## 1. Introduction

Nonlinear kinetics in the coordinate or velocity space or in the phase space has been used systematically over the last few decades. A popular approach involves studying the temporal evolution of a particle system while it interacts with a medium in thermodynamic equilibrium. In Ref. [

1], the most general nonlinear kinetics in phase space was considered in the limit of the diffusive approximation. In this limit, the interaction between the particles and the medium is viewed as a diffusive process in the velocity space. It was shown that the

particle probability density function $f=f(t,r,v)$ (henceforward particle density function) obeys the following evolution equation

where

$D=D(r,v)$ is the diffusion coefficient, while

$V=V\left(r\right)$ and

$U=U\left(v\right)$ represent the external potential and the particle kinetic energy respectively. In addition,

$\partial f/\partial r$,

$\partial f/\partial v$, denote the gradients of the particle density with respect to velocity and space, and

m is the particle mass. The constant

$\beta $ is given by

$\beta =1/{k}_{\mathrm{B}}T$ where

${k}_{\mathrm{B}}$ is the

Boltzmann constant and

T the temperature of the system.

The above equation can be viewed as a

generalized Kramers equation that describes a nonlinear kinetics in the phase space. The nonlinearity is due to the presence of the two arbitrary functions

$a\left(f\right)$ and

$b\left(f\right)$ that depend on the particle density function

f. In the case where

$a\left(f\right)=f$ and

$b\left(f\right)=1$, Equation (

1) reduces to the ordinary Kramers equation that describes the standard linear kinetics, e.g., [

2].

Equation (

1) has been employed to study the kinetic foundations of nonlinear statistical systems governed by

$\kappa $-entropy. Generalized statistical mechanics, based on

$\kappa $-entropy [

1,

3,

4], preserves the main features of ordinary Boltzmann-Gibbs statistical mechanics. For this reason, it has attracted the interest of many researchers over the last 16 years, who have studied its foundations and mathematical aspects [

5,

6,

7,

8,

9,

10,

11,

12], the underlying thermodynamics [

13,

14,

15,

16,

17], and specific applications of the theory in various scientific and engineering fields. A non-exhaustive list of application areas includes quantum statistics [

18,

19,

20], quantum entanglement [

21,

22], plasma physics [

23,

24,

25,

26,

27], nuclear fission [

28], astrophysics [

29,

30,

31,

32,

33,

34,

35], geomechanics [

36], genomics [

37], complex networks [

38,

39], economy [

40,

41,

42,

43] and finance [

44,

45,

46,

47].

It is important to note that the right hand side of Equation (

1), which introduces nonlinear effects into kinetics, describes an unconventional diffusion process in the velocity space. This process can clearly also be studied in the coordinate space, where it describes an

anomalous diffusion process. Such processes have long been observed and studied in various fields of condensed and soft matter physics [

48].

Knowledge of the master equation that is associated with a given Fokker–Planck partial differential equation is crucial for two reasons. First, as stated above the master equation provides a uniquely defined discretization scheme for numerical integration. Second, the master equation is physically meaningful, since it defines the process dynamics on lattices. Hence, the study of the lattice dynamics allows an improved understanding of the dynamics in continuous space.

For example, consider the Fokker–Planck equation in the form (

1) that involves two arbitrary functions

$a\left(f\right)$ and

$b\left(f\right)$. Numerical integration of this equation requires discretization in the spatial variable. The resulting master equation cannot contain any other functions besides

$a\left(f\right)$ and

$b\left(f\right)$, that are already present in the original partial differential equation. Hence, any lattice (finite-size) dynamical effects that are lost in the continuum limit cannot be re-introduced at this level. In addition, the master equation derived from the discretization of the Fokker–Planck equation depends on the particular differencing scheme (e.g., forward differences versus central differences). To avoid these shortcomings, it is worth investigating the use of physical principles

at the lattice scale to univocally define the discrete kinetics and directly derive the master equation. The master equation thus obtained is clearly useful, not only for deriving the nonlinear Fokker–Planck Equation (

1) in the continuum limit, but also as the starting point for the direct numerical integration of the Fokker–Planck equation.

The present paper shows that the

Kinetic Interaction Principle (

KIP) which governs the particle kinetics [

1] univocally defines a very simple lattice-based master equation, which yields the nonlinear Fokker–Planck equation in the continuous limit. Both the master equation and the ensuing Fokker–Planck equation involve physically motivated terms.

The paper is organized as follows. In

Section 2, we introduce the specific form of the Fokker–Planck equation that we will investigate. We then recall the KIP that underlies the kinetics of a particle system, and we apply it to derive the most general nonlinear master equation in the nearest-neighbour approximation. Furthermore we express the master equation in the form of a discrete continuity equation and introduce the nonlinear particle current. In

Section 3, we calculate the particle current in the continuous limit and derive the respective Fokker–Planck Equation (

2). In

Section 4 we focus on standard finite-difference-based discretization schemes of the Fokker–Planck equation, and we show that in the nonlinear case the KIP-based master equation cannot be derived from such discretization schemes. In

Section 5, we focus on an important subclass of nonlinear kinetics, described by Equation (

2), which is related to ordinary Fickian diffusion. Finally, in

Section 6 we present our concluding remarks and suggest future extensions of this research to lattices in higher dimensions.

## 4. Fokker–Planck Equation and Discretization Schemes

If we insert the continuum limit of the Fokker–Planck

$j(t,x)$ current, given by Equation (

39), in the continuity Equation (

24) we obtain the

nonlinear Fokker–Planck equationThe above equation recovers the original Fokker–Planck equation given by Equations (

2) and (

3). Thus, the derivations in the preceding section show how the nonlinear Fokker–Planck equation can be obtained from the discrete Fokker–Planck current of Equation (

23), which is derived from the simple, KIP-based master Equation (

7).

In the following, we investigate the impact of the discretization of the time derivative in Fokker–Planck equation, as well as the impact of the space derivative discretization in the linear and nonlinear kinetic regimes. We also present an extension of the one-dimensional formalism to two spatial dimensions.

We will express the Fokker–Planck Equation (

40) as

There are various numerical schemes for the discretization of partial differential equations such as the above, including finite-difference schemes, methods based on finite elements, finite-volume methods, spectral approaches, multigrid methods, etc. [

64]. Each approach leads to a different set of equations that depends on the mathematical assumptions made. Different discretization schemes for the one-dimensional Fokker–Planck equation are presented in [

65].

It is most straightforward to compare the master Equation (

7) with discretized Fokker–Planck equations that are obtained by means of finite-difference methods. Discretized versions of the functions

$f(t,x)$ are denoted by means of

${f}_{i}^{n}$, where the index

i refers to the lattice site and the index

n to the time instant.

#### 4.1. Temporal Discretization

Three of the most common finite-difference methods lead to the following discretization schemes of (

41a)

where

$\Delta t$ is the time step.

On the other hand, the discretization of the master Equation (

7) leads to

If we compare the master Equations (

43) with (42) we notice that the right-hand side of the former involves the function

${F}_{i}^{n}$ only at the current time step. In contrast, the backward Euler and the Crank-Nicolson schemes involve the values of the function,

${F}_{i}^{n+1}$, at the next time step as well.

In the forward Euler discretization the time difference is determined by the function

${F}_{i}^{n}$, which is in general different from

${\tilde{F}}_{i}^{n}.$ The function

${F}_{i}^{n}$ is obtained by discretizing the spatial partial derivatives in

$F(t,x)$. Using the forward spatial derivative defined in (13), the function

${\tilde{F}}_{i}^{n}$ can be shown to depend on the values of

$a(\xb7),b(\xb7),U(\xb7)$ at the sites labeled by

$i,i+1,i+2$. In contrast, the time difference of the discretized master Equation (

43) depends on the lattices sites

i and

$i\pm 1$.

#### 4.2. Linear Kinetics Regime

The linear kinetics regime of (

2) is obtained if

$a\left(f\right)=f$ and

$b\left(f\right)=1$. After recalling that

$\gamma \left(f\right)=a\left(f\right)\phantom{\rule{0.166667em}{0ex}}b\left(f\right)$,

$\mathsf{\Omega}\left(f\right)=a\left(f\right)\phantom{\rule{0.166667em}{0ex}}b\left(f\right)\phantom{\rule{0.166667em}{0ex}}\frac{\partial}{\partial f}ln\frac{a\left(f\right)}{b\left(f\right)}$, it follows that

$\gamma \left(f\right)=f$ and

$\mathsf{\Omega}\left(f\right)=1$. Hence, the linear Fokker–Planck equation derived from (

2) assumes the form:

with

$A=D\left(x\right)\phantom{\rule{0.166667em}{0ex}}\beta \phantom{\rule{0.166667em}{0ex}}\frac{\partial U\left(x\right)}{\partial x}$.

To discretize the above equation in space, we introduce at the place of the first and second order partial derivatives the following

symmetric finite differences:

where

$\Delta x$ is the uniform discretization step in space.

Then, the linear Fokker–Planck Equation (

45) transforms into the linear master equation

with the nearest-neighbor transition rates

${w}_{i+1}^{-}$ and

${w}_{i-1}^{+}$ defined by

Similarly, the nearest-neighbor master Equation (

7) is given in the linear regime by the following equation

After observing that

${w}_{i}\approx {w}_{i}^{+}+{w}_{i}^{-}$, the linear master Equation (

48) derived by means of the KIP is essentially equivalent to the master Equation (

47a) obtained from the discretization of the linear Fokker–Planck Equation (

45).

We can thus conclude that in the case of linear kinetics the KIP-based master equation leads in the continuum limit to a Fokker–Planck equation. Discretization of the Fokker–Planck equation using the centered-difference scheme yields the original master equation.

In the case of linear kinetics we can use either the forward or the backward Euler method, or the Cranck-Nicolson method, since the coefficients of the Fokker–Planck equation do not depend on time. In addition, the linear dependence on the probability density function ensures that even for the more demanding Crank-Nicolson scheme, it suffices to solve a linear system.

#### 4.3. Nonlinear Kinetics

In the case of nonlinear kinetics, we consider the Fokker–Planck equation as given in (

3). Expansion of the terms that involve the spatial derivatives lead to the following expression

Using the centered finite difference approximation of the spatial derivatives for the functions

$f(t,x)$,

$D\left(x\right)$ and

$A\left(x\right)$ we obtain the following master equation

where the density-dependent coefficients

${\omega}^{\mp}\left({f}_{i}\right)$,

$\omega \left({f}_{i}\right)$, and

${\theta}_{i}\left({f}_{i}\right)$ are given by

The master Equation (

7) that is based on the KIP with nearest-neighbor transitions represents different kinetics than the master Equation (50) that is obtained from the discretization of the nonlinear Fokker–Planck Equation (

3). The kinetics described by the KIP-based master equation is physically motivated at the microscopic level. On the other hand, it is not straightforward to interpret in physical terms the kinetics described by (50). More precisely, the first two terms on the right-hand side of (

50a) can be viewed as representing transitions into the site

i from the site

$i-1$. The population of the departure site enters linearly the transition rate in the first term, but the population of the arrival site

i can be a highly nonlinear and complicated function. In the second term, the dependence on the departure site is also nonlinear in addition to that of the arrival site. Analogous remarks hold for the third and fourth terms which represent transitions from the site

$i+1$ towards the site

i. The last two terms represent exiting transitions from the sites

i. However, there is no obvious physical mechanism that explains why the transition rate for particles exiting the site

i should be proportional to the product of the densities at both sites

$i-1$ and

$i+1$ (as dictated by the term

${\theta}_{i}\left({f}_{i}\right)\phantom{\rule{0.166667em}{0ex}}{f}_{i+1}\phantom{\rule{0.166667em}{0ex}}{f}_{i-1}$). Clearly, the master Equation (50) does not have an obvious microscopic interpretation, even though in the continuum limit it yields the nonlinear Fokker–Planck Equation (

3).

Based on the above observations, in order to integrate numerically the nonlinear Fokker–Planck equation it is recommended to start with the spatial discretization provided by the master Equation (

7). For the time step, the forward or backward Euler methods are simpler alternatives than the Crank–Nicolson method which leads to a nonlinear system of algebraic equations with respective computational cost. More importantly, in order to study the statistical properties of a physical system governed by lattice nonlinear kinetics, the master Equation (

7) is clearly the starting point, independently of the method used to integrate it (which is not the focus of the current study).

How should we understand the difference between the master Equation (

7) and that produced by the discretization of the nonlinear Fokker–Planck Equation (

3), which after all is obtained from the initial master Equation (

7) in the continuum limit? The key is that in process of taking the continuum limit information is lost. The master Equation (

7) contains the density functions

${f}_{i}$,

${f}_{i-1}$,

${f}_{i+1}$ of three sites

$i,i\pm 1$ through the set of functions

${I}_{M}=\{a\left({f}_{i}\right),b\left({f}_{i}\right),a\left({f}_{i-1}\right),b\left({f}_{i-1}\right),a\left({f}_{i+1}\right),b\left({f}_{i+1}\right)\}$. The nonlinear Fokker–Planck Equation (

3) on the other hand, contains the function

$f\left(x\right)$ through its dependence on

$a\left(f\right)$,

$b\left(f\right)$ and the partial derivatives

$\partial f/\partial x$,

${\partial}^{2}f/\partial {x}^{2}$. Clearly the set

${I}_{M}$ that is used in the master Equation (

7) contains more information than the set

${I}_{FP}$ = {

$a\left(f\right)$,

$b\left(f\right)$,

$\partial f/\partial x$ and

${\partial}^{2}f/\partial {x}^{2}$}.

Due to this loss of information in the transition from the discrete (lattice-based) to the continuum model, discretization of the latter generates the master Equation (50), which is radically different from the master Equation (

7), although both master equations yield the same Fokker–Planck equation in the continuum limit. It is important to stress that more than one master equations, including (

7) imposed by the KIP, yield the same Fokker–Planck equation in the continuum limit. In contrast, applying the standard discretization rules to the Fokker–Planck equation, the master equations derived are not physically motivated and they differ from the master equation generated by the KIP. Only in the case of the linear Fokker–Planck equation, the master equation obtained by discretization coincides with the KIP-based master equation.

## 6. Conclusions

Let us summarize the main results of the present study. Starting from the Kinetic Interaction Principle introduced in [

1], we obtained the evolution equation of a general statistical system defined on a one-dimensional lattice. The KIP-based master equation in the continuum limit yielded the already known nonlinear-Fokker–Planck Equation (

2), which is widely used in the literature to describe anomalous diffusion in condensed matter physics and nonconventional statistical physics. Our derivation provides a better, physical understanding of the underlying many-body lattice dynamics at the microscopic level.

A second result is related to the possibility of directly using the KIP-based master equation as the numerical discretization scheme for the solution of the Fokker–Planck partial differential Equation (

2). This approach resolves the ambiguity that results from different numerical discretization schemes of the Fokker–Planck equation. The proposed discretization scheme based on the master equation is physically motivated and follows from the KIP that describes microscopic nonlinear dynamics. On the contrary, discretization schemes based on finite differences follow from mathematically considerations. In addition, the master equations obtained from the

nonlinear Fokker–Planck equation by applying different discretization schemes cannot be obtained starting from the KIP. The master equation proposed in this paper and obtained by means of the KIP is thus uniquely defined by microscopic interactions.

Finally, after noting that the Fokker–Planck current expresses the sum of two distinct contributions, i.e., the

nonlinear drift current and the

generalized Fick current as shown in Equations (

51)–(

53), we have demonstrated that some important nonlinear kinetics formulations proposed in the literature can be successfully described by

ordinary Fickian diffusion. This can be accomplished by introducing a nonlinear drift term that is combined with Fickian diffusion. This combination leads to the same stationary solution of the Fokker–Planck equation as an equation that involves non-Fickian diffusion and linear drift.

It is straightforward to extend the KIP-based approach for the derivation of physically inspired master equations to lattices in two and three dimensions. For example, in the case of a two dimensional square lattice whose sites are labeled by the integer indices

$i,j$, the master Equation (

7) with nearest-neighbor transitions becomes