \rangle / D \equiv \langle x \rangle$. This result can be demonstrated analytically using different methods, such as the Boltzmann equation, entropy maximization, etc., as discussed in the following sections. Here we start with a geometrical derivation of the exponential distribution based on the micro-canonical hypothesis.

Since the total amount of wealth $X = \sum _i x_i$ is conserved, the system is isolated and its state evolves on the positive part of the hyperplane defined by Eq. (11.1) in the configuration space. The surface area $S_N(X)$ of an equilateral N-hyperplane of side X is given by

$$\begin{aligned} S_N(X) = \frac{\sqrt{N}}{(N - 1)!} \, X^{N - 1} . \end{aligned}$$

(11.4)

If the ergodic hypothesis is assumed, each point on the N-hyperplane is equiprobable. The probability density $f(x_i)$ of finding agent i with value $x_i$ is proportional to the $(N-1)$- dimensional area formed by all the points on the N-hyperplane having the ith coordinate equal to $x_i$. If the ith agent has coordinate $x_i$, the $N-1$ remaining agents share the wealth $X - x_i$ on the $(N-1)$-hyperplane defined by

$$\begin{aligned} x_1 + x_2 \cdots + x_{i-1} + x_{i+1} \cdots + x_N = X - x_i, \end{aligned}$$

(11.5)

whose surface area is $S_{N - 1}(X - x_i)$. Defining the coordinate $\theta _N$ as

$$\begin{aligned} \sin \theta _N = \sqrt{\frac{N - 1}{N}}, \end{aligned}$$

(11.6)

then it can be shown that

$$\begin{aligned} S_N(X) = \! \int _{0}^{N} \! S_{N - 1}(X - x_i) \frac{dx_i}{\sin \theta _N}. \end{aligned}$$

(11.7)

Hence, the surface area of the N-hyperplane for which the ith coordinate is between $x_i$ and $x_i+dx_i$ is proportional to $S_{N - 1}(X-x_i)dx_i/\sin \theta _N$. Taking into account the normalization condition, one obtains

$$\begin{aligned} f(x_i) = \frac{1}{S_N(E)} \frac{ S_{N-1}(E - x_i) }{ \sin \theta _N } = (N - 1) E^{-1} \left( 1 - \frac{x_i}{E} \right) ^{N - 2} \rightarrow \frac{1}{\langle x \rangle } \exp \left( - x_i / \langle x \rangle \right) \, , \end{aligned}$$

(11.8)

where the last term was obtained in the limit of large N introducing the mean wealth per agent $\langle x \rangle = X/N$. From a rigorous point of view the Boltzmann factor $\exp ( - x_i / \langle x \rangle )$ is recovered only in the limit $N\gg 1$ but in practice it is a good approximation also for small values of N. This exponential distribution has been shown to agree well with real data in the intermediate wealth range (Dragulescu and Yakovenko 2001a, b) but in general it does not fit real distributions, neither at very low nor at very high values of wealth. Thus, some improvements of this minimal model are required.

3 KEMs with Saving: The Maxwell Velocity Distribution Approach ($\varGamma$-Distribution)

We now turn to a more general version of kinetic exchange model. In this model, a saving propensity parameter $\lambda$, with $0 \le \lambda < 1$, is assigned to agents, representing the minimum fraction of wealth saved during a trade. Models of this type have also been proposed in the social science by Angle in the 80s (Angle 1983, 1986, 1993, 2002) and were rediscovered as an extension of the model considered above in Chakraborti and Chakrabarti (2000), Chakraborti (2002), Chakraborti and Patriarca (2008). As a working example, for clarity here we consider a simple model (Chakraborti and Chakrabarti 2000) in which the evolution law is defined by the trading rule

$$\begin{aligned} x_i'= & {} \lambda x_i + \varepsilon (1-\lambda ) (x_i + x_j) \, , \nonumber \\ x_j'= & {} \lambda x_j + \bar{\varepsilon } (1-\lambda ) (x_i + x_j) \, , \end{aligned}$$

(11.9)

where $\varepsilon$ and $\bar{\varepsilon } = 1 - \varepsilon$ are two random numbers from a uniform distribution in (0, 1). In this model, while the wealth is still conserved during each trade, $x_i' + x_j' = x_i + x_j$, only a fraction $(1 - \lambda )$ of the wealth of the two agents is reshuffled between them during the trade. The system now relaxes toward an equilibrium state in which the exponential distribution is replaced by a $\varGamma$-distribution (Abramowitz and Stegun 1970)—or at least it is well fitted by it (this was also noted in Angle 1986). The $\varGamma$-distribution $\gamma _{\alpha ,\theta }(\xi )$ has two parameters, a scale-parameter $\theta$ and a shape-parameter $\alpha$, and it can be written as

$$\begin{aligned} \gamma _{\alpha ,\theta }(x) = \frac{1}{\theta \varGamma (\alpha )} \, \left( \frac{x}{\theta }\right) ^{\alpha -1} \exp ( - x/\theta ) \, , \end{aligned}$$

(11.10)

where $\varGamma (\alpha )$ is the $\varGamma$-function. Notice that the $\varGamma$-distribution only depends on the ratio $x/\theta$; namely, $\theta \gamma _{\alpha ,\theta }(x)$ is a dimensionless function of the rescaled variable $\xi = x/\theta$. In the numerical simulations of the model, in which one assigns the initial average wealth $\langle x \rangle$ which is constant in time, the equilibrium distribution f(x) is just the $\varGamma$-distribution with the $\lambda$-dependent parameters

$$\begin{aligned} \alpha (\lambda )= & {} 1 + \frac{3 \lambda }{1 - \lambda } = \frac{1 + 2\lambda }{1 - \lambda } \, , \end{aligned}$$

(11.11)

$$\begin{aligned} \theta (\lambda )= & {} \frac{\langle x \rangle }{\alpha } = \frac{1 - \lambda }{1 + 2\lambda } \, \langle x \rangle \, . \end{aligned}$$

(11.12)

As the saving propensity $\lambda$ varies from $\lambda = 0$ toward $\lambda = 1$, the parameter $\alpha$ continuously assumes all the values between $\alpha = 1$ and $\alpha = \infty$. Notice that for $\lambda = 0$ the model and correspondingly the equilibrium distribution reduce to those of the model considered in the previous section.

At first sight it may seem that, in going from the exponential shape wealth distribution (obtained for $\lambda = 0$) to the $\varGamma$-distribution (corresponding to a $\lambda > 0$) the link between wealth-exchange models and kinetic theory is in a way lost, but, in fact, the $\varGamma$-distribution $\gamma _{\alpha ,\theta }(x)/\theta$ represents just the canonical Boltzmann equilibrium distribution for a perfect gas in $D = 2\alpha$ dimensions and a temperature (in energy units) $\theta = k_\mathrm {B} T$, where T is the absolute temperature. This can be easily shown—in the case of an (integer) number of dimensions D—also from the Maxwell velocity distribution of a gas in d dimensions. Setting in the following $\theta = k_\mathrm {B} T$, the normalized Maxwell probability distribution of a gas in D dimensions is

$$\begin{aligned} f(v_1, \ldots , v_D) = \left( \frac{m}{2\pi \theta }\right) ^{D/2} \exp \left( - \sum _{i=1}^{D} \frac{m v_i^2}{2\theta } \right) \, , \end{aligned}$$

(11.13)

where $v_i$ is the velocity of the ith particle. The distribution (11.13) depends only on the velocity modulus v, defined by $m v^2 / 2 = \sum _{i=1}^{D} m v_i^2 /2$, and one can then integrate the distribution over the $D-1$ angular variables to obtain the velocity modulus distribution function f(v). With the help of the expression for the surface $\sigma _D(r)$ of a hypersphere of radius r in D dimensions,

$$\begin{aligned} \sigma _D(r) \equiv \sigma _D^1 \, r^{D - 1} = \frac{2\pi ^{D/2}}{\varGamma (D/2)} \, r^{D-1} \, , \end{aligned}$$

(11.14)

where $\sigma _D^1$ is the expression for a unit-radius sphere, one obtains

$$\begin{aligned} f(v) = \frac{2}{\varGamma (D/2)} \left( \frac{m}{2\theta }\right) ^{D/2} \, v^{D-1} \exp \left( - \frac{m v^2}{2 \theta } \right) \, , \end{aligned}$$

(11.15)

and then, by changing variable from the velocity v to the kinetic energy $x = m v^2/2$,

$$\begin{aligned} f(x) = \frac{1}{\varGamma (D/2) \theta } \left( \frac{x}{T} \right) ^{D/2-1} \exp \left( -\frac{x}{T}\right) \, , \end{aligned}$$

(11.16)

which is just the distribution in Eq. (11.10) if one sets $\alpha = D/2$.

Notice that in order to construct a KEM with a given effective temperature $\theta$ and dimension D, it is not sufficient to fix the saving parameter $\lambda$. Following Eqs. (11.11)–(11.12), one has to assign both $\lambda$ and the average energy $\langle x \rangle$. Using the relation $\alpha = D/2$, Eqs. (11.11)–(11.12) can be rewritten as

$$\begin{aligned} D(\lambda )= & {} 2 \left( 1 + \frac{3 \lambda }{1 - \lambda } \right) = \frac{2 ( 1 + 2\lambda )}{1 - \lambda } \, , \end{aligned}$$

(11.17)

$$\begin{aligned} \theta (\lambda )= & {} \frac{2 \langle x \rangle }{D} \, . \end{aligned}$$

(11.18)

Inverting these equations, one has a simple recipe for finding the suitable values of $\lambda$ and $\langle x \rangle$ that set the system dimension D and temperature $\theta$ to the required values, e.g. first fixing $\lambda$ from D and then $\langle x \rangle$ using D and $\theta$,

$$\begin{aligned} \lambda= & {} \frac{ D - 2 }{ D + 4 } \, , \end{aligned}$$

(11.19)

$$\begin{aligned} \langle x \rangle= & {} \frac{D \theta }{2} \, . \end{aligned}$$

(11.20)

Here the second equation can be recognized as an expression of the equipartition theorem.

It can be noticed that if $\lambda$ is fixed (depending on the corresponding D) while the value of $\langle x \rangle$ is kept constant in all the simulations with different $\lambda$’s, then from Eq. (11.18) a system with larger dimension D will have lower temperature $\theta$ (and therefore a shape of the probability distribution function with smaller width). From these equations one can also notice the existence of a minimum value for the system dimension, $D_\mathrm {min} = 2$, corresponding to the minimum value $\lambda = 0$. As $\lambda$ increases in the interval $\lambda \in (0,1)$, D also increases monotonously diverging eventually for $\lambda \rightarrow 1$. This is the specific result of the model considered in this section and the minimum value $D_\mathrm {min}$ is different in different KEMs.

4 KEMs in D Dimensions: A Variational Approach

As an alternative, equivalent, and powerful approach, one can use the Boltzmann approach based on the minimization of the system entropy in order to obtain the equilibrium distribution (Chakraborti and Patriarca 2009). The method can provide both the exponential distribution as well as the $\varGamma$-distribution obtained in the framework of wealth-exchange models with a saving parameter $\lambda > 0$, a natural effective dimension $D > 2$ being associated to systems with $\lambda > 0$.

The representative system is assumed to have D degrees of freedom, $q_1,\ldots ,q_D$ (e.g. the particle momenta in a gas), and a homogeneous quadratic Hamiltonian X,

$$\begin{aligned} X(q_1, \ldots , q_D) \equiv X(q^2) = \frac{1}{2} (q_1^2 + \cdots + q_D^2) = \frac{1}{2} \, q^2 \, , \end{aligned}$$

(11.21)

where $q = (q_1^2 + \cdots + q_D^2)^{1/\,2}$ is the distance from the origin in the D-dimensional q-space. As an example, the D coordinates $q_i$ can represent suitably rescaled values of the velocities so that Eq. (11.21) provides the corresponding kinetic energy function. The expression of the Boltzmann entropy of a system described by D continuous variables $q_1, \ldots , q_D$, is

$$\begin{aligned} S_D[q_1,\ldots ,q_D] = -\int dq_1\ldots \int dq_D \, f_D(q_1,\ldots ,q_D) \ln [f_D(q_1,\ldots ,q_D)] \, . \end{aligned}$$

(11.22)

The system is subjected to the constraints on the conservation of the total number of systems (i.e. normalizing to one for a probability distribution function) and of the total wealth (implying a constant average energy $\bar{x}$), expressed by

$$\begin{aligned}&\int dq_1\ldots \int dq_D \, f_D(q_1,\ldots ,q_D) = 1 \, , \end{aligned}$$

(11.23)

$$\begin{aligned}&\int dq_1\ldots \int dq_D \, f_D(q_1,\ldots ,q_D) X(q_1, \ldots , q_D) = \bar{x} \, , \end{aligned}$$

(11.24)

They can be taken into account using the Lagrange method, i.e. by a variation (with respect to the distribution $f_D(q_1,\ldots ,q_D)$) of the functional

$$\begin{aligned} S_\mathrm {eff}[f_D] = \int dq_1\ldots \int dq_D f_N(q_1,\dots ,q_D) \{ \ln [f_D(q_1,\ldots ,q_D)] + \mu + \beta X(q^2) \} ,\qquad \qquad \quad \end{aligned}$$

(11.25)

where $\mu$ and $\beta$ are two Lagrange multipliers. The invariance of the Hamiltonian, depending only on the modulus q, allows the transformation from Cartesian to polar coordinates. Integrating over the $(D-1)$ coordinates spanning the solid angle with the help of the expression (11.14) for the surface of the hyper sphere, one obtains

$$\begin{aligned} S_\mathrm {eff}[f_1] = \int _{0}^{+\infty } dq \, f_1(q) \left[ \ln \left( \frac{f_1(q)}{\sigma _D^1 \, \, q^{D-1}} \right) + \mu + \beta X(q) \right] \, \end{aligned}$$

(11.26)

where the probability density $f_D(q_1,\ldots ,q_D)$ in the D-dimensional space was expressed with the reduced probability density $f_1(q)$ in the one-dimensional q-space,

$$\begin{aligned} f_1(q) = \sigma _D^1 \, \, q^{D-1} f_D(q) \, . \end{aligned}$$

(11.27)

Finally, transforming from q to the energy variable $x = q^2/\,2$, one obtains the probability distribution function

$$\begin{aligned} f(x) = \frac{dq(x)}{dx} \left. f_1(q)\right| _{q = q(x)} = \frac{\left. f_1(q)\right| _{q = q(x)}}{\sqrt{2x}} \, , \end{aligned}$$

(11.28)

where $q(x) = \sqrt{2x}$ from Eq. (11.21). In terms of the new variable x and distribution f(x), from Eq. (11.26) one obtains the functional

$$\begin{aligned} S_\mathrm {eff}[f] = \int _{0}^{+\infty } dx \, f(x) \left[ \ln \left( \frac{f(x)}{\sigma _D^1 \,\, x^{D/\,2-1}} \right) + \mu + \beta x \right] \, , \end{aligned}$$

(11.29)

Varying this functional with respect to f(x), $\delta S_\mathrm {eff}[f]/\delta f(x) = 0$, leads to the equilibrium $\varGamma$-distribution in Eq. (11.10) with rate parameter $\beta = 1 / \theta$ and the same shape parameter $\alpha = D/\,2$.

5 Kinetic Theory Approach to a D-Dimensional Gas

The deep analogy between kinetic wealth-exchange models of closed economy systems, where agents exchange wealth at each trade, and kinetic gas models, in which energy exchanges take place at each particle collisions, was clearly noticed in Mandelbrot (1960). The analogy can be justified further by studying the microscopic dynamics of interacting particles in the framework of standard kinetic theory.

In one dimension, particles undergo head-on collisions, in which they can exchange the total amount of energy they have, i.e. a fraction $\omega = 1$ of it. Alternatively, one can say that the minimum fraction of energy that a particle saves in a collision is in this case $\lambda \equiv 1 - \omega = 0$. In the framework of wealth-exchange models, this case corresponds to the model of Dragulescu and Yakovenko mentioned above (Dragulescu and Yakovenko 2000), in which the total wealth of the two agents is reshuffled during a trade.

In an arbitrary (larger) number of dimensions, however, this does not take place, unless the two particles are travelling exactly along the same line in opposite verses. On average, only a fraction $\omega = (1-\lambda ) < 1$ of the total energy will be lost or gained by a particle during a collision, that is most of the collisions will be practically characterized by an energy saving parameter $\lambda > 0$. This corresponds to the model of Chakraborti and Chakrabarti (2000), in which there is a fixed maximum fraction $(1-\lambda ) > 0$ of wealth which can be reshuffled.

Consider a collision between two particles in an N-dimensional space, with initial velocities represented by the vectors $\mathbf{v}_{(1)} = (v_{(1)1}, \ldots , v_{(1)N})$ and $\mathbf{v}_{(2)} = (v_{(2)1}, \ldots , v_{(2)N})$. For the sake of simplicity, the masses of the all the particles are assumed to be equal to each other and will be set equal to 1, so that momentum conservation implies that

$$\begin{aligned} \mathbf{v}_{(1)}'= & {} \mathbf{v}_{(1)} + \varDelta \mathbf{v} \, , \nonumber \\ \mathbf{v}_{(2)}'= & {} \mathbf{v}_{(2)} - \varDelta \mathbf{v} \, , \end{aligned}$$

(11.30)

where $\mathbf{v}_{(1)}'$ and $\mathbf{v}_{(2)}'$ are the velocities after the collisions and $\varDelta \mathbf{v}$ is the momentum transferred. Conservation of energy implies that $\mathbf{v}_{(1)}'^{\,2} + \mathbf{v}_{(2)}'^{\,2} = \mathbf{v}_{(1)}^2 + \mathbf{v}_{(2)}^2$ which, by using Eq. (11.30), leads to

$$\begin{aligned} \varDelta \mathbf{v}^2 + (\mathbf{v}_{(1)} - \mathbf{v}_{(2)}) \cdot \varDelta \mathbf{v} = 0 \, . \end{aligned}$$

(11.31)

Introducing the cosines $r_i$ of the angles $\alpha _i$ between the momentum transferred $\varDelta \mathbf{v}$ and the initial velocity $\mathbf{v}_{(i)}$ of the i th particle ($i = 1,2$),

$$\begin{aligned} r_i = \cos \alpha _i = \frac{\mathbf{v}_{(i)} \cdot \varDelta \mathbf{v}}{v_{(i)} \, \varDelta v} \, , \end{aligned}$$

(11.32)

where $v_{(i)} = |\mathbf{v}_{(i)}|$ and $\varDelta v = |\varDelta \mathbf{v}|$, and using Eq. (11.31), one obtains that the modulus of momentum transferred is

$$\begin{aligned} \varDelta v = - r_1 v_{(1)} + r_2 v_{(2)} \, . \end{aligned}$$

(11.33)

From this expression one can now compute explicitly the differences in particle energies $x_i$ due to a collision, that are the quantities $x_i' - x_i \equiv (\mathbf{v}_{(i)}'^{\,2} - \mathbf{v}_{(i)}^2)/\,2$. With the help of the relation (11.31) one obtains

$$\begin{aligned} x_1'= & {} x_1 + r_2^2 \, x_2 - r_1^2 \, x_1 \, , \nonumber \\ x_2'= & {} x_2 - r_2^2 \, x_2 + r_1^2 \, x_1 \, . \end{aligned}$$

(11.34)

The equivalence to KEMS should now appear clearly. First, the number $r_i$’s are squared cosines and therefore they are in the interval $r \in (0,1)$. Furthermore, they define the initial directions of the two particles entering the collision, so that they can be considered as random variables if the hypothesis of molecular chaos is assumed. In this way, they are completely analogous to the random coefficients $\varepsilon (1-\lambda )$ [or $(1 - \varepsilon )(1-\lambda )$] appearing in the formulation of KEMs, with the difference that they cannot assume all values in (0, 1), but are limited in the interval $(0,1-\lambda )$. However, in general the $r_i^2$’s are not uniformly distributed in (0, 1) and the most probable values $\langle r_i^2 \rangle$ drastically depend on the space dimension, which is at the base of their effective equivalence with the KEMs: the greater the dimension D, the smaller the $\langle r_i^2 \rangle$, since the more unlikely it becomes that the corresponding values $\langle r_i \rangle$ assume values close to 1 and the more probable that instead they assume a small value close to 1 / D. This can be seen by computing their average—over the incoming directions of the two particles or, equivalently, on the orientation of the initial velocity $\mathbf{v}_{(i)}$ of one of the two particles and of the momentum transferred $\varDelta \mathbf{v}$, which is of the order of 1 / D.

The 1 / D dependence of $\langle r_i^2 \rangle$ well compares with the wealth-exchange model with $\lambda > 0$, in which a similar relation is found between the average value of the corresponding coefficients $\varepsilon (1 - \lambda )$ and $\bar{\varepsilon }(1 - \lambda )$ in the evolution equations (11.9) for the wealth exchange and the effective dimensions $D(\lambda )$, Eq. (11.17): since $\varepsilon$ is a uniform random number in (0, 1), then $\langle \varepsilon \rangle = 1/\,2$ and inverting $D = D(\lambda )$, Eq. (11.17), one finds $\langle (1 - \varepsilon ) (1 - \lambda ) \rangle = \langle \varepsilon (1 - \lambda ) \rangle = (1-\lambda )/\,2 = 3/(D + 4)$.

6 Numerical Simulations of a D-Dimensional Gas

Besides the various analytical considerations discussed above, the close analogy with kinetic theory allows one to resort to molecular dynamics simulations also to study a D-dimensional system. It is instructive to obtain the very same Boltzmann energy distributions discussed above from molecular dynamics (MD) simulations. For the sake of simplicity we consider the distribution of kinetic energy of a gas, since it is known that at equilibrium it relaxes to the Boltzmann distribution with the proper number D of dimensions of the gas independently of the inter-particle potential.

For clarity we start with the case $D = 2$. In fact, as discussed above when considering the model defined in Eq. (11.9), the case of the minimum dimension $D = 2$ is characterized by an equilibrium distribution which is a perfect exponential. We have performed some numerical simulation of a Lennard-Jones gas in $D = 2$ dimensions using the leapfrog algorithm (Frenkel and Smit 1996), with a small system of $N = 20$ particles in a square box of rescaled size $L = 10$, for a simulation time $t_\mathrm {tot} = 10^4$, using an integration time step $\delta t = 10^{-4}$ and averaging the energy distribution over $10^5$ snapshots equidistant in time. Reflecting boundary conditions were used and a “repulsive Lennard-Jones” U(r) interaction potential between particles was assumed,

$$\begin{aligned} U(r)&= \varepsilon \left[ \left( R / r \right) ^6 - 1 \right] ^2 \,&\mathrm {for} ~~~r < R \, , \end{aligned}$$

(11.35)

$$\begin{aligned} \,&= 0 \, , \,&\mathrm {for} ~~~r \ge R \, , \end{aligned}$$

(11.36)

representing a purely repulsive potential decreasing monotonously as the interparticle distance r increases, as far as $R = 1$, where the potential becomes (and remains) zero for all larger values of r.

Fig. 11.1

Example of kinetic energy distribution for a gas in $D = 2$ dimensions in the linear (left) and semilog (right) scale. In the particular case of $D = 2$, the distribution is a perfect exponential

As examples, the results of the kinetic energy distribution in $D = 2$ dimensions are shown in Fig. 11.1. The corresponding results for a gas in a cubic box in $D=1$ and $D = 3$ dimensions, with the same parameters are shown in Figs. 11.2 and 11.3. Notice that in all figures the “MD” curve represents the result of the molecular simulation, while the “D = ...” curve is the corresponding $\varGamma$-distribution with shape parameter $\alpha = D/2$ and scale parameter $\theta = T = 1$.

In $D = 1$ dimensions, Newtonian dynamics predicts that the velocity distribution does not change with time in a homogeneous gas, since at each collision the two colliding particles simply exchange their momenta. Therefore, only in the case $D = 1$, we have added a Langevin thermostat at $T = 1$ (damping coefficient $\gamma = 0.5$) in order to induce a thermalization of the system.

Fig. 11.2

Example of kinetic energy distribution for a gas in $D = 1$ dimensions in the linear (left) and semilog (right) scale

Fig. 11.3

Example of kinetic energy distribution for a gas in $D = 3$ dimensions in the linear (left) and semilog (right) scale. In the case of $D = 3$ dimensions, the Boltzmann distribution is proportional to $\sqrt{x}\exp (-\beta x)$

7 Conclusions

KEMs represent one more approach to the study of prototypical statistical systems in which N units exchange energy and for this reason they certainly have a relevant educational dimension (Patriarca and Chakraborti 2013). This dimension is emphasized in this contribution by presenting different approaches to the same model and to obtain the corresponding canonical Boltzmann distribution in D dimensions.

There are other reasons for their relevance and for the interest they have attracted:

• (a) KEMs have the peculiarity that the system dimension D can be easily tuned continuously. Letting it assume real values by changing the parameters regulating the energy exchanges, makes them interesting to study various other statistical systems.

• (b) KEMs have by now been used in interdisciplinary physics in various topics such as modeling of wealth exchange and opinion dynamics;

• (c) but they also appear in condensed matter problems such as fragmentation dynamics.