How to initialize a second class particle?
We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile which, in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with non-concave, non-convex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of P. A. Ferrari and C. Kipnis’ arguments (“Second class particles in the rarefaction fan”, Ann. Inst. H. Poincaré, 31, 1995) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle-antiparticle pair.
Keywords. second class particle, limit distribution, rarefaction fan, shock, hydrodynamic limit, collision probability.
Acknowledgement. The authors thank valuable discussions with Pablo A. Ferrari on the problem, with Ellen Saada on the hydrodynamic limit of asymmetric processes and with Cédric Bernardin on symmetric processes. We also thank the anonymous referee for his/her comments. M. Balázs acknowledges support from the Hungarian National Research, Development and Innovation Office, NKFIH grants K100473 and K109684.
This paper studies the behavior of second class particles in a wide class of one-dimensional attractive particle systems. The evolution of such particles can be obtained by coupling two systems (of first class particles) coordinate-wise in such a manner that their initial configurations only differ at finitely many places. Second class particles interact with the underlying process and perform highly nontrivial motion which is only partially understood in general. In asymmetric models they are known, in first order, to follow the characteristic lines of the limiting hydrodynamic equation of the density. In three classical cases: translation-invariant stationary, rarefaction fan, and shock scenario this results in a law of large numbers with the characteristic velocity, a random admissible characteristic velocity of the rarefaction fan, and the speed of the shock, respectively. These make the second class particle a relevant microscopic object that captures macroscopic properties of the ambient system. Fluctuations show superdiffusive scaling for translation-invariant stationary, rarefaction fan, deterministic shock initial data and diffusive scaling for random shock initial data. Many of the previous properties have been proven rigorously for the most-studied totally asymmetric simple exclusion process (TASEP) and in some cases for other processes as well. However, they are conjectured to hold in a wide range of particle systems. Second class particles in symmetric systems have not been much explored, in some simple cases diffusive behavior is known.
We build on the seminal paper  by P. A. Ferrari and C. Kipnis which made use of a translation argument to investigate the second class particle of the TASEP starting from the rarefaction fan. Their argument compares a step initial product Bernoulli distribution with its translated version and notices that the joint realization of these two can be understood as a coupled initial distribution with possibly a second class particle at the origin. This program crucially relied on the fact that the second class particle of simple exclusion is a uniquely defined object as it can only conceive by coupling a process of zero particles with one of one particle at the site of the second class particle. When dealing with systems of more choices for one site occupation numbers, the second class particle stops being a uniquely determined object. Stochastic domination of the natural measures associated with attractive models still holds but the actual realization of a coupled pair has some details to fix besides its marginals. In particular, it is not clear whether two models with slightly different densities can be coupled using zero or one second class particles per site only, or more than one of them on a site have to be assumed with positive probability. Actually, this latter is the case for popular stationary distributions as the ones of Geometric or Poisson marginals (e.g., for zero-range processes).
First, we build up a natural initial distribution () for the second class particle in step initial configurations (with different densities on the left and right) which allows for an extension of P. A. Ferrari and C. Kipnis’ arguments. Our construction works even when coupling with zero or one second class particles only fails. This is where the main novelty of the paper lies: to force zero or one second class particles with the correct one-site marginals for the coupled pair, one has to introduce negative weights in the coupling measure. As it turns out negative weights only belong to configurations without a second class particle, and this non-physical coupling measure always assigns positive weights to states with a second class particle. By normalizing on these states only, the proper probability distribution a.s. has then the second class particle, which will also turn out to be canonical in many sense.
Under the initial distribution we connect the displacement of the second class particle to easier quantities of the ambient system. Using recent results of hydrodynamics we can then proceed to prove limit distribution results on the rescaled position of the second class particle. Both asymmetric and symmetric systems are handled under the natural scaling that fits the respective scenario. The limit distributions then relate to the solution of the hydrodynamic equation with step initial condition. There are two particular and interesting instances, to the best of our knowledge not much explored in the literature, of second class particle-behavior:
in asymmetric models with non-concave and non-convex hydrodynamic flux, shocks and rarefaction fans can coexist and the limit distribution of the second class particle reflects this fact by developing both continuous and discrete components at the same time; and
central limit theorem for the second class particle is pointed out in a symmetric system where, as opposed to simple symmetric exclusion, it is not a simple random walker.
As a by-product of our arguments we are able to relate the one-site marginal of the first class particles at the site of the second class particle to the distribution of a model without the second class particle. Under certain initial distributions this results in a time-stationary one-site marginal – a quite unexpected result. Finally, we push the arguments, in line with , to give a lower estimate on the survival probability of a second class particle-antiparticle pair in general models.
Earlier results. A review and several open problems appeared in [16, 23] many of which are completely solved by the present paper. A law of large numbers for the position of the second class particle of exclusion and zero range processes with shock initial condition was obtained by F. Rezakhanlou . Note that his initial setup of the second class particle slightly differs from ours. As described above, in case of the rarefaction fan (and for the TASEP) the first and fundamental paper was . T. Mountford and H. Guiol  then sharpened  by proving that the convergence takes place almost surely. Recently P. Gonçalves has translated the results of  for the totally asymmetric constant rate zero-range process in  via a direct coupling between exclusion and zero-range. P. A. Ferrari, P. Gonçalves and J. B. Martin  have very elegant arguments on collision probabilities in exclusion processes. Many results on the behavior of the second class particle in the TASEP have been reproven by P. A. Ferrari and L. P. R. Pimentel  and by P. A. Ferrari, J. B. Martin and L. P. R. Pimentel , translating the problem into one of competition interfaces in last passage percolation. E. Cator and S. Dobrynin  have studied Hammersley’s process in continuous space in which limit theorems were obtained for the second class particle starting from the rarefaction fan. D. Romik and P. Śniady  pointed out an elegant algebraic connection between the motion of second class particles in a variant of the TASEP and an evolution, so-called “jeu de taquin”, defined on infinite Young tableaux through which the distributional limit was proved. TASEP equipped with higher order particles (like third, fourth, etc. class particles), known as the multi-type TASEP, was investigated in  by G. Amir, O. Angel and B. Valkó, where the joint distribution of the speeds of higher order particles were also identified. This in particular includes collision probabilities and the formation of convoys. Analytic formulæ were obtained by C. A. Tracy and H. Widom  for the second class particle starting from the rarefaction fan of the ASEP.
Organization of the paper. We start with discussing initial distributions in Subsection 2.1 which form a crucially important part of our arguments. We then proceed with describing the dynamics in Section 2.2 with additional requirements in Section 5. The second class particle, our main object, is introduced in Section 2.3. We early on state the main results of this paper in Section 3 for which the precise hydrodynamic statements we need are postponed to Section 6 due to organizational purposes. Remarks on the initial distribution (Section 4.1), the fundamental identity behind our results (Section 4.2), and a theorem on the background distribution of the site of the second class particle (Section 4.3) are also slightly postponed. We outline and discuss several examples of models in Section 7. Proofs follow in Section 8.
2.1 State space and initial distribution
The model class we investigate originates in the work of Cocozza-Thivent , extensions and several examples we cover first appeared in the papers [6, 36]. We consider general, nearest neighbor stochastic interacting particle systems on the configuration space with such that . In particular can as well be an infinite subset of . The quantity denotes the number of (signed) first class particles sitting at the lattice point at time . We adopt this interpretation even if happens to be negative.
Our main object of investigation is the second class particle which comes up from couplings of systems of first class particles. In particular it lives in the space , so before describing the dynamics of the above systems the appropriate choice of the initial measure on will be discussed. This measure to be defined later turns out to be canonical and is indeed one of the crucial points of this paper.
We start with a general assumption on one-site marginals which will be the basis of building product initial distributions of configurations in and of coupled pairs of configurations in .
Let be a family of probability measures on , where is a bounded subset of that satisfies the following properties:
it is parametrized by its mean, that is holds for every ; and
for each , where , the measure stochastically dominates , that is holds for every .
This assumption is very mild, for e.g. any deterministic marginals of the form satisfy it with a . We will present a more general set of measures in Section 5 that also satisfies the above assumption.
In the sequel, we will refer to as the set of densities. With , we define the product distribution
on . Whenever this will be called the microscopic Riemannian density profile or simply the step initial condition.
Next, we turn to special distributions on . Fix two densities of and we define the measure on as
where . It is an easy exercise to check that this indeed defines a probability distribution. We will comment on its origin later in Section 4.1. Notice that holds -a.s. for its two marginals. By a slight abuse of notation we also set
as diagonal measures on . We can now define the initial probability distribution as a site-wise product coupling measure on the space :
Later, we will start a coupled pair of systems of first class particles under the initial distribution , and we denote the associated probability and expectation by and , respectively. Though the precise notion of the second class particle will be defined in Subsection 2.3, here we notice in advance that a.s. has a second class particle that initially starts from the origin.
2.2 Dynamics of the models
A continuous time Markov jump dynamics is attached on top of the configuration space that allows the particles to execute right as well as left jumps with respective instantaneous rates and . Formally, with the Kronecker symbol ( stands for the indicator function throughout the article), the transitions are of the form
where are given deterministic functions. Conditioned on a given configuration, the above steps take place independently for each with the above respective rates. Throughout the article we assume non-degeneracy for the rates, that is for every : () if and only if and ( and ). This also makes sure that the process a.s. keeps the state space . Sometimes we will let one of the left or right jump rates be zero (totally asymmetric case).
Now, the (formal) infinitesimal generator of our Markov process acts on a cylinder function (one that depends only on a finite number of coordinates of ) as
If and are bounded functions on then the above Markov process can be constructed on in an appropriate manner having generator (see [28, Chapter 1]). In other cases, existence of the dynamics can only be established by posing further (growth) conditions on the rates (see ,  and further references therein). Within the scope of this article we do not intend to deal with this issue in general, though we will discuss some models with unbounded rates in Section 7. From now on we assume that the processes can be constructed with appropriate initial data in with the above dynamics. In the next subsection we introduce the attractiveness assumption which will further tighten the model class.
2.3 Second class particles
Pick two configurations and both in aligning them coordinate-wise. Then one can define the number and the sign of signed second class particles at position in the configuration pair . In particular, if
then we say that a single positive (negative) second class particle is placed at the origin in . To allow second class particles evolve in time we use the basic, “particle-to-particle”, coupling, that is for each time and lattice point , a hop to the right can occur in both systems:
with rate ; while “compensating” right jumps occur according to the following rules with respective rates:
Here and denote the positive and negative part function, respectively. The coupling tables for the left jumps can be obtained analogously. Note that a second class particle can hop only if a compensating step occurs. Also notice that under the basic coupling the marginal processes, that is and , follow the same stochastic evolution rules (2.5). Now, recall the following notion from [28, Definition 2.3, pp. 72].
Definition 1 (Attractiveness).
We say that the dynamics defined by the infinitesimal generator of (2.6) is attractive, if the initial dominance implies the one for all times under the basic coupling.
From now on we will always assume that the processes we consider are attractive. It is not hard to see that this is equivalent to saying that the rate () is monotone non-decreasing (non-increasing) in its first and monotone non-increasing (non-decreasing) in its second variable.
In attractive processes, the above basic coupling tables reveal some extra properties for the second class particles. In particular, having initial configurations as in (2.7), a.s. there will always be a single second class particle in the system, the position of which will be denoted by at time . More generally, one can see that the total number of (positive and negative) second class particles is non-increasing in time.
3 Main results
The results below heavily rely on the hydrodynamic description of particle systems. For the time being we skip the rather technical details of hydrodynamic limit theory. We refer to Subsections 6.1 and 6.2 which are devoted to the precise definitions and statements on hydrodynamics where all the missing elements are fully expounded. Our first two results concern the limit distribution of the position of the second class particle.
Theorem 1 (Speed of the second class particle in asymmetric models).
any attractive process with bounded one site occupation numbers () and we have no further assumptions on the measure ; or
a misanthrope process with bounded rate functions (but not necessarily bounded occupations). In this case is restricted to be a stationary marginal.
Then we have the limit
for every that is a continuity point of , where is the unique entropy solution of the conservation law with step initial datum and hydrodynamic flux function .
Next a couple of comments. First, we underline that for systems with bounded occupation numbers the limit (3.1) holds for any choice of marginal distribution satisfying Assumption 1. On the other hand, we note that the misanthrope family of processes forms a large and important part of attractive particle systems. The rate functions of these satisfy further combinatorial identities which enable one to give a full description of the translation invariant stationary distributions. The corresponding results will be recapitulated in Section 5 (see Theorem 6).
The hydrodynamic flux (which will be defined later in (6.2) in Section 6) roughly speaking describes the average signed rate of jumping particles across a bond in stationarity. In some models strict concavity or convexity of has been established and it is then well understood that the Riemann (or step) initial condition () develops shock or rarefaction fan solutions depending on the order of and and on concavity or convexity of . In a shock, the limiting probability (3.1) is of - form which means convergence of the scaled second class particle position to the deterministic velocity of the shock. In a rarefaction fan we have convergence to a random velocity. This randomness is uniform for the totally asymmetric simple exclusion process, as is a linear function of its first – spatial – argument which has been first observed by P. A. Ferrari and C. Kipnis  but this distribution might vary with other models. We highlight that there are attractive models with product-form stationary distributions but with non-concave, non-convex hydrodynamic fluxes. In the associated conservation laws coexistence of shocks and rarefaction fans is possible, in which cases our result shows that the limit distribution of the velocity of the second class particle is mixed with a discrete mass and a continuous counterpart (see for e.g. the -type model of Section 7).
Our arguments are general enough to include symmetric models as well which have interesting consequences for the second class particle in this case. The analogous result follows, and we will illustrate its significance with the symmetric zero range processes at the end of Section 7. See the definition of a gradient process and the related quantity in Section 6.2.
Theorem 2 (Speed of the second class particle in symmetric models).
Suppose , Assumption 1 and let be a symmetric gradient process which is attractive. Then we have the limit
for every continuity point of provided that holds for every , where is the unique weak solution to the parabolic partial differential equation with step initial datum and diffusivity coefficient .
Symmetric gradient processes and their hydrodynamic properties (diffusivity) will be rigorously discussed in Subsection 6.2.
Finally, we focus on the interaction of two second class particles of opposite charges dropped into the system initially. Denote by the total number of second class particles present in the system at time . For the long-time behavior of we have the following result.
Theorem 3 (Collision probability of second class particles).
Assume that and are finite numbers. Let be any pair of attractive systems starting from the deterministic initial configurations
and evolving according to the basic coupling. Then
where , while denotes the associated probability of .
In particular, if the dynamics is totally asymmetric () then holds. On the other hand, considering one of the misanthrope processes (described by Theorem 6) we have provided that is a continuity point of , where is the hydrodynamic flux (defined in (6.2)) and is the unique entropy solution to with step initial datum ().
The previous assertion tells that two second class particles of distinct charges initially placed at lattice points and will never meet with positive probability provided that the constant of (3.3) is positive. For the asymmetric simple exclusion process with and rate functions and , where , we recover the result [18, Theorem 2], if . Indeed, we know exactly from [17, Theorem 2.3] that for each
4 Additional results
In this section we state additional results, following from very general coupling arguments, that give further insight to phenomena under the initial distribution (2.4). We first indicate where this initial distribution comes from. Then an intermediate step towards main Theorems 1 and 2, without any reference to hydrodynamics, is shown. Finally, we proceed with an invariance property of the model at the site of the second class particle.
4.1 The distribution
The following will demonstrate why the measure (2.4) serves as a natural choice for initial distribution.
Suppose that Assumption 1 holds and let , where are fixed. Then there exists a joint probability measure with and as respective marginals and with if and only if
holds for every . In this case can be obtained as
Under the narrower assumptions of Proposition 1, we can set up another measure, namely
which we can call the unconditional version of , since this latter can be obtained from by conditioning on the existence of a single second class particle at the origin.
Some, but not all, interacting particle systems have translation-invariant product stationary distributions. For those with product measures, it seems natural to choose the marginals and to be these stationary marginals. As two classical examples, the product of Geometric and Poisson distributions on are stationary for zero-range processes with constant and linear rate functions, respectively, to be discussed in Section 7 in more details. Notice, as the following Proposition 2 also demonstrates, that the additional requirement (4.1) of Proposition 1 might be too restrictive in some cases where , hence , might not exist as a probability measure.
4.2 The distribution of the second class particle
We spell out our fundamental result which will combine with hydrodynamics (to be explicated in Section 6) to give the main Theorems 1 and 2. It connects the law of the displacement of a single second class particle with that of a (first class) particle occupation variable. Fix , and recall the initial distributions (2.1) for a single model and (2.4) for a pair with the second class particle.
Theorem 4 (Displacement distribution of the second class particle).
Suppose that a family of measures fulfills Assumption 1. Then for any and we have
Note that does not have to be related to the stationary distributions of or of in any way. Also observe that Theorem 4 holds regardless of whether the family of measures satisfy the property detailed in Proposition 1 above.
Furthermore, notice that we had no further assumptions on the rates and , hence both asymmetric and symmetric processes are included in the above assertion. Indeed, a careful overview of our technique (see the proof of Theorem 4) reveals that (4.3) also holds for those models with long range jumps or with (non-)finite range dependent rates.
A rather classical result immediately follows from Theorem 4, namely the quantity has uniform lower and upper bounds and , respectively, in the space . Also observe that for each fixed , the function is monotone non-increasing in .
4.3 The site of the second class particle
Simple exclusion is special in many ways. One of its simplifying feature is due to : there is no choice for the configuration at the site of the second class particle. We deterministically have and for all . There are more options when , and the next theorem gives an interesting result on the site of the second class particle in such models. We take any function for which either condition , or holds. Then we define and further assume .
Theorem 5 (Background as seen from the position of the second class particle).
Suppose that a family of measures fulfills Assumption 1. Then we have the identity:
In plain words, this theorem tells that the law of for a , i.e. the particle occupation number at the position of the second class particle, can fully be captured by that of of starting from and then . In particular, if and are stationary distributions for the dynamics (2.6) then the background marginal one-site process , as seen from the position of the lone second class particle, is stationary. This can be thought of as another fact proving the intrinsicality of the marginal . Notice though that Theorem 5 does not say anything about the distribution of any site other than that of the second class particle, those are in general not stationary. A few very special cases of joint stationary distributions seen by the second class particle are described in [15, 5, 8] and references therein.
5 The misanthrope family
In this section we briefly discuss a special class of attractive particle systems called the misanthrope family where our main results naturally apply. We again underline that there is a much larger class of processes (and initial measures) that we also cover.
First, define the Gibbs measures as
where is a generic real parameter, which is often referred to as the chemical potential; is any function with appropriate asymptotic growth; finally, the statistical- or partition sum is .
Assume that forms a bunch of probability measures with finite variance, where is some open set of the reals. Then satisfies Assumption 1. In particular, there is a bijection between the parameters and the densities ; and for , or equivalently for , the measure stochastically dominates .
Due to the bijection claimed in the previous assertion we will change freely between the representations of the measure (5.1) either by the chemical potential or by the density .
We emphasize that is not necessarily a stationary marginal of the dynamics (2.6) in general. Following ideas of Cocozza-Thivent , for attractive systems, where is indeed stationary, a nice characterization theorem was established by M. Balázs et al., which we recall in the following.
Theorem 6 (M. Balázs et al.).
where is such that whenever , and is monotone non-decreasing on . (The empty sum is as usual defined to be zero.) Suppose furthermore that:
there are symmetric functions such that
where holds whenever or is finite, otherwise they are non-zero except when or is set to be zero (totally asymmetric case);
for any and :
Then the density parametrized product measure is extremal among the translation-invariant stationary distributions of the process with rates and infinitesimal generator of (2.6).
The stationarity part of Theorem 6 has been carried out thoroughly in , in which all the extremal translation-invariant stationary distributions were covered by examining the convergence region of the partition sum . For the ergodicity we will briefly comment on how Lemmas 7.2 and 7.3 of  established for the bricklayers’ process can be modified to be handy for any process. First, it is not hard to see that Lemma 7.2 can be extended to the cases when (in any order) a positive and a negative second class particle start from next to each other. This results in that the probability of them colliding before any given time is (strictly) positive. Here the only required property of the underlying process is the continuity of its semigroup. Then in Lemma 7.3 ergodicity is carried out by showing that any invariant function w.r.t. is constant. Now, by using (the extended version of) Lemma 7.2 it can be easily pointed out that adding (or ) to adjacent occupation numbers, whenever this change keeps the state space, does not modify the value of an invariant . It follows that interchanging any two adjacent sites does not change the value of under . The argument is then completed by the application of the Hewitt–Savage - law.
for . Now, fixing and taking the limit as we obtain
where it is easy to see that for distributed as . (The empty sum is defined to be zero.) Observe that this probability measure is just the marginal at the origin of the initial distribution that was used in [12, Theorem 2.2] to start a single second class particle from that position. Thus our treatment is in correspondence with results from . As a side remark we mention without details that via a second order Taylor expansion as one can formally recover the covariance formula in [12, Theorem 2.2] directly from (4.3). Bounding the error terms that arise is straightforward when , making this argument rigorous.
This section is devoted to briefly recall the main notions and results from hydrodynamics of asymmetric and symmetric particle systems as well. Some of the results below use the misanthrope class (see the previous Section 5) while others are more general.
6.1 Hydrodynamics of asymmetric models
The idea behind the hydrodynamic limit for asymmetric systems is that, in hyperbolic scaling (i.e. same scale for space and time), the rescaled microscopic average density of interacting particles behaves as a deterministic density field obeying the conservation law
where is the (macroscopic) density with initial condition . The function is called the hydrodynamic flux and is
Here denotes the expectation w.r.t. the extremal stationary distribution for a density .
The rescaled empirical measure of a sequence of random configurations is defined as
A deterministic bounded Borel measurable function on is the density profile of , if converges to as for all , in probability as a random object, and in the topology of vague convergence as a measure, meaning that
is required to hold for each and with all continuous test function of compact support.
Definition 2 (Hydrodynamic limit).
We note that the hydrodynamic limit just defined is also referred to as the weak conservation of local equilibrium (cf. [25, Chapter 4]). Finally, we introduce one more notation: for a fixed denote by the shift operator which acts on a configuration as and on a measure as , respectively.
In the following, we will make the choice of (2.1) as a common initial distribution for the sequence of processes to be rescaled in the hydrodynamic limit. It follows that the limiting process has the Riemannian (step) initial density profile
Under Assumption 1 and mild assumptions on the flux function there exists a unique entropy solution to the problem (6.1) with (6.3) as initial condition. It is also known that for each , this weak solution is continuous apart from a finite set of jump discontinuities (shocks), where we define to be left-continuous. For concepts and results in hyperbolic conservation laws, which were omitted here, we refer to  and further references therein (see also ).
In what follows some exact results on hydrodynamics will be collected concerning the above setting. The first general result is from , valid in the misanthrope framework of Section 5 (see Theorem 6).
Theorem 7 (F. Rezakhanlou).
Take any process from the misanthrope family equipped with bounded rates. Set the initial measure to be of stationary marginals. Then exhibits a hydrodynamic limit , where is the unique entropy solution to (6.1) with hydrodynamic flux (6.2) and with initial datum (6.3). In addition, the limit
also holds for every continuous of compact support and any cylinder function .
In the above result we are much restricted for the marginals of the initial measure to be chosen properly. However, this was far more generalized by C. Bahadoran et. al. in  for systems of bounded particle numbers per site. In particular, this result does not require the special algebraic structure of Section 5.
Theorem 8 (C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada).
Our ultimate goal would be to conclude that the rescaled quantity also converges, where denotes the integer part function. This, however, does not appear to be an immediate consequence of the above theorems. But C. Landim  has elaborated a set of assumptions under which this consequence eventually holds (note also [25, Proposition 0.6, Chapter 6]). We are going to recapitulate this result below to be formulated in our special context with sharper conclusions, outlining its proof in Section 8.
Suppose that the process with infinitesimal generator (2.6) exhibits a density parametrized, stochastically ordered and continuous family of translation-invariant stationary distributions, where is such that . Fix a cylinder function , being either bounded or monotone non-decreasing, such that . Assume furthermore that the convergence
takes place for every and continuous of compact support with some uniformly bounded family of functions for which is monotone non-increasing for each fixed and for every continuity point of : . Then we have
for every continuity point of .
6.2 Hydrodynamics of symmetric models
In our context, being symmetric means for each . Note that attractiveness is still up, that is is required to be monotone non-increasing (non-decreasing) in its first (second) variable. We say that a symmetric attractive process is gradient if there exists a cylinder function for which
holds for every and (recall the shift operator after Definition 2). Usually it is more convenient and simpler to deal with attractive gradient systems. For such systems the key quantity turns out to be the diffusivity coefficient which is defined to be , where is a stationary distribution of the process with density . Note the difference between and the hydrodynamic flux (6.2) being its hyperbolic counterpart.
The concepts of hydrodynamics of the previous subsection can be exactly repeated here, except that this time the relevant scaling is diffusive instead of hyperbolic. Hence the macroscopic behavior of the density field is described by a parabolic partial differential equation of the form
where is the step function defined in (6.3). In general it is not so obvious for (6.8) to have a unique bounded (classical or weak) solution due to the discontinuity of and the smoothness of . We skip investigating this issue by assuming that there always exists a unique weak solution to (6.8). The hydrodynamic limit of gradient systems is well known (see , [35, Chapter 8] and many references therein, particularly  and ) and methods partially extend to non-gradient systems  as well.
7 Particular examples
We have selected some particular models in order to demonstrate the versatility of our results. Our general framework contains several well studied examples like the (totally) asymmetric simple exclusion process or the class of zero-range processes. We first list asymmetric and then symmetric processes with additional descriptions. Once for all we fix two reals for which and hold.
Generalized exclusion processes
Many systems with bounded occupations lie in this class but we only illustrate two of them.
The first one is the -type model of , which is a totally asymmetric process with and rates
and . The dynamics consists of the following simple rules: two adjacent holes can produce an antiparticle-particle pair (creation), (anti)particles can hop to the (negative) positive direction (exclusion), and when a particle meets an antiparticle they can annihilate each other (annihilation). The process is attractive if and only if and lies in the range of Theorem 6.
The hydrodynamic behavior, but not the second class particles, of the model has been thoroughly investigated by M. Balázs, A. L. Nagy, B. Tóth and I. Tóth . In that article the hydrodynamic flux was explicitly calculated, which turned out to be neither concave nor convex in some region of the parameter space. Hence the entropy solution of the hydrodynamic equation can produce various mixtures of rarefaction fans and shock waves. By (3.1) it implies that the limit distribution of the second class particle can have both continuous and discrete parts which will be demonstrated in the following. Using the results of  we can basically evaluate (3.1) of Theorem 1 in each case but we highlight only two of them below. For sake of simplicity we let and .
In the region : the hydrodynamic flux is concave . In particular for , Figure 1 demonstrates how the one parameter family of limit distributions of the second class particle evolves in time as . We notice that for all the cumulative distribution function is continuous but has a vertical “slope” at the origin. Thus its density is unbounded around zero.
In the region : the hydrodynamic flux is neither concave nor convex . As a particular example, for the model can develop a (non-linear) rarefaction fan – shock – rarefaction fan profile in the hydrodynamic limit. The second class particle then may stick into the shock with probability or it follows a continuously chosen characteristics in one of the regions of the rarefaction fan. Figure 2 demonstrates this behavior as .
Relying again on  we finally note that one can explicitly calculate the estimate of Theorem 3: the collision of two second class particles starting from the rarefaction fan has at least probability in this model.
Another example we highlight is the -exclusion process, where is any positive integer. Set and let the rates be
In particular for we obtain the asymmetric simple exclusion process with the family of Bernoulli product measures as extremal translation-invariant stationary distributions which work well for Assumption 1 and thus recover the result of . For much less is known (the assumptions of Theorem 6 cease to hold). In particular, it is not known whether its density parametrized translation-invariant extremal stationary distributions span the range . They are proved to exist for some closed parameter set (see [3, Corollary 2.1]). The structure of these measures is also unknown. The model, however, exhibits a hydrodynamic limit resulting in a conservation law with a concave flux (see Theorem 8 and also ). For only some qualitative properties have been established (see ). Nevertheless, one can still apply Theorem 1 with any product initial distribution that satisfies Assumption 1.
Zero range processes
Let and . The jump rates are defined as and , where is a monotone non-decreasing function with at most linear growth and with . This family satisfies all the assumptions of Theorem 6, hence hydrodynamics follows via Theorem 7. For sake of brevity we only spell out two totally asymmetric examples (). The hydrodynamic flux is then given by . The two most well-known special cases we consider are the ones of constant and linear rates: and , respectively. In the former case the extremal translation-invariant stationary distributions are of product form with geometric site-marginals while in the latter the Poisson distribution takes over this role. is and , respectively.