Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part III Extensions and applications to kinetic theory and transport

Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part III Extensions and applications to kinetic theory and transport

Massimiliano Giona Dipartimento di Ingegneria Chimica DICMA Facoltà di Ingegneria, La Sapienza Università di Roma via Eudossiana 18, 00184, Roma, Italy
Email: massimiliano.giona@uniroma1.it
Antonio Brasiello Dipartimento di Ingegneria Industriale Università degli Studi di Salerno via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy Silvestro Crescitelli Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale Università degli Studi di Napoli “Federico II” piazzale Tecchio 80, 80125 Napoli, Italy
Abstract

This third part extends the theory of Generalized Poisson-Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker-Planck-Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac’s program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.

1 Introduction

This third and last part of the work on Generalized Poisson-Kac (GPK) processes and their physical applications extends the analysis developed in parts I and II [1, 2], developing the generalization of GPK theory to a broad spectrum of stochastic phenomenologies. With respect to the theory developed in [1, 2], two lines of attack characterize this extension: (i) the inclusion of nonlinearities, and (ii) the extension to a continuum of states.

Nonlinearities can be treated in two different ways. The first class of nonlinear models assumes that the state (position) variable can influence the basic parameters characterizing stochastic GPK perturbations. In the case of GPK perturbations, this reflects into the functional dependence of , and on . In the case of a position dependent system of stochastic velocities, i.e, , GPK models correspond to nonlinear Langevin equations [3, 4], since the latter provide the Kac limit (in the Stratonovich interpretation of the stochastic integral) for this class of systems. The functional dependence of the transition rates , or of the entries of the transition probability matrix on , provides new phenomena, as it emerges from the analysis of their Kac limits.

The second way to include nonlinearities, analogous to the McKean approach to Langevin equations [5] leads to GPK microdynamic equations which depend on the statistical characterization of the process itself (in the present case, the system of partial probability density functions ). This leads to the concept of nonlinear Fokker-Planck-Kac equation (this diction stems from the Langevin counterpart [6]), the dynamic properties of which can be extremely rich.

The extension from a discrete number of states to a continuum of stochastic states is fairly straightforward within the formalism developed in part I (see also the discussion in part I on the multidichotomic approach). Moreover, the coupling of nonlinear effects with a continuum of stochastic states permits to derive the classical nonlinear Boltzmann equation of the kinetic theory of gases [7] within the GPK formalism. This result deserves particular attention as it shows, unambiguously, that the Boltzmann equation admits a fully stochastic explanation. In some sense, this result completes the original Kac’s program in kinetic theory [8, 9, 10, 11], originated from the article [12] aimed at providing an extended Markov model for interpreting the celebrated Boltzmann equation of kinetic theory. For a discussion on extended Markov models see [13].

Finally, the article outlines the bridge between the stochastic description of particle microdynamics based on GPK equations, and transport theory of continuous media. This connection is developed with the aid of some classical problems. In developing a transport theory from GPK microdynamics the role of the primitive statistical formulation of GPK processes, based on the system of partial probability densities , clearly emerges (for a discussion see also Section 2 in part I), and it is mapped into a corresponding system of partial concentrations/velocity fields. This part of the article is of primary interest in extended thermodynamic theories of irreversible processes [14, 15, 16], as it provides a novel way to develop these theories enforcing the assumption of finite propagation velocity for thermodynamic processes, and overcoming the intrinsic limitations of models based on the higher-dimensional Cattaneo equation (see part I for details).

The article is organized as follows. Section 2 develops the extensions of GPK models (nonlinearity, continuum of stochastic states), presenting for each class of models a physical example. Section 3 derives the connection (equivalence) between a nonlinear GPK process admitting a continuum of stochastic states and the Boltzmann equation, discussing some implications of this result. Section 4 addresses the connection between GPK microdynamics and the associated transport formalism in continua by considering several problems ranging from dynamo theory [17, 18] to mass and momentum balances, including a brief description of chemical reactions.

2 Generalizations

The theory of GPK processes can be generalized in several different directions that provide, from one hand, a valuable system of stochastic modeling tools of increasing complexity and, from the other hand, the possibility of interpreting a broader physical phenomenology. In the remainder of this Section we introduce the various generalizations by considering first one-dimensional Poisson-Kac processes, and subsequently extending the theory to GPK processes.

2.1 Nonlinear GPK processes and Poisson fields

In order to define nonlinear GPK processes it is convenient to introduce the concept of Poisson fields. A Poisson field , is a Poisson process over the real line such that its transition rate depends on and eventually on time . If the Poisson field is said to be stationary, while if depends explicitly on time is referred to as a non-stationary field.

Let , and let be a positive real-valued function. A nonlinear GPK process is defined via the stochastic differential equation

 dx(t)=v(x(t))dt+b(x(t))(−1)χ(x(t),t) (1)

where is a deterministic bias. The presence of a position dependent stochastic velocity , and the dependence on of the transition rate defining the Poisson field , makes this model conceptually similar to the nonlinear Langevin equations [3, 4].

We have assume that does not depend explicitly on time . This condition can be easily removed, but the generalization to time-dependent involves more lengthy calculations of the Kac limit, the full development of which is left to the reader.

For the process associated with eq. (1), the partial probability density functions fully characterize its statistical properties. These quantities satisfy the balance equations

 ∂tp±(x,t)=−∂x(v(x)p±(x,t))∓∂x(b(x)p±(x,t))∓λ(x,t)[p+(x,t)−p−(x,t)] (2)

and the “diffusive” probability flux is given by . Let , where . Set where . In terms of normalized quantity the constitutive equation for the diffusive flux becomes

 ∂tϕ(x,t)=−∂x(v(x)ϕ(x,t))−(b(c))2∂x(ˆb(x)p(x,t))−2λ(x,t)ϕ(x,t) (3)

Let , where . In the limit , the constitutive equation for becomes

 ϕ(x,t)=−Dnomˆλ(x,t)∂x(ˆb(x)p(x,t)) (4)

that, substituted into the balance equation for , provides

 ∂tp(x,t)=−∂x(v(x)p(x,t))+Dnom∂x[ˆb(x)ˆλ(x,t)∂x(ˆb(x)p(x,t))] (5)

which represents the Kac limit for the nonlinear Poisson-Kac process considered. In terms of the original quantities and , the Kac limit can be expressed equivalently as

 ∂tp(x,t)=−∂x[v(x)p(x,t)]+12∂x[b(x)∂xb(x)λ(x,t)p(x,t)]+12∂x[b(x)b(x)λ(x,t)∂xp(x,t)] (6)

Eq. (6) corresponds to an advection-diffusion equation characterized by an effective velocity

 veff(x,t)=v(x)−b(x)∂xb(x)2λ(x,t) (7)

and by an effective diffusivity

 Deff(x,t)=b2(x)2λ(x,t) (8)

The above-derived Kac limit should be compared with the statistical description of a classical Langevin equation driven by Wiener fluctuations

 dx(t)=vS(x(t),t)dt+√2DS(x,t)∘dw(t) (9)

where are the increments of a one-dimensional Wiener process in the time interval , to be interpreted “a la Stratonovich”. In eq. (9), “” indicates the Stratonovich recipe for the stochastic integrals. The Fokker-Planck equation associated with eq. (9) is given by

 ∂tp(x,t)=−∂x[vS(x,t)p(x,t)]+12∂x[p(x,t)∂xDS(x,t)]+∂x[DS(x,t)∂xp(x,t)] (10)

The reason for the choice of the Stratonovich rather than the Ito calculus follows from the Wong-Zakai theorem [19, 20]: Poisson-Kac processes are stochastic dynamical systems excited by a.e. differentiable smooth perturbations, converging in the Kac limit to ordinary Brownian motion. According to the Wong-Zakai result, that in the present case corresponds to the Kac limit, these processes should converge in the Kac limit to the Stratonovich formulation of the Langevin equation (9), where and should coincide with and , respectively. Below, we discuss this convergence that, in point of fact, is slightly more subtle than expected.

Two cases should be considered. Case (A): does not depend on . It follows from the comparison of eqs. (6) and (10) that the Kac limit of eq. (1) coincides with eq. (9) provided that

 vS(x)=v(x),DS(x,t)=Deff(x,t) (11)

and this can be viewed as a corollary of the Wong-Zakai theorem. In this case, the Poisson-Kac process is a stochastic mollification of the Langevin-Stratonovich equation (9). Case (B): depends explicitly on . Also in this case

 DS(x,t)=Deff(x,t) (12)

but the equivalence between the convective contributions provides the relation

 vS(x,t)=v(x)−DS(x,t)2∂xlogλ(x,t) (13)

A physical justification of this phenomenon is addressed at the end of this paragraph.

The generalization to nonlinear GPK processes in is straightforward. Define a -state finite Poisson field a stochastic process parametrized with respect to , attaining different possible states, such that the transition structure between the states is described by the time-continuous Markov chain

 ∂tP(χ)α(x,t)=−λα(x,t)P(χ)α(x,t)+N∑γ=1Kα,γ(x,t)P(χ)γ(x,t) (14)

, where is the probability of the occurrence of at position and time . In equation (14) , and are the entries of a symmetric transition matrix, , for any , . Moreover, let us assume that

 Aα,γ(x,t)=Kα,γ(x,t)λγ(x,t) (15)

represents an irreducible left-stochastic matrix function for any and . Given vector valued functions , satisfying the zero-bias condition

 N∑α=1bα(x)=0 (16)

identically for , a nonlinear GPK process is described by the stochastic differential equation

 dx(t)=v(x(t))dt+bχN(x,t)(x)dt (17)

Its statistical characterization involves partial probability density functions , satisfying the balance equations

 ∂tpα(x,t) = −∇⋅(v(x)pα(x,t))−∇⋅(bα(x)pα(x,t)) (18) − λα(x,t)pα(x,t)+N∑γ=1Kα,γ(x,t)pγ(x,t)

Using the representation in terms of and , it is straightforward to construct a stochastic simulator of eq. (18) analogous to that defined in Section 4 of part I for linear GPK processes.

It is worth observing that there is a substantial difference between nonlinear Poisson-Kac/GPK process of the form (1) or (17) and the nonlinear Langevin equations driven by Wiener perturbations, such as eq. (9). In the latter case, the -dynamics does not influences the statistical properties of the stochastic Wiener forcing and implies solely a modulation of the intensity of the stochastic perturbation, that depends on , via the factor as in eq. (9). Conversely, in the case of Poisson-Kac/GPK processes, there is a two-way coupling between the -dynamics of the Poissonian perturbation, as the evolution of influences the statistics of the Poissonian field, whenever the transition rate or the transition rate vector depend explicitly on and , respectively.

This observation, physically explain the apparently “anomalous” correspondence relation (13), as this model does not fall within the range of application of the Wong-Zakai theorem.

2.2 Continuous GPK processes

A further generalization of GPK theory is the extension to a continuous number of states. Such a continuous extension is not suitable within the framework of multi-dichotomic processes discussed in Section 3 of part I, and this constitutes the main shortcoming of this class of models in the applications to statistical physical problems.

Next, consider the one-dimensional case. Let be a time-continuous Markov process attaining a continuum of states belonging to a domain . Its statistical description involves the transition rate kernel , which is a positive symmetric kernel

 K(α,β)=K(β,α),K(α,β)≥0 (19)

Given , it is possible to introduce the transition rates

 λ(α)=∫DK(β,α)dβ>0 (20)

and the transition probability kernel

 A(α,β)=K(α,β)λ(β) (21)

The transition probability kernel possesses the following properties: (i) normalization, i.e.,

 ∫DA(β,α)dβ=1 (22)

i.e., it is a left-stochastic kernel, and (ii) it is assumed that is irreducible, meaning that solely the constant function is the left eigenfunction of , associated with the Frobenius eigenvalue . In other terms, the multiplicity of the Frobenius eigenvalue is .

Indicating with , for any , the evolution of this probability density follows the Markovian character of the transition dynamics

 ∂tp(Ξ)(α,t)=−(∫DK(β,α)dβ)p(Ξ)(α,t)+∫DK(α,β)p(Ξ)(β,t)dβ (23)

Let , the stochastic velocity function satisfying the zero-bias property

 ∫Db(x,α)dα=0 (24)

for any . Within this setting, it is possible to introduce the stochastic differential equation

 dx(t)=v(x)dt+b(x,Ξ(t))dt (25)

which represents the microdynamic description of a one-dimensional Continuous GPK process (CGPK). The statistical description of a CGPK process involves the partial probability density functions continuously parametrized with respect to the state variable of the stochastic forcing ,

 p(x,t;α)dxdα=Prob[X(t)∈(x,x+dx),Ξ(t)∈(α,α+dα)] (26)

such that the overall probability density function for and its diffusive flux are respectively given by

 p(x,t)=∫Dp(x,t;α)dα,Jd(x,t)=∫Db(x,α)p(x,t;α)dα (27)

In the continuous setting represents the primitive statistical description of a CGPK process, and its evolution equation is expressed by

 ∂tp(x,t;α) = −∂x(v(x)p(x,t;α))−∂x(b(x,α)p(x,t,;α)) (28) + ∫DK(α,β)[p(x,t;β)−p(x,t;α)]dβ

where the symmetry of the kernel has been applied.

Assume for simplicity that does not depend on , and that the stochastic velocity function is a monotonic function of . Under these conditions one can simply map the states of using the transformation , thus defining the new stochastic process defined as

 b(Ξ(t))=Θ(t),Ξ(t)=b−1(Θ(t)) (29)

Let . In this way, the stochastic differential equation defining CGPK process attains the more compact expression

 dx(t)=v(x(t))dt+Θ(t)dt (30)

where . Letting the inverse of , and define

 ¯¯¯p(x,t;c)=p(x,t;α)|α=f(c)df(c)dc (31)
 ¯¯¯¯¯K(c,c′)=K(α,β)|α=f(c),β=f(c′)df(c)dc (32)

the CGKP process, parametrized with respect to the values attained by stochastic velocity , is described by the transformed probability density functions associated with eq. (30), that satisfy the balance equations

 ∂t¯¯¯p(x,t;c) = −∂x(v(x)¯¯¯p(x,t;c))−c∂x¯¯¯p(x,t;c) (33) + (∫C¯¯¯¯¯K(c′,c)dc′)¯¯¯p(x,t;c)−∫C¯¯¯¯¯K(c,c′)¯¯¯p(x,t;c′)dc′

The transformation of the transition kernel (32) is such that the transition rates coincide with the corresponding ones expressed with respect to the parameter , namely

 ¯¯¯λ(c)=∫C¯¯¯¯¯K(c′,c)dc′=∫DK(β,f(c))dβ=λ(α)|α=f(c) (34)

while the transformed transition probability matrix kernel is defined as

 ¯¯¯¯A(c′,c)=¯¯¯¯¯K(c′,c)¯¯¯λ(c)=A(β,α)|α=f(c),β=f(c′)df(c′)dc′ (35)

Observe that if is symmetric, this is no longer true for in the presence of a nonlinear expression of .

In the case , and symmetric, the analysis greatly simplifies, and eq. (33) reduces to

 ∂t¯¯¯p(x,t;c) = −∂x(v(x)¯¯¯p(x,t;c))−c∂x¯¯¯p(x,t;c) (36) + ∫D¯¯¯¯¯K(c,c′)[¯¯¯p(x,t;c′)−¯¯¯p(x,t;c)]dc′

since . It can be recognized that eq. (36) represents a one-dimensional linearized Boltzmann equation. The zero-bias condition (24) becomes

 ∫Bcdc=0 (37)

which indicates that the domain must be the union of symmetric intervals with respect to , eventually reduces to a single interval . The overall probability density function satisfies the balance equation

 ∂t¯¯¯p(x,t)=−∂x(v(x)¯¯¯p(x,t;c))−∂x¯¯¯¯Jd(x,t) (38)

where the diffusive flux fulfills the constitutive equation

 ∂t¯¯¯¯Jd(x,t)=−∂x(v(x)¯¯¯¯Jd(x,t))−∂x∫Bc2¯¯¯p(x,t;c)dc −∫Bc¯¯¯λ(c)¯¯¯p(x,t;c)dc+∫B∫Bc¯¯¯¯¯K(c,c′)¯¯¯p(x,t,c′)dcdc′ (39)

where .

The Kac limit can be ascertained in the continuous case using techniques and arguments analogous to those developed in part I. To give an example, assume , , and that there exits a such that

 ∫Bc¯¯¯¯¯K(c,c′)dc=δc′ (40)

For , keeping fixed the nominal diffusivity , the infinitely fast recombination implies the equipartition amongst the partial probability waves,

 ¯¯¯p(x,t;c)=¯¯¯p(x,t)2b(c)+o(¯¯¯λ−1) (41)

and the effective diffusivity can be obtained in closed form

 Deff=12(1−δ)¯¯¯λb(c)∫Bc2dc=2Dnom3(1−δ) (42)

Other cases of interest are addressed in paragraph 2.4.

2.3 Nonlinear FPK models

In paragraph 2.1 we have indicated with the diction nonlinear GPK those processes in which, either the stochastic velocity vectors, or the transition rates/transition probability matrix, or both, depend on the state variable . There is another, significant, source of nonlinearity in stochastic models occurring whenever the statistical properties of the stochastic perturbation influence the dynamics of the stochastic process itself, so that the stochastic microdynamics depends on the collective behavior of , i.e., on its probability density functions. This is the case of the nonlinear Fokker-Planck equation introduced by McKean for Langevin-Wiener stochastic models [5]. For review see [6]. In the case of Poisson-Kac and GPK model this means that the stochastic velocity vectors and the transition rates are functionals of the probability densities describing , i.e., of the partial probability density functions . We refer to this situation as a Nonlinear Fokker-Planck-Kac model (NFPK).

The paradigm of Nonlinear FPK processes is represented by the one-dimensional Poisson-Kac model

 dx(t)=v(x(t))dt+b(−1)χ(x,t,{p±(x(t),t)})dt (43)

where is a Poisson field the transition rate of which is a positive functional of the partial probability waves associated with . For example, it can depend linearly on , such as

 λ(x,{p±(x(t),t)})=λ0(x)+∫Ra+(x)p+(x,t)dx+∫Ra−(x)p−(x,t)dx (44)

where are functions of , or can be a function of the partial moment hierarchy associated with . For the NFPK process (43), the evolution equations for the partial probability waves become nonlinear, namely

 ∂tp±(x,t)=−∂x[(v(x)±b)p±(x,t)]∓λ(x,{p±(x(t),t)})[p+(x,t)−p−(x,t)] (45)

Alternatively, another class of NGPK processes can be defined in the case the advective contribution represents a functional of the partial probability waves. An example of this class of model is addressed in the next paragraph.

The generalization to GPK processes in is straightforward. We develop an example in the next Section, combining all the extension discussed so far in order to address a physically relevant problem, namely the stochastic nature of the collisional Boltzmann equation.

2.4 Examples

In this paragraph we analyze three prototypical examples covering the range of generalization of GPK processes treated in the previous paragraphs.

To begin with, consider a one-dimensional Poisson-Kac process in , defined by the stochastic differential equation

 dx(t)=b(−1)χ(x,t)dt (46)

where the Poisson field is characterized by the following transition-rate function

 λ(x)=λ0(1+|x|) (47)

Figure 1 depicts the behavior of the mean square displacement vs , obtained from the stochastic simulation of eqs. (46)-(47), using particles starting from at , for two different values of : (line a) and (line b). In this problem , so that .

The mean square displacement admits a crossover behavior,

 σ2x(t)∼{t2t

where the crossover time is about at , and order of at . At short time-scales grows quadratically with time , due to the finite propagation velocity (a.e. smoothness) characterizing Poisson-Kac and GPK processes. Asymptotically, i.e., for , the mean square displacement exhibits an anomalous scaling in time with an exponent equal to .

This result can be easily interpreted using elementary scaling analysis, as the transition rate expression (47) corresponds to a position dependent diffusion coefficient . Consequently, , and therefore , implying .

This example admits another byproduct: in the framework of nonlinear GPK processes it is possible to generate anomalous diffusion scalings just by a suitable choice of the transition rate function characterizing the Poissonian field or .

As a second example, consider a continuous GPK process in corresponding to the dynamics, in the overdamped regime, of a particle moving in the potential

 U(x)=U(x,y)=x44−x22+y22=Ux(x)+Uy(y) (49)

the contour plot of which is depicted in figure 2 panel (a). This expression corresponds to the superposition of a bistable potential along the -coordinate, and of a harmonic potential in the -coordinate. Assume a friction factor a.u., and let a continuous Markov process attaining values in , characterized by a uniform transition rate and by a uniform transition kernel

 K(α,β)=λ2πα,β∈[0,2π) (50)

Consider the Continuous GPK process defined by the stochastic differential equation

 dx(t)=−∇xU(x(t))dt+b(Ξ(t))dt (51)

where the stochastic velocity function is defined by

 b(α)=b(cos(α),sin(α))α∈[0,2π) (52)

For this problem, the balance equation for the partial probability waves reads

 ∂tp(x,t;α) = ∇x⋅(∇xU(x)p(x,t;α))−b(α)⋅∇xp(x,t;α) (53) − λp(x,t;α)+λ2π∫2π0p(x,t;β)dβ

In this case, the diffusive flux is given by . For the dyadic tensor associated with one has

 ∫2π0b(α)b(α)dα=b2I2 (54)

where is the identity matrix, and consequently the Kac limit provides . The numerical simulation of this stochastic process possessing a continuum of states is simple: (i) the intervals between two consecutive transitions are distributed exponentially with probability density , (ii) whenever a transition occurs from a state to a new state, say , the determination of the new state is chosen randomly from a uniform probability distribution in .

Figure 2 panels (b)-(d) depict the contour plot of the stationary overall probability density functions for several values of and , obtained from stochastic simulations of eq. (51) using an ensemble of particles. The corresponding stationary marginal distributions with respect to the coordinate are depicted in figure 3 panel (a). At low values of and , panel (b) in figure 2 and curve (a) in figure 3, the stationary probability density concentrates in an external shell far away from the minima of the potential. This is a peculiar feature of undulatory transport models in the presence of conservative potentials, which may possess an invariant region such that , where is the external normal unit vector at points (see also the discussion in part II on invariant regions).

As decreases, displays the typical bimodal shape characterizing bistable motion with diffusion, panel (d) in figure 2 and lines (b) and (d) in figure 3.

As increases, keeping constant the value of the effective diffusivity, the stationary density approaches the Boltzmannian distribution. An example is depicted in figure 2 panel (d) and it is clearly evident from figure 3 panel (b) that the marginal stationary distribution , at and , practically coincides with its Boltzmann-Kac limit (observe that ).

Finally, consider a Nonlinear FPK process expressed in the form of a Shimizu-Yamada model [21]. More precisely, consider the stochastic one-dimensional differential equation

 dx(t)=v(x(t),{p±(x(t),t)})dt+b(−1)χ(t)dt (55)

where is a usual Poisson process possessing constant transition rate , and the drift velocity depends on the partial probability densities as

 v(x,{p±(x,t)})=−x+ε[m(1)+(t)+m(1)−(t)]+vf=−x+εm(1)(t)+vf (56)

where are the partial first-order moments of the process, and , two constant parameters. This model corresponds to the dynamics of a stochastic particle in a harmonic potential (in the overdamped regime), subjected to an external constant bias and to an additional contribution proportional to the overall first-order moment .

The balance equation for the partial waves attains the form

 ∂tp±(x,t)=∂x[(x−εm(1)(t)−vf∓b)p±(x,t)]∓λ[p+(x,t)−p−(x,t)] (57)

where . As initial condition for the partial probability waves consider

 p±(x,0)=δ(x)2 (58)

Figure 4 panel (a) depicts the evolution of the overall density function in the case of a pure harmonic oscillator (linear case) at , , for which the density function becomes localized at with a variance equal to . Panel (b) refers to , i.e., to a truly Nonlinear FPK model.

In order to derive the salient qualitative properties of this model consider the first elements of the partial moment hierarchy. For the zero-th order moments, the symmetric initial condition (58) implies

 m(0)+(t)=m(0)−(t)=12t≥0 (59)

The evolution equations for the first-order moments attain the form

 ∂tm(1)±(t)=−m(1)±(t)+[m(1)(t)+vf±b]m(0)±(t)−λ[m(1)+(t)−m(1)−(t)] (60)

Summing the equations for and , and accounting for eq. (59) one obtains

 ∂tm(1)(t)=vf (61)

which implies

 m(1)(t)=vft (62)

Therefore, the probability profile moves at constant speed . As regards the partial second-order moments

 ∂tm(2)±(t)=−2m(2)±(t)+2[m(1)(t)+vf±b]m(1)±(t)−λ[m(2)+(t)−m(2)−(t)] (63)

From eqs. (59)-(63), the overall variance fulfills the balance equation

 ∂tσ2(t)=−2σ2(t)+2b[m(1)+(t)−m(1)−(t)] (64)

Taking the difference between the evolution equations of the first-order partial moments provides

 ∂t[m(1)+(t)−m(1)−(t)]=−[m(1)+(t)−m(1)−(t)]+b−2λ[m(1)+(t)−m(1)−(t)] (65)

Asymptotically, the difference between the partial first-order moments converges towards the value

 m(1)+,∞−m(1)−,∞=b2λ+1 (66)

which implies that where

 σ2∞=b22λ+1=Deff1+1/2λ (67)

The variance of the overall probability density wave attains asymptotically a constant value equal to . The NFPK considered above describes the evolution of a nonlinear soliton traveling with constant speed and possessing a constant variance (as can be observed from the profiles in figure 4 panel (b)). Figure 5 depicts the comparison of numerical simulation results for and the asymptotic expression (67).

The shape of the propagating solitons depends significantly on the transition rate . For high values of , a nearly Gaussian soliton propagates as expected from the Kac limit, see figure 4 panel (b). However, for small values of , profiles completely different from the Gaussian one can occur. This phenomenon is depicted in figure 6 panels (a) and (b), corresponding to and , respectively. The resulting probability density profiles of the propagating solitons, depicted in this figure, are rescaled to unit zero-th order moment, i.e.