Nonlinear Kinetics on a Lattice Featuring Anomalous Diffusion

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker-Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker-Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker-Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker-Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker-Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in [G. Kaniadakis, Physica A 296, 405 (2001)], univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker-Planck equation in its most general form.


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(r) and U = U(v) represent the external potential and the particle kinetic energy respectively.In addition, ∂ f /∂r, ∂ f /∂v, denote the gradients of the particle density with respect to velocity and space, and m is the particle mass.The constant β is given by β = 1/k B T where k 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( f ) and b( f ) that depend on the particle density function f .In the case where a( f ) = f and b( f ) = 1, Eq. ( 1) reduces to the ordinary Kramers equation that describes the standard linear kinetics, e.g.[2].
It is important to note that the right hand side of Eg. (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( f ) and b( f ).Numerical integration of this equation requires discretization in the spatial variable.The resulting master equation cannot contain any other functions besides a( f ) and b( f ), 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 in order 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 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 Sect.2, we introduce the specific form of the Fokker-Planck equation that we will investigate.We then recall the KIP underlying the kinetics of a particle system and express the most general nonlinear master equation, in nearest-neighbour approximation.Furthermore we express the master equation in the form of a discrete continuity equation and introduce the nonlinear particle current.In Sect.3, we calculate the particle current in the continuous limit and derive the respective Fokker-Planck equation (2).In Sect.4, we focus on an important subclass of nonlinear kinetics, described by Eq. ( 2), which is related to the ordinary Fickian diffusion.Finally, in Sect. 5 we present our concluding remarks.

Nonlinear Fokker-Planck Kinetics in One Dimension
In the following, we focus on the one-dimensional non-linear diffusive process that is described by the equation where f = f (t, x), U = U(x) and D = D(x), while x can indicate either the velocity or the coordinate space variable.Eq. ( 2) is just the nonlinear Fokker-Planck equation.Upon setting , and A = D(x) β ∂U(x)  ∂x , this equation can also be expressed in the equivalent but more familiar form where the first summand inside the bracket on the right-hand side yields the drift term while the second summand yields the diffusive term.
Several special cases of the nonlinear Fokker-Planck equation ( 3) have long been known.The best known and most frequently studied case in the literature is that of diffusion in porous media, which corresponds to A = 0, D = constant and Ω( f ) = f n [49].Eq. ( 3) (expressed in velocity space) has also been used in statistical physics, by setting 3) describes the kinetics of bosons (positive sign) or fermions (negative sign) [50], whereas when Ω( f ) = f n , it describes particle kinetics in non-extensive statistical mechanics [51].
In the case of bosonic and fermionic statistics, Eq. ( 3) was obtained with standard methods used in linear kinetics, i.e., by applying the Kramers-Moyal expansion to a balance equation or alternatively starting from a master equation [50].The same techniques were also used to derive Eq. (3) in the case of nonextensive statistical mechanics [52,53].
The most general form of Eq. ( 3) involves arbitrary functions γ( f ) and Ω( f ).The study of this general form is important for classifying the stationary and thermodynamically stable solutions [1,[55][56][57].On the other hand, the derivation of the nonlinear Fokker-Planck equation from physical principles is crucial for understanding the nature of the γ( f ) and Ω( f ) functions that determine the solutions.
The non-linear Fokker-Planck equation in the most general possible from (2) has been obtained by applying the Kramers-Moyal expansion (that takes into account only transitions between near neighbors) to a balance equation based on the Kinetic Interaction Principle (KIP) [1].
Subsequently, Eq. ( 3) was obtained in [54] from a master equation that involves nearest-neighbor transitions, the rates of which are determined by two arbitrary functions a( f ) and Y( f , f ).The first function, a( f ), depends on the particle density function f and is related to the γ( f ) function through the simple relation γ( f ) = f a( f ).On the other hand, Y( f , f ) depends on two distinct particle density functions, f and f and is related to the Ω( f ) function by means of a more complicated expression.

The Master Equation
Let us consider the particle kinetics in a one-dimensional lattice gas comprising N identical particles.Under the hypothesis that only transitions involving the nearest-neighbour lattice sites are allowed during the evolution of the particle system, the probability density function f i = f (t, x i ) obeys the following master equation where π(t, x i → x j ) is the transition probability from site i to site j.
The transition probability related to the hopping of a particle from site i to site j depends on the nature of the interaction between the lattice and the particle, and this dependence is taken into account through a factor called transition rate w ij .The transition probability can of course depend on the particle density of the starting i and arrival j sites.
In ref. [1], it was postulated that the f i and f j populations, of the starting and arrival sites affect the transition probability through the two arbitrary functions a( f i ), and b( f j ), so that π(t, x i → x j ) assumes the following factorized form and this particular dependence on the particle population of the starting and arrival sites defines the Kinetic Interaction Principle.

The Kinetic interaction Principle
Let us now consider the transition probability from site i to site j.The a( f i ) function must satisfy the obvious condition a(0) = 0, because in the case where the starting site is empty no particles can transit toward the arrival site.On the other hand the b( f j ) function must satisfy the condition b(0) = 1, because in the case where the arrival site is empty the transition probability cannot be influenced by the arrival site.
In linear kinetics, a( f i ) = f i and b( f j ) = 1, is usually posed [60].In nonlinear kinetics the b( f j ) function plays an important role, because it can stimulate or inhibit the particle transition to site j and can simulate collective interactions in many body physics.For instance, the expression b( f j ) = 1 − f j can account for the Pauli exclusion principle that governs a fermionic particle system [50,58,59].Analogously, the expression b( f j ) = 1 + f j accounts for bosonic systems.Other more complicated expressions of the b( f j ) factor can take into account collective interactions introduced by the Haldane generalized exclusion principle originating from the fractal structure of the single particle Hilbert space.
Another case of nonlinear kinetics corresponding to a( f i ) = f i e −u f i where u > 0 and b( f j ) = 1 was investigated in [61], motivated by observations of two different grain growth regimes in sintering studies [62].In this type of master equation the value of u controls the equilibrium state: a normal diffusive regime is obtained for u below a threshold, while an abnormal diffusion regime that exhibits the Matthew effect of accumulated advantage is obtained at higher u.
If transitions are allowed only among the nearest lattice neighbors, i.e. i → j = i ± 1, after setting w ij = w ± i , where w + i corresponds to the transition i → i + 1 and w − i corresponds to the transition i → i − 1, the KIP assumes the following form while the master equation (4) becomes

Incoming and Outgoing Lattice Currents
Let us express the master equation (7) in the form where the outgoing, j + i , and incoming, j − i , currents are defined as follows Taking into account the continuity relation between the incoming current at the node i and the outgoing current at node i − 1, that is, the difference j + i − j − i between the outgoing and the incoming currents can be expressed exclusively in terms of the outgoing current (or alternatively in terms of the incoming current), according to where the forward difference ∆ + g i and the backward difference ∆ − g i of the discrete function g i are defined through

Continuity Form of the Master Equation
Based on the definition of the incoming and outgoing currents given above, the master equation ( 8) assumes the following compact form The two forms of the master equation, as given by Eq. ( 15), are asymmetric with respect to the currents.By employing the symmetry ∆ − j + i = ∆ + j − i , the master equation can be expressed in a symmetric form that involves both the outgoing j + i and incoming j − i currents, as well as both the forward ∆ + and backward ∆ − differences, i.e., In order to write the latter equation in an even more symmetric form we introduce the discrete Fokker-Planck current, j i , according to the average When this definition is employed, the incoming j − i and the outgoing j + i currents can be expressed in terms of the j i current and after taking into account the master equation ( 8), we obtain Finally, the following first-and second-order symmetric finite difference operators are introduced and after taking into account Eq. ( 19), the master equation ( 16) assumes the form The latter expression of the master equation is particulary suitable for obtaining the continuous limit as ∆x → 0. First, we introduce the particle distribution function f (t, x), according to and the Fokker-Planck current j(t, x), through Subsequently, after taking into accounts the limits 2 , both Eq. ( 16) and Eq. ( 22) reduce to the continuity equation The above continuity equation is the nonlinear Fokker-Planck equation obtained in the continuous limit, starting from the master equation (7).In order to write the nonlinear Fokker-Planck equation explicitly, as last step we have to calculate the Fokker-Planck current j(t, x), starting from its definition (17), and this will be the task of the next section.

Lattice Expression of the Fokker-Planck Current
In order to calculate the continuous limit of the discrete Fokker-Planck current defined in Eq. ( 24), we express j i , defined through Eqs. ( 9), ( 10) and (17), in terms of the functions a(•) and b(•) thus obtaining In order to express the current j i in a form that can be more efficiently used to compute the continuous limit, we first note that the transition rates w ± i∓1 can be expanded in terms of the nearest-neighbour rates w ± i as follows Furthermore, we assume that the functions a(•) and b(•) are differentiable functions of f i .Then, the a( f i+1 ) function can be approximated using the nearest-neighbour approximation of f i+1 in terms of f i and the leading-order Taylor expansion of a( f i+1 ) around f i which lead to Following the same procedure, one obtains the approximate expressions for the functions a(•), b(•), at the nearest-neighbour sites of a given site i After substitution of the above expressions for a( f i±1 ) and b( f i±1 ) in the equation Eq. ( 26) of the discrete Fokker-Planck current j i , one obtains

Continuum Limit Expression of the Fokker-Planck Current
The above expression for the discrete current can be simplified in the continuum limit, ∆x → 0, by taking into account that there is no difference between the first-order forward, backward, and symmetric finite-differences, i.e., Hence, the discrete Fokker-Planck current j i assumes the form The diffusion coefficient, defined by means of and the drift coefficient, defined by means of are usually introduced into Fokker-Plank kinetics.Following these replacements, the discrete Fokker-Planck current given by Eq. ( 33) assumes the equivalent form Furthermore, the introduction of the kinetic energy function U i according to allows expressing the current in the following form or equivalently as The discrete current j i , as given by the latter equation, is expressed in a form that permits straightforward evaluation of the ∆x → 0 continuous limit.Thus, the nonlinear Fokker-Planck current in the continuum limit is given by the following equation

Continuum Limit and Fokker-Planck Equation
Finally, by inserting the continuum limit of the Fokker-Planck j(t, x) current, given by Eq. ( 40), in the continuity equation (25) we obtain the nonlinear Fokker-Planck equation which coincides with Eq. ( 2).Thus, the derivations in this section show how the nonlinear Fokker-Planck equation can be obtained from the discrete Fokker-Planck current of Eq. ( 24), which is derived from the simple, KIP-based master equation given by Eq. ( 7).

Nonlinear Drift and Fick Currents
Let us now write the Fokker-Plank equation (41) in the following form where the nonlinear drift current j drift = j drift (t, x) and the nonlinear Fick current j Fick = j Fick (t, x) are defined according to The term Dφ( f ) represents the nonlinear diffusion coefficient.If φ( f ) = 1, then the ordinary Fick current j Fick = −D ∂ f ∂x is obtained.The ordinary Fick current appears in (i) linear kinetics for a( f ) = f and b( f ) = 1, and (ii) in the classical models of boson or fermion kinetics, for a( f ) = f and b( f ) = 1 ± f respectively.
This important result, pertaining to the ordinary diffusion that underlies the kinetics of classical bosons and fermions, naturally leads to the question whether there exist other nonlinear kinetics which are governed by standard Fickian diffusion.

Nonlinear Kinetics with Fickian Diffusion
In the following, we focus on the Fokker-Planck equation (41) in the case of Fickian diffusion i.e., if the φ( f ) function defined in Eq. ( 45) is subject to the φ( f ) = 1 condition.In nonlinear kinetics, the generalized logarithm Λ( f ) is introduced through In terms of the generalized logarithm, the stationary and stable solution, f s , of Eq. ( 41) assumes the form where µ is an arbitrary constant.
It is easy to express the a( f ) and b( f ) functions in terms of the Λ( f ) function.First, we write Eq. ( 46) in the form Then, based on Eq. ( 45) and taking into account the φ( f ) = 1 condition, we obtain Finally, the solutions of the two equations above for a( f ) and b( f ) lead to The nonlinear Fokker-Planck kinetics described by Eq. ( 41), when a( f ) and b( f ) are given by Eq. ( 50) and ( 51), simplifies to where the function γ( f ) represents the inverse of the derivative of the generalized logarithm, given by 1 The Eq. ( 52) describes a process that undergoes standard Fickian diffusion dominated by a nonlinear drift.The latter imposes the non-standard form given by Eq. ( 47) to the stationary distribution function.

The Case of Kappa Kinetics
As a working example, let us briefly consider the κ-kinetics [1], where the generalized logarithm is given by ln κ ( f ) being the κ-logarithm defined by means of the equation The inverse of the logarithm function, i.e., the κ-exponential, is given by In this case the a( f ) and b( f ) functions assume the expressions while the respective Fokker-Planck equation becomes It is remarkable that the latter equation is different from the one proposed in [1].Both equations are correct and describe two different nonlinear kinetics that admit the same stationary solution The main difference between the two kinetics is that the present Fokker-Planck equation describes a process which undergoes ordinary Fickian diffusion in the presence of non-Brownian drift, while the Fokker-Planck equation of ref. [1] describes a process that undergoes non-Fickian diffusion in the presence of ordinary Brownian drift.

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 yields 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.
As a first result, we used the KIP-based master equation to obtain the nonlinear Fokker-Planck equation Eq. ( 2), This 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 that will be used to solve the Fokker-Planck partial differential equation (2).This approach resolves the ambiguity that results from different numerical discretization schemes of the Fokker-Planck equation.
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 Eqs. ( 42)-( 44), we have demontrated that some important nonlinear kinetics formulations proposed in the literature can be successfully described by ordinary Fickian diffusion.