Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

# Generalized Poisson-Kac processes and hydrodynamic modeling of systems of interacting particles I - Theory

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
###### Abstract

This article analyzes the formulation of space-time continuous hyperbolic hydrodynamic models for systems of interacting particles moving on a lattice, by connecting their local stochastic lattice dynamics to the formulation of an associated (space-time continuous) Generalized Poisson-Kac process possessing the same local transition rules. The hyperbolic hydrodynamic limit follows naturally from the statistical description of the latter in terms of the system of its partial probability density functions. Several cases are treated, with particular attention to: (i) models of interacting particles satisfying an exclusion principle, and (ii) models defined by a given interparticle interaction potential. In both cases, the hydrodynamic models may display singularities, dynamic phase-transitions and bifurcations (as regards the flux/concentration-gradient constitutive equations), whenever the Kac limit of the model (infinite propagation velocity limit) is considered.

## 1 Introduction

The study of systems of interacting particles represents a central issue in the thermodynamics of irreversible processes and in transport theory since the seminal work by Boltzmann on the kinetic theory of dilute gases [1, 2].

In many cases the analysis of this problem can be simplified by considering particle motion on a discrete lattice. In this way, local particle dynamics is expressed as a system of transition probabilities for particle hopping between the nearest neighburing sites of the lattice. For the setting of this class of problems the reader is referred to [3, 4, 5, 6].

In lattice problems, interactions depend either on sterical and quantum effects, or by the explicit representation of the interaction potential. Sterical and quantum effect imply some form of exclusion principle, whenever no more that a single particle or at most a finite number of particles with different values of the some internal degree of freedom (spin) can be simultaneously present at the same lattice site. Interaction potentials, be them short or long-ranged, influences the hopping transition matrix in a continuous way.

One of the central issues in the physical understanding of these particle systems is the description of their collective statistical properties, i.e., the transition from the local probabilistic lattice dynamics at the level of the single lattice site to a continuous space-time evolution for the associated concentration field (probability density function), accounting for the collective motion of a statistical particle ensemble.

The transition from the lattice motion to the continuous and collective description of particle dynamics, which is the key problem in statistical physics, involves essentially two different and conceptually separate steps: (i) the collective description of the interaction amongst particles in the form of constitutive equations for the probability density flux, expressed as generic nonlinear functional of the particle concentration field, of its spatial gradient and, in principle of its spatial derivatives of any order, and (ii) the continuum limit of a lattice particle problem, the time evolution of which is defined at discrete time instants, in the form of a physical system defined in a continuous space-time.

The first problem is in general extremely difficult and its solution often requires suitable physical approximations on the representation of particle interactions in terms of functionals involving the particle probability density function (one-particle density). The classical example of this type of approximation is the stosszahlansatz in the Boltzmannian description, in which the effects of the binary collisions are treated (invoking the hypothesis of molecular chaos) as loss and gain terms in the evolution equation for the one-particle distribution function and can be assumed proportional to the product of the two one-particle distribution functions , performing a collisional event with velocities and . A similar approximation characterizes the kinetic theory of other systems such e.g. a gas of electrons (plasma), where a self-consistent continuous approximation for the electric field is adopted in the Vlasov equation [2].

An example of the latter problem (transition from a lattice to a continuum description) is the statistical formulation in a spece-time continuum of lattice random walk for system of independent particles, i.e., in the absence of exclusion principles or potential contributions [7]. The technical issue in this case in the transformation of the discrete Markov process describing the evolution for the probability density function of particles evolving onto the lattice (characterized by a discrete spacing between nearest neighbouring sites) at discrete times (corresponding to a physical time interval between subsequent events), into a continuous group (or semigroup) of tranformations parametrized with respect to the physical time acting on the probability density functions , continuously parametrized with respect to the space coordinate .

The latter problem involves the so called hydrodynamic limit, defined for lattice spacing and characteristic time tending to zero, assuming a suitable scaling ansatz between the two characteristic space-time parameters, expressed in the form of a limit behavior

 limτ→0δα(τ)τ=constant (1)

where is some characteristic exponent defining the scaling ansatz. For a thorough discussion on the mathematical physical aspects of the setting and formulation of the hydrodynamic limit for lattice particle dynamics and on the functional form of the resulting hydrodynamic models for prototypical interacting particle systems, the reader is referred to the classical monographs on this topic [8, 9].

In principle, different choices of the scaling assumption (1) provides different hydrodynamic models as analyzed in [7], and briefly reviewed in Section 2. Some choices of the scaling ansatz (1), and specifically the diffusive scaling corresponding to destroy some fundamental physical properties associated with lattice propagation, and forces the hydrodynamic formulation of the statistical properties of the system to be described by parabolic models (first-order in time, second-order in space derivatives) that, by nature, violates fundamental physical conditions (finite propagation velocity, deriving from the Minkowskian metrics of the space-time).

The latter hydrodynamic approach (leading to parabolic models) is fully rigorous from the mathematical point of view. Nevertheless, it superimposes and intermingles two qualitative different physical properties: (i) the existence of long-term (emerging) statistical features in a lattice particle systems, with (ii) the formulation of a continuous space-time description of its statistical evolution, defined technically from the operation of letting with the constraint imposed by the scaling assumption.

From the physical point of view, the assessment of a continuous limit is in principle independent of the finite/infinitesimal values of and . More precisely, there are situations, in which the lattice description is an approximation of the continuous evolution of a particle system in which the values of the parameters and do possess a well defined physical meaning, and are not allowed to attain vanishing values. A typical situation of this sort is a diluted particle gas system, where, near equilibrium, corresponds to the mean-free path between two subsequent collisions depending on the temperature and on the pressure , while the characteristic lattice time scale is related to the root mean square speed depending solely on temperature. A diffusive scaling ansatz () would implies , which violates the equilibrium gas law in diluted condition for fixed volume and particle number.

The analysis developed in [7] for the random walk of independent particles on a lattice suggests another possibility for deriving a space-time continuous statistical description of a system of particles on a lattice for any finite value of and , respectful of the local lattice dynamics. The tool for achieving this program, at least for lattice dynamics of independent particles, is the connection of the original lattice equation of motion with an associated Generalized Poisson-Kac process possessing the same transition probabilities amongst local directions of motion, out which a space-time continuous statistical description of the original lattice process follows.

The scope of the present work is to develop a similar program for systems of interacting particles, which is a much more challenging task as the local dynamic rules for particle motion depend on the state of the whole particle ensemble. These collective effects can be formally treated by invoking a molecular chaos assumption similar to the Boltzmannian “stosszahlansatz” (see Section 4).

Once the statistical description of systems of interacting particles has been embedded in the theory of GPK processes new physical phenomenologies can be unveiled, associated with: (i) the Kac limit of the resulting hyperbolic hydrodynamic description whenever the characteristic propagation velocity is hypothesized to diverge (this occurs for particle systems subjected to exclusion principles); (ii) a new class of dynamic phase transitions can occur in the presence of interparticle potentials, related to multiplicity and bifurcations in the constitutive equations for the concentration flux in terms of the concentration gradient.

Throughout this article the theory is developed for system of interacting particles in one-dimensional spatial problems, in order to simplify the notation and highlight in the simple possible way the new and rich phenomenology that can occur. The numerical investigation of the main qualitative phenomenologies highlighted in this article is addressed in [10],

The article is organized as follows. Starting from a brief conceptual summary of the result presented in [7], section 2 reviews the formalism of Generalized Poisson-Kac processes, and its application to achieve a hyperbolic continuous statistical description of interacting particle systems. Section 3 analyzes the construction of the corresponding GPK processes for systems of particles satisfying an exclusion principles. The analysis is limited to the case of a tagged particle in a mean field characterized by a given (and fixed) particle concentration. Section 4 extends the analysis to the nonlinear case. The class of models considered corresponds to exclusion models where the exclusion principle is satisfied probabilistically. This concept is introduced in this Section and thoroughly explained. The resulting nonlinear hyperbolic hydrodynamic models display very interesting and singular features in the Kac limit. Finally section 5 develops the formalism of hyperbolic hydrodynamic models in the presence of interaction potentials.

## 2 Stochastic processes with finite propagation velocity and hydrodynamic behavior

In a recent work [7], Giona analyzed a very simple example of lattice particle dynamics: the random walk of independent particles on a one-dimensional lattice in the case of asymmetric transitions amongst the two nearest neighboring sites (Asymmetric Lattice Random Walk, ALRW) and its continuous statistical description. The discrete lattice dynamics is characterized by the lattice spacing between nearest neighboring sites and by the constant hopping time between two subsequent events.

The starting observations motivating this revisitation of ALRW are:

• the definition of a space-time continuous process associated with ALRW does not require the limit for and tending to zero. This is because a time-continuous formulation of the process requires solely the local interpolation of particle trajectories between subsequent time instants , and and subsequent positions , , and eventually the assumption of some level of uncertainty in the initial particle position .

• The long-term emergent statistical properties of the process are well defined for any (finite and non vanishing) values of and . Consequently, a space-time continuous hydrodynamic model for this process should be defined independently of any lattice limit , and of any scaling ansatz connecting and in this limit.

• In a smooth, time-continuous, formulation of the process, the ratio , corresponding to the local propagation velocity, should be constant and bounded.

• A time-continuous hydrodynamic model, subjected to the above mentioned constraint on the local propagation velocity, should be able to describe the whole process dynamics, from the early stages, at which particles perform a ballistic motion, to the long-term dispersive features, corresponding to a linear Einsteinian scaling of the mean square displacement, for any value of and .

It has been shown in [7] that the formulation of such a “smooth” hydrodynamic model is possible and it is grounded on the formulation of a space-time continuous stochastic process, analogous to ALRW, belonging to the class of Generalized Poisson-Kac processes [11, 12, 13, 14]. Here the diction “smooth” has been used to indicate that the local propagation velocity is bounded, contrarily to the classical limit formulation grounded on a diffusive scaling asumption , leading to a stochastic description based on almost nowhere differentiable Wiener processes. In the next paragraph, the basic concept of GPK theory are reviewed.

### 2.1 Generalized Poisson-Kac processes

The introduction of Generalized Poisson-Kac processes (GPK for short) stems originally from two main physical reasons: (i) to generalize the class of stochastic models proposed by Marc Kac in one-dimensional spatial systems [15], possessing finite propagation velocity and driven by a simple Poisson process, to any spatial dimension and to any number of stochastic states (including the limit towards a continuum of states); (ii) the setting of stochastically consistent transport models of hyperbolic nature suitable for describing physical transport processes possessing finite propagation velocity. Here, “stochastically consistent” means that there exists a stochastic process admitting these models as its statistical description. This issue is closely connected to the fact, that while the original one-dimensional model considered by Kac provides a stochastic interpretation for the one-dimensional Cattaneo equation , where and are positive constants and , , [16], there are no stochastic processes in with admitting the higher dimensional Cattaneo model as the evolution equation for their probability density function . This property follows also from the observation that the Green function for the Cattaneo hyperbolic transport model in , does not present positivity and attains negative values [17] (which is deprecable in a probabilistic context). The definition of GPK processes is closely connected with the class of higher-dimensional stochastic models studied by Kolesnik [18, 19, 20]

A GPK process in is defined by a finite number of stochastic states, by a family of constant velocity vectors , , by a vector of transition rates , . , and by a transition probability matrix , , , . The generator of stochasticity is a finite -state Poisson process attaining distinct values , and such that the probabilities , satisfy the Markov chain dynamics

 dˆPh(t)dt=−λhˆPh(t)+N∑k=1Ah,kλkˆPk(t) (2)

From the above setting it follows that a GPK process in is defined by the stochastic differential equation

 dx(t)=bχN(t;Λ,A)dt (3)

This means that according to the transition mechanism of state recombination specified by the -state finite Poisson process , defined by and , the velocity vector defining eq. (2) switches amongst the possible realizations .

Since is bounded, the process possesses finite propagation velocity and the trajectory of each realization of a GPK process is with probability 1 an almost everywhere smooth function of time consisting of smooth line segments. It is therefore differentiable at all the time instant, but at the transition points, where switches from one state to another, still possessing well defined left and right derivatives at the transition points (Lipschitz continuity).

The statistical description of a GPK process involves partial probability density functions , ,

 ph(x,t)dx=Prob[X(t)∈(x,x+dx),χN(t)=h] (4)

where , , is the measure element, and means that for each , , . The partial probability densities satisfy the system of first-order differential equations

 ∂ph(x,t)∂t=−bh⋅∇ph(x,t)−λhph(x,t)+N∑k=1Ah,kλkpk(x,t) (5)

Eq. (5) represents the complete statistical description of a GPK process: it plays the same role of the classical parabolic Fokker-Planck equation for Langevin models driven by Wiener noise. The difference with the latter case is that, for GPK processes, a system of partial probability densities, accounting also for the local state of the stochastic perturbation should be defined, owing to the non strictly Markovian structure of the process. The overall probability density function of the process is , and satisfies the conservation equation

 ∂p(x,t)∂t=−∇⋅Jp(x,t) (6)

where the probability density flux is expressed by

 Jp(x,t)=N∑h=1bhph(x,t) (7)

and the constitutive equation for , follows from the definition (7) and from the balance equations (5).

Depending on the structural properties of the GPK, i.e., on and , a variety of different stochastic models can be constructed and the reader is referred to [12] for a structural characterization of these processes. Consider below the simple case where all the stochastic velocity vectors possess the same modulus , i.e., , , where are unit vectors and all the transition rates are equal i.e., , . Under this conditions, it is natural to formulate the Kac limit of a GPK process, i.e., the asymptotics of the GPK process in the case , keeping fixed the ratio

 b202λ0=Dnom (8)

where is referred to as the “nominal diffusivity” of the GPK process. The Kac limit corresponds to the limit behavior of a GPK process in the case its propagation velocity diverges and the same does the transition rate, under the scaling hypothesis (8). Under this conditions, and assuming reasonable no-bias constraints on the system of velocity vectors (see [12] for details), the balance equations (5) for the partial probability density functions collapse into a single parabolic equation for the overall probability density

 ∂p(x,t)∂t=∇⋅(D∇p(x,t)) (9)

where is the effective diffusivity tensor. If the system possesses enough symmetries, is isotropic, i.e., , and the Kac limit of the process is characterized by the single overall probability density function , solution of the diffusion equation

 ∂p(x,t)∂t=D0∇2p(x,t) (10)

where is the scalar effective diffusivity, depending linearly on , i.e. , where .

In the case of ALRW (), the number of states is , corresponding to the movements towards the two (left and right) neighboring sites of any lattice site. Correspondingly, the velocity vectors are expressed by , , where . As regards the transition probabilities, if and are probabilities of moving to the right/left site respectively, letting , it follows that the transition probability matrix is given by

 A=⎛⎝1+r21+r21−r21−r2⎞⎠ (11)

The GPK process associated with the ALRW dynamics on the real line is thus expressed by

 dx(t)=bχ2(t;λ0I,A)dt (12)

where the transition rate vector is isotropic and characterized by the value . The expression for in terms of the lattice parameters can be obtained from the long-term linear scaling of the mean square displacement in the simplest case of symmetric motion () for which follows.

The system of hyperbolic first-order equation for the partial probability densities , represents a continuous hydrodynamic model for the statistical properties of ALRW, and the classical hydrodynamic limit (see e.g. [21]) can be regarded as the Kac limit of this hyperbolic model.

The latter observation provides a novel way of interpreting the classical parabolic hydrodynamic limit of lattice particle dynamics: not as the limit for space-time discretized characteristic scales ( and ) tending to zero (as eq. (12) is already defined in a space-time continuum ), but as the limit for the characteristic propagation velocity of the process tending to infinity, assuming also that the transition rate would diverge . In the latter (Kac) limit, the scaling relation (8) is essential in ensuring the existence of this limit. For further details see [7].

### 2.2 The program

From the analysis developed above, it follows a conceptual program towards the construction of continuous hydrodynamic models of systems of interacting particles. This program is reviewed schematically in figure 1, and follows the same approach applied in [7] to ALRW.

The central issue is the association with a local lattice dynamics of its corresponding continuous GPK process, possessing the same transition probability structure of the original lattice model. Once this step is performed, the derivation of the different forms of continuous hydrodynamic models follows directly from GPK theory. In the remainder of this article, this program is outlined and developed for prototypical models of interacting particle systems.

## 3 Model systems and mean field analysis of tagged particle diffusion

In this Section we consider typical random walk models with exclusion, meaning that at each lattice site no more than one particle or a finite number of them, possessing different characteristic properties (spin), can be present simultaneously.

For several prototypical models we first derive the mean-field behavior of a tagged particle i.e., the properties of the particle diffusive motion by assuming that the average particle concentration is given. Subsequently, we provide the formalization of the same process within the GPK formalism.

Throughout this Section, we consider one-dimensional spatial models.

### 3.1 Fermionic random walk with exclusion

This model has been addressed by Colangeli et al. [22] and represents, in the absence of other interactions, a form of Kawasaki model [23]. Particles behave as fermions, and the direction of the velocity , corresponds to their spin. At each lattice site, at most two particles can be simultaneously present with opposite spins (i.e., oppositive velocity directions).

The dynamic of the exclusion interaction is as follows:

1. first, a velocity switch is considered, meaning that if solely a particle is present at a given site it switches its direction with probability ;

2. the next step is the advective step: particles at a given site move towards the nearest neighboring sites consistently with their velocity directions, i.e., with the values of their spins, and compatibly with the exclusion principle. For instance, a particle at site possessing velocity moves towards provided that the arrival site does not contain already a particle with positive velocity.

As stated at the beginning of this Section, consider the self-diffusion dynamics of a tagged particle, assuming that the average fraction of positively and negatively oriented particles is equal to .

The random walk model of a tagged particle following the recipe stated above, (which is a mean-field approximation), can be described by considering at time both the particle position and its spin variable .

The dynamics for the spin variable is given by:

 sn+1=ξn+1sn (13)

starting at time from , where are uncorrelated random variables attaining values , according to the probabilistic scheme

 ξn+1={1Prob(1+π)/2−1Prob(1−π)/2 (14)

For instance , corresponding to the probability of a velocity switching, equals the probability that a switching event occurs (which is ) times the probability that the arrival site does not contain already a particle with oppositive velocity (which equals ). Observe that the random variables , are uncorrelated with each other, i.e.,

 ⟨ξhξk⟩={⟨ξ2h⟩k=h⟨ξh⟩⟨ξk⟩k≠h (15)

As regards the initial condition , one has

 s0={1Prob1/2−1Prob1/2 (16)

so that .

The dynamics of particle position is then expressed by

 xn+1=xn+sn+1ηn+1 (17)

where are random variables attaining values according to the rule

 ηn+1={0Probπ1Prob(1−π) (18)

For instance, correponds to the probability that the arrival site contains already a particle with the same spin, and therefore equals . Also the variables are uncorrelated with each other,

 ⟨ηhηk⟩={⟨η2h⟩k=h⟨ηh⟩⟨ηk⟩k≠h (19)

and independent of the -variables, . Setting , it follows from eqs. (13) and (17) that

 xn=n∑h=1shηh (20)

for , where

 sh=s0h∏k=1ξk (21)

It follows that

 ⟨sn⟩=0,⟨shsk⟩=⟨s0s|k−h|⟩ (22)

and therefore

 ⟨xn⟩=0,⟨x2n⟩=n∑h=1n∑k=1⟨shsk⟩⟨ηhηk⟩ (23)

Since

 ⟨ηhηk⟩={1−πk=h(1−π)2k≠h (24)

and

 ⟨s20⟩ = 1 ⟨s0sm⟩ = m∏k=1⟨ξk⟩=πm,m≥1 (25)

the expression for the mean square displacement can be explicited in the form

 ⟨x2n⟩ = n∑h=1(1−π)+2n∑h=1h−1∑k=1πh−k(1−π)2 (26) = (1−π)n+2(1−π)2n∑h=1πhh−1∑k=1π−k

Since , for any real , it follows that

 n∑h=1πhh−1∑k=1π−k = n∑h=1πh1/π−1/πh1−1/π (27) = −π1−π(1πn∑h=1πh−n)

For any , converges to and consequently,

 ⟨x2n⟩ = (1−π)n+2(1−π)2π1−πn+O(1) (28) = (1−π)(1+2π)n+O(1)

Eq. (28) indicates that the effective self-diffusion coefficient for this random walk scheme in the mean-field approximation equals

 Dsd(π)=(1−π)(1+2π)2 (29)

The interesting feature of this result is that displays a non monotonic behavior as a function of : for small , increases above the value , while for , . This phenomenon is depicted in figures 2 and 3. Figure 2 shows the mean square displacement vs time obtained for stochastic simulations of eqs. (13) (17), using an ensemble of particles, while figure 3 compares the values of the self-diffusivity obtained from the simulations against the theoretical prediction (29).

Next, consider the same process in the framework of the theory of GPK processes, still assuming a mean-field approximation. While there are only two different spin states as regards the lattice model, there are four different velocity/spin states in its GPK counterpart: namely the two states in which particles possess spin states , and an effective velocity , and the two “ghost states” , at which the velocity is vanishing while the value of the spin state is . Let us label these four states with

Let be a uniform transition rate. The stochastic GPK model is thus given by

 dx(t)=bχ4(t;λ01,A)dt (30)

where the stochastic velocity vector corresponds to the second row of table 1,

 b=⎛⎜ ⎜ ⎜⎝b0−b000⎞⎟ ⎟ ⎟⎠ (31)

is a 4-state finite Poisson process characterized by a uniform transition rate and by the transition probability matrix given by

 A=⎛⎜ ⎜ ⎜ ⎜⎝π(1−π)(1−π)2π(1−π)(1−π)2(1−π)2π(1−π)(1−π)2π(1−π)π2π(1−π)π2π(1−π)π(1−π)π2π(1−π)π2⎞⎟ ⎟ ⎟ ⎟⎠ (32)

Let us clarify the structure of the transition probability matrix. Consider as initial state, the state “”. The transition from this state to state “”, corresponding to a moving particle with opposite velocity, can occur solely if the initial site does not contain any other particle, and this happens in the mean-field approximation with probability and the nearest neighbouring site can be reached without violating the exclusion principle, which occurs with probability . The probability is therefore equal to . The transition from state “” to state “”, corresponding to a rest particle with the same spin can occur solely if the initial site contains a particle with oppositive spin, and the nearest neighboring site is occupied by a particle with the same spin. Both these events occurs with probability , and are indpendent of each other, so that . The transition from state “” to state “” corresponding to a rest particle with opposite spin, can occurs solely if the initial state does not contain any other particle (occurring with probability ), and the nearest neighbouring site is occupied by a particle possessing the same spin (occurring with probability ), so that . As regards , its expression follows from the probabilistic closure condition . An analogous derivation can be applied to determine all the other entries of the matrix .

The statistical description of the process (30) involves four partial probability density functions , , , associated with the four states of , fulfilling the balance equations

 ∂pi(x,t)∂t=−bi∂pi(x,t)∂x−λ0pi(x,t)+λ04∑j=1Ai,jpj(x,t) (33)

and the overall probability density function is obviously . Let us define the two probabilistic -vectors , as

 pb=(p1p2),p0=(p3p4) (34)

is the vector of the partial probability density associated with moving states, i.e., with states corresponding to an effective particle motion, while groups together the partial probabilities pertaining to the rest states. With this notation, the balance equations for the partial probability waves can be compactly expressed as

 ∂pb∂t = −b0Lx[pb]−λ0pb+λ0A1pb+λ0A1p0 ∂p0∂t = −λ0p0+λ0A2pb+λ0A2p0 (35)

 Lx=(∂/∂x00−∂/∂x) (36)

and the two matrices , read

 (37)

The balance equation for the overall probability density follows from (33) by summing over the states (i.e., over the index ),

 ∂p(x,t)∂t=−∂Jp(x,t)∂x (38)

where .

Next, consider the Kac limit of this model, corresponding to keepind fixed the ratio to a fixed nominal diffusivity . Letting , the second equation (35) provides the ration between and ,

 (I−A2)p0=A2pb (39)

i.e., . Indicating with and , from eq. (39), and from the identity one obtain the relation between , and , namely

 p0(x,t)=π1−πpb(x,t),pb(x,t)=(1−π)p(x,t) (40)

that, substituted into the first equation (35), yields a balance equation involving solely the partial probability density associated with moving particles

 ∂pb∂t = −b0Lx[pb]−λ0[I−A1−A1(I−A2)−1A2]pb (41) = −b0Lx[pb]−λ0Λeff(π)pb

After some elementary algebra, the matrix takes the form

 Λeff(π)=(ℓ(π)−ℓ(π)−ℓ(π)ℓ(π)),ℓ(π)=11+2π (42)

With respect to the partial probability densities associated with moving states , the statistical description of the process reduces to a classical Poisson-Kac model

 ∂p1∂t = −b0∂p1∂x−λ0ℓ(π)[p1−p2] ∂p2∂t = b0∂p2∂x+λ0ℓ(π)[p1−p2]

the Kac limit of which provides the expression for probability flux entering the balance equation (38)

 Jd(x,t) = −b02λ01ℓ(π)∂pb(x,t)∂x=−D01−πℓ(π)∂p(x,t)∂x (43) = −D0(1−π)(1+2π)∂p(x,t)∂x

Setting the nominal diffusivity , one obtains from eq. (43) the expression for the mean-field self-diffusivity (29) derived from the original stochastic model. Several observations deserve some attention:

• in the derivation of eq. (43) we have first considered the limit for in the second equation (35) for the probability densities for the non-moving particles, and the result obtained is then substituted back in the first equation (35) for , deriving the self-diffusion from the Kac limit of this equation. We have use this, more physically oriented, approach to obtain in order to derive eq. (3.1) corresponding to the quasi steady-state approximation for the dynamics of the partial probability density functions associated with non-moving particles. If one perform simultaneously the Kac limit, (i.e., , keeping fixed the nominal diffusiviy ) one still obtains eq. (43).

• The analysis of the above problem involving interacting particles through an exclusion principle indicates that, once the microdynamics of the interacting particles has been specified (in the present case within the mean-field approximation), it is rather straightforward to define and derive the corresponding stochastic GPK model, in the present case eq. (30), specified by the number of GPK states, by the stochastic velocities of each state , by the transition rate vector and by the transition probability matrix .

• Observe that the number of states in the GPK model may be different, and in general greater than the number of spin states of the original system. In the present case, the number of different spin configurations is , while . This is because two additional states are required to discriminate between moving and non-moving particles in order to account for the exclusion principle;

• As discussed with the aid of the present case study, it is fairly easy to derive the structure of and and their dependence on the partial probability density functions (in the presence case on the concentration , since the simpler case of a mean-field approximation is considered) from the rules of particle interaction. The analysis developed in this Section is limited to the mean-field case. The general problem is treated in the next Section.

• Given the stochastic GPK model (in the present case eq. (30)), the hydrodynamic limit of this model follows directly from GPK theory, in the present case eq. (33). Out of it, the Kac-limit of the latter, provides the classical parabolic transport model. Therefore, and this represents a very powerful by-product of GPK theory, there are several classes of hydrodynamic limits of the same interacting particle systems, depending, once is fixed, on the characteristic time scales of the stochastic process, i.e., essentially on the value of . In some cases, due to the presence of particle interactions, while the hyperbolic hydrodynamic limit exists, the Kac limit of the corresponding model could not exist.

### 3.2 Simple exclusion random walk

Let us consider another classical exclusion random walk without spin. In this model, particles on a lattice move towards the nearest neighbouring site (with equal probability towards the left or right neighboring site) solely if no other particle is simultaneosly occupying it.

In the mean-field approximation, indicating with particle concetration, the random walk model takes the form

 xn+1=xn+rn+1ηn+1 (44)

where the random variables , are specified by

 (45)

where and are uncorrelated with each other, , and satisfy eq. (19). Consequently, starting from ,

 xn=n∑h=1rhηh (46)

, and while for the mean square displacement

 ⟨x2n⟩=n∑h=1n∑k=1⟨rhrk⟩⟨ηhηk⟩=n∑h=1⟨η2h⟩=(1−π)n (47)

Thus, for the self diffusion coefficient of a tagged particle one obtains, in the mean-field approximation,

 Dsd(π)=1−π2 (48)

Next, consider the GPK modeling. The GPK version of the process involves three states: state “1” corresponding to particles moving forward along the -axis, state “2” to particles moving backward, and state “3” corresponding to resting particles, that do not perform any motion due to the exclusion principle. Indicating with and the characteristic velocity and transition rate of the GPK process, it follows that

 (49)

and the stochastic GPK version of the model is formally analogous to eq. (30), namely

 dx(t)=bχ3(t;λ01,A)dt (50)

where the transition probability matrix depends on the mean-field concentration and is given by

 A=⎛⎜ ⎜ ⎜⎝1−π21−π21−π21−π21−π21−π2πππ⎞⎟ ⎟ ⎟⎠ (51)

The statistical description of eq. (50) involves three partial probability densities , and , where the latter corresponds to the density of resting particles. The overall probability density is , and indicate with the probability density function of the moving particles. The balance equations for the partial densities, accunting for eqs. (49) and (51) are

 ∂p1(x,t)∂t = −b0∂p1(x,t)∂x−λ0p1(x,t)+λ0a(π)p(x,t) ∂p2(x,t)∂t = b0∂p2(x,t)∂x−λ0p2(x,t)+λ0a(π)p(x,t) (52) ∂p3(x,t)∂t = −λ0p3(x,t)+λ0πp(x,t)

where , from which one obtains that the conservation equation for the overall density is still expressed by eq. (38) with . As regards the probability flux , from the first two equations (52) one obtains

 ∂Jd(x,t)∂t=−b20∂pb(x,t)∂x−λ0Jd(x,t) (53)

In the Kac limit, eq. (53) provides

 Jd(x,t)=−2D0∂pb(x,t)∂x (54)

where, as usual, corresponds to the nominal diffusivity of the GPK scheme. In the Kac limit, from the third equation (52) one obtains , thus

 pb(x,t)=(1−π)p(x,t) (55)

which inserted into eq. (54) provides

 Jd(x,t)=−2D0(1−π)∂p(x,t)∂x (56)

which implies for the self-diffusion that coincides with eq. (48), by setting the nominal diffusivity equal to .

### 3.3 TASEP model

To conclude, let us consider another simple and paradigmatic example, namely the Totally Asymmetric Simple Exclusion Process (TASEP) on the real line. In this model, particles move solely in the forward direction satisfying an exclusion principle, corresponding to one particle per site, at most. The mean field dynamics of TASEP, letting be the mean-field particle concentration, is described by the dynamics

 xn+1=xn+ξn+1 (57)

where the random variables are uncorrelated with each other and described statistically by

 ξn+1={0Probπ1Prob1−π (58)

Consequently,

 ⟨ξhξk⟩={1−πh=k(1−π)2h≠k (59)

If , the integral representation of the dynamics is , thus

 ⟨xn⟩=n∑h=1⟨ξh⟩=(1−π)n (60)

and

 ⟨x2n⟩=n∑h=1n∑k=1⟨ξhξk⟩=(1−π)2n2+π(1−π)n (61)

The mean square displacement attains the expression , thus the mean-field self-diffusion coefficient is given by

 Dsd(π)=π(1−π)2 (62)

It vanished both for (infinite dilution) and for corresponding to total exclusion. In both cases the dynamics is strictly (and trivially) deterministic.

Let us analyze the GPK formulation of TASEP. It implies the occurrence of two state: state “1” which is the mobile state, and state “2” which is the stationary (non-moving) state. Correspondingly, , and . The transition rates are uniform and equal to . As regards the transition probability matrix, TASEP dynamics indicates the following dependence on the mean field concentration

 A=(1−π1−πππ) (63)

The process is described stastically by the two partial probability density functions satisfying the hyperbolic equations

 ∂p1(x,t)∂t = −b0∂p1(x,t)∂x−λ0πp1(x,t)+λ0(1−π)p2(x,t) ∂p0(x,t)∂t = λ0πp1(x,t)−λ0(1−π)p2(x,t) (64)

In the limit , , thus the second equation (64) provides

 p2(x,t)=π1−πp1(x,t),p1(x,t)=(1−π)p(x,t) (65)

where is the overall concentration, the dynamics of which is given by

 ∂p(x,t)∂t=−b0∂p1(x,t)∂x (66)

Inserting in it the expression (65), obtained for , one finally arrive to the hyperbolic model for

 ∂p(x,t)∂t=−b0(1−π)∂p(x,t)∂x (67)

providing an effective mean velocity equal to

 veff=b0(1−π) (68)

Observe that eqs. (67)-(68) does not correspond to any Kac limit, but solely to the limit of infinitely fast recombination kinetics ().

In order to extract from the GPK process defined statistically by eq. (64) the value for the effective diffusivity and to perform a Kac limit of the process, let us consider TASEP dynamics in the inertial frame moving with the effective velocity . Let be the position coordinate in this moving reference system

 x′=x−b0(1−π)t (69)

In the moving system, let , the two partial probability densities characterized by the velocities

 b′1=b0−b0(1−π)=b0π,b′2=−b0(1−π) (70)

which satisfty the balance equations

 ∂p′1(x′,t)∂t = =−b0π∂p′1(x′,t)∂x′−λ0[πp′1(x′,t)−(1−π)p′2(x′,t)] ∂p′2(x′,t)∂t =