Hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices
We investigate the hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices. We construct a suitable scaling limit by using a discrete harmonic map. As we shall observe, the quasi-linear parabolic equation in the limit is defined on a flat torus and depends on both the local structure of the crystal lattice and the discrete harmonic map. We formulate the local ergodic theorem on the crystal lattice by introducing the notion of local function bundle, which is a family of local functions on the configuration space. The ideas and methods are taken from the discrete geometric analysis to these problems. Results we obtain are extensions of ones by Kipnis, Olla and Varadhan to crystal lattices.
The purpose of this paper is to discuss the hydrodynamic limit for interacting particle systems in the crystal lattice. Problems of the hydrodynamic limit have been studied intensively in the case where the underlying space is the Euclidean lattice. We extend problems to the case where the underlying space has geometric structures: the crystal lattice. The crystal lattice is a generalization of classical lattice, the square lattice, the triangular lattice, the hexagonal lattice, the Kagomé lattice (Figure1) and the diamond lattice. Before explaining difficulties for this extension and entering into details, we motivate to study these problems.
There are many problems on the scaling limit of interacting particle systems, which have their origins in the statistical mechanics and the hydrodynamics. (See ,  and references therein.) The hydrodynamic limit for the exclusion process is one of the most studied models in this context. Here we give only one example for exclusion processes in the integer lattice, which is a prototype of our results, due to Kipnis, Olla and Varadhan (). From the view point of physics and mathematics, it is natural to ask for the scaling limit of interacting particle systems evolving in more general spaces and to discuss the relationship between macroscopic behaviors of particles and geometric structures of the underlying spaces. In this paper, we deal with the crystal lattice, which is the simplest extension of the Euclidean lattice . Although the crystal lattice has periodic global structures, it has inhomogeneous local structures.
On the other hand, crystal lattices have been studied in view of discrete geometric analysis by Kotani and Sunada (, , , and the expository article ). They formulate a crystal lattice as an abelian covering graph, and then they study random walks on crystal lattices and discuss the relationship between asymptotic behaviors of random walks and geometric structures of crystal lattices. In , they introduce the standard realization, which is a discrete harmonic map from a crystal lattice into a Euclidean space to characterize an equilibrium configuration of crystals. In , they discuss the relationship between the Albanese metric which is introduced into the Euclidean space, associated with the standard realization and the central limit theorem for random walks on the crystal lattice. Considering exclusion processes on the crystal lattice, one is interested to ask what geometric structures appear in the case where the interactions depend on the local structures.
Given a graph, the exclusion process on it describes the following dynamics: Particles attempt to jump to nearest neighbor sites, however, they are forbidden to jump to sites which other particles have already occupied. So, particles are able to jump to nearest neighbor vacant sites. Then, the problem of the hydrodynamic limit is to capture the collective behavior of particles via the scaling limit. If we take a suitable scaling limit of space and time, then we observe that the density of particles is governed by a partial differential equation as a macroscopic model. Here it is necessary to construct a suitable scaling limit for a graph and to know some analytic properties of the limit space.
A crystal lattice is defined as an infinite graph which admits a free action of a free abelian group with a finite quotient graph . We construct a scaling limit of a crystal lattice as follows: Let be a positive integer. Take a finite index subgroup of , which is isomorphic to when is isomorphic to . Then we take the quotient of by -action: . We call this finite quotient graph the -scaling finite graph. The quotient group acts freely on . Here we consider exclusion processes on . To observe these processes in the continuous space, we embed into a torus. We construct an embedding map from into a torus by using a harmonic map in the discrete sense in order that the image converges to a torus as goes to the infinity. (Here the convergence of metric spaces is verified by using the Gromov-Hausdorff topology, however, we do not need this notion in this paper.) Then we obtain exclusion processes embedded by into the torus.
In this paper, we deal with the simplest case among exclusion processes: the symmetric simple exclusion process and its perturbation: the weakly asymmetric simple exclusion process. In the latter case, we obtain a heat equation with nonlinear drift terms on torus as the limit of process of empirical density (Theorem3.1 and Examples below). We observe that the diffusion coefficient matrices and nonlinear drift terms can be computed by data of a finite quotient graph and a harmonic map . (See also examples in Section 2.2.) The hydrodynamic limit for these processes on the crystal lattice is obtained as an extension of the one on . So, first, we review the outline of the proof for , following the method by Guo, Papanicolaou and Varadhan in . Since the lattice is naturally embedded into , the combinatorial Laplacian on the scaled discrete torus converges to the Laplacian on the torus according to this natural embedding. The local ergodic theorem is the key step of the proof since it enables us to replace local averages by global averages and to verify the derivation of the limit partial differential equation. It is formulated by using local functions on the configuration space and the shift action on the discrete torus. The proof of the local ergodic theorem is based on the one-block estimate and the two-blocks estimate. Roughly speaking, the one-block estimate is interpreted as the local law of large numbers and the two-blocks estimate is interpreted as the asymptotic independence of two different local laws of large numbers.
Second, we look over the outline of the proof for the crystal lattice. There are two main points with regard to the difference between and the crystal lattice, that are the convergence of the Laplacian and the local ergodic theorem. Although the crystal lattice is embedded into an Euclidean space by a harmonic map , the combinatorial Laplacian on the image of the -scaling finite graph does not converge to the Laplacian on the torus straightforwardly. It is proved by averaging each fundamental domain by -action because of the local inhomogeneity of the crystal lattice. Thus, it is necessary to obtain the local ergodic theorem compatible with the convergence of the Laplacian. Furthermore, it is also necessary to obtain the local ergodic theorem compatible with the local inhomogeneity of the crystal lattice. For these reasons, we have to modify the local ergodic theorem in the case of crystal lattices. To formulate the local ergodic theorem in the crystal lattice, we introduce the notion of -periodic local function bundles. A -periodic local function bundle is a family of local functions on the configuration space which is parametrized by vertices periodically. Moreover, we introduce two different ways to take local averages of a -periodic local function bundle. The first one is to take averages per each fundamental domain as a unit. The second one is to take averages on each -orbit. The local ergodic theorem in the crystal lattice is formulated by using -periodic local function bundles, two types of local averages and the -action on the -scaling finite graph . In fact, we use only special -periodic local function bundles to handle the weakly asymmetric simple exclusion process. The proof of this local ergodic theorem is also based on the one-block estimate and the two-blocks estimate. Proofs of these two estimates are analogous to the case of the discrete torus since we use the fact that the whole crystal lattice is covered by the -action of a fundamental domain in the first type of the local average and we restrict to a -orbit in the second type of the local average. In this paper, we call the local ergodic theorem the replacement theorem and prove it in the form of the super exponential estimate. The derivation of the hydrodynamic equation is the same manner as the case of the discrete torus.
Let us mention related works. Interacting particle systems are categorized into the gradient system and the non-gradient system, according to types of interactions. We call the system the gradient system when the interaction term is represented by the difference of local functions. Otherwise, we call the system the non-gradient system. We mention a recent work on the non-gradient system by Sasada . The symmetric simple exclusion process is a model of the gradient system. Our problems essentially correspond to problems for the gradient system since the hydrodynamic limit for the weakly asymmetric simple exclusion process is reduced to the one for the symmetric simple exclusion process, following . As for the hydrodynamic limit on spaces other than the Euclidean lattice, Jara investigates the hydrodynamic limit for zero-range processes in the Sierpinski gasket (). As for the crystal lattice, there is another type of the scaling limit. In , Shubin and Sunada study lattice vibrations of crystal lattices and calculate one of the thermodynamic quantities: the specific heat. They derive the equation of motion by taking the continuum limit of the crystal lattice. As a further problem, we mention the following problem: Recently, attentions have been payed for interacting particle systems evolving in random environments (e.g., ,  and ). For example, the quenched invariance principle for the random walk on the infinite cluster of supercritical percolation of with is proved by Berger and Biskup (). Their argument is based on a harmonic embedding of percolation cluster into . Our use of the harmonic map and local function bundles will play a role in the systematic treatment of particle systems in more general random graphs. Furthermore, the hydrodynamic limit on the inhomogeneous crystal lattice is considered as the case where the crystal lattice has topological defects. This problem would be interesting in connection with material sciences.
This paper is organized as follows: In Section 2, we introduce the crystal lattice and construct the scaling limit by using discrete harmonic maps. In Section 3, we formulate the weakly asymmetric simple exclusion process on the crystal lattice and state the main theorem (Theorem 3.1). In Section 4, we introduce -periodic local function bundles and show the replacement theorem (Theorem 4.1). We prove the one-block estimate and the two-blocks estimate. In Section 5, we derive the quasi-linear parabolic equation, applying the replacement theorem and complete the proof of Theorem 3.1. Section 6 is Appendix;A. We prove some lemmas related to approximation by combinatorial metrics to complete the scaling limit argument. Section 7 is Appendix;B. We refer an energy estimate of a weak solution and a uniqueness result for the partial differential equation to this appendix.
Landau asymptotic notation. Throughout the paper, we use the notation to mean that as . We also use the notation to mean that as .
2 The crystal lattice and the harmonic realization
In this section, we introduce the crystal lattice as an infinite graph and its realization into the Euclidean space.
2.1 Crystal lattices
Let be a locally finite connected graph, where is a set of vertices and a set of all oriented edges. The graph may have loops and multiple edges. For an oriented edge , we denote by the origin of , by the terminus and by the inverse edge of . Here we regard as a weighted graph, whose weight functions on and are all equal to one.
We call a locally finite connected graph a -crystal lattice if a free abelian group acts freely on and the quotient graph is a finite graph . More precisely, each defines a graph isomorphism and the graph isomorphism is fixed point-free except for . In other words, a -crystal lattice is an abelian covering graph of a finite graph whose covering transformation group is .
2.2 Harmonic maps
Let us construct an embedding of a -crystal lattice into the Euclidean space of dimension .
Given an injective homomorphism such that there exits a basis ,
then we define a harmonic map associated with .
Fix an injective homomorphism as above. We call an embedding , a -periodic harmonic map if satisfies the followings: is -periodic, i.e., for any and any , and is harmonic, i.e., for any , , where .
We note that a -periodic harmonic map depends on and call a periodic harmonic map in short when we fix some .
For , we take a lift of , and define . By the -periodicity, does not depend on the choices of lifts. For , let us define a -matrix by
Here the matrix is symmetric and positive definite. We call the matrix the diffusion coefficient matrix.
The one dimensional standard lattice.
The one dimensional standard lattice which we identify the set of vertices with and the set of (unoriented) edges with the set of pairs of vertices . Now acts freely on by the additive operation in and the quotient finite graph consists of one vertex and one loop as un oriented graph. When we regard as an oriented graph, we add both oriented edges to and the quotient graph consists of one vertex and two oriented loops (Figure 2).
Let us choose a canonical injective homomorphism . In our formulation, choose a basis in and define by setting for so that . By identifying the set of vertices of with , we define an embedding map , . This embedding map is a -periodic harmonic map. In this case, .
Let us give another example of periodic harmonic map for the one dimensional standard lattice . Now we define a -action on in the following way: For , , define . Then this induces a free -action on and the quotient graph consists of two vertices and two edges between them as an unoriented graph. Let be the injective homomorphism as the same as in Example 0a. We define an embedding map by setting , . Then is a periodic harmonic map. The image of is not isomorphic to the previous one (Figure 3). In this case, .
The square lattice.
The square lattice has the standard embedding in and this embedding is shown to be periodic and harmonic in our sense in the following. We identify the set of vertices of the square lattice with and the set of edges with the set of pairs of vertices . Now acts freely on by the additive operation in and the quotient graph is the bouquet graph with one vertex and two unoriented loops. When we regard as an oriented graph, we add both oriented edges to and the quotient finite graph is the bouquet graph with one vertex and four oriented loops.
Let us choose a canonical injective homomorphism . That is, choose a basis in and define by setting for so that . By identifying the set of vertices of with , we define an embedding map , . This embedding map is a -periodic harmonic map. In this case,
Let us give another example of periodic harmonic map for the square lattice . Choose a basis in and define by setting for so that . In the same way as above Example 1a, we define an embedding map , . (Figure 4.) This embedding map is a -periodic harmonic map. In this case,
Let us give an example an embedding map which is periodic but not harmonic. We choose an action of on the square lattice in the following way: Again, we identify the set of vertices of with . For , , define . Then this induces a free -action and the quotient graph consists of two vertices, two edges between them and one loop on each vertex (two loops) as an unoriented graph. Let be the same as in Example 1a. We define an embedding map by setting , for . Then is periodic but not harmonic since for , .
The hexagonal lattice.
The hexagonal lattice admits a free -action with the quotient graph consisting of two vertices and three edges as an unoriented graph. We define a fundamental subgraph by setting the set of vertices and the set of (unoriented) edges . Then the hexagonal lattice has a subgraph isomorphic to and is covered by copies of the subgraph translated by the -action. Choose a basis in and define by setting for so that . We define an embedding map by setting , , and for . (Figure 5.) Then is a periodic harmonic map. In this case,
The Kagomé lattice.
The Kagomé lattice admits a free -action with the quotient graph consisting of three vertices and six edges (two edges between each pair of vertices) as an unoriented graph. We define a fundamental subgraph by setting the set of vertices and the set of (unoriented) edges . Then the Kagomé lattice has a subgraph isomorphic to and is covered by copies of the subgraph translated by the -action. Choose a basis in and define as the same as in Example 2. We define an embedding map by setting , , , , for . (Figure 6.) Then is a periodic harmonic map. In this case,
The notion of periodic harmonic map on -crystal lattice is studied by Kotani and Sunada and including the standard realization which they introduced in  as a special case. They use harmonic maps to characterize equilibrium configurations of crystals. In fact, a periodic harmonic map is characterized by a critical map for some discrete analogue of energy functional. The standard realization is not only a critical map but also the map whose energy itself is minimized by changing flat metrics on torus with fixed volume. (More precisely, see). The existence of periodic harmonic map for every injective homomorphism producing lattices in and the uniqueness up to translation is proved in Theorem 2.3 and Theorem 2.4 in .
2.3 Scaling Limits
Let us construct the scaling limit of the crystal lattice. Suppose that is isomorphic to . Let be an arbitrary positive integer and the subgroup isomorphic to . The subgroup acts also freely on and its quotient graph is also a finite graph . Then acts freely on . We call the -scaling finite graph. The map
satisfies that for all and all since is -equivariant. We have the torus , equipped with the flat metric induced from the Euclidean metric. The torus depends on , however, we do not specify it in the following. Then the map induces the map
We call the -scaling map. (Figure7.)
Next, we observe convergence of the combinatorial Laplacian on . Since the degrees of might be different, depending on each , we consider “average” of the combinatorial Laplacian on a fundamental domain.
Let be the degree of a vertex , i.e., the cardinality of the set . Define the combinatorial Laplacian associated with acting on the space of continuous functions by
for and . We show that the combinatorial Laplacian converges to the Laplacian on in the following sense: For every twice continuous derivative functions , for every , for each , take an arbitrary sequence of vertices such that is a lift of and as , then by the Taylor formula,
Since is harmonic,
Since , the last term is equal to , where is a diffusion coefficient matrix and and .
3 Hydrodynamic limit for exclusion processes
We formulate the symmetric simple exclusion process and the weakly asymmetric simple exclusion process in crystal lattices. As we see below, the former is a particular case of the latter.
Let be the -scaling finite graph of . We denote the configuration space by . We denote the configuration space for the whole crystal lattice by . Each configuration is defined by with or and by in the same way.
We consider the Bernoulli measure and on , , respectively, for . They are defined as the product measures of the Bernoulli measure on , where .
Let be the -space of -valued functions on . The action of on lifts on by setting for and . The group also acts on by for . For and , we denote by the configuration defined by exchanging the values of and , i.e.,
For each , we define the operator by setting . We see that and for . The above notations also indicate corresponding ones for the configuration space on the whole crystal lattice.
The symmetric simple exclusion process is defined by the generator acting on as
The weakly asymmetric simple exclusion process is defined as a perturbation of the symmetric simple exclusion process. We denote by the space of continuous functions with continuous derivatives in and the twice continuous derivatives in . For each function , the weakly asymmetric simple exclusion process on is defined by the generator acting on as
Here is the -scaling map. The meaning of the perturbation is as follows: We introduce a “small” drift depending on space and time in particles. In the original process, a particle jumps with rate from to ( is an edge) at time , while in the perturbed process, a particle jumps approximately with rate
Therefore, the external field which is now gives a small asymmetry of the order in the jump rate. Notice that we obtain the symmetric simple exclusion process when is constant.
The weakly asymmetric simple exclusion process which we introduced here does not include the well-studied case where for one dimensional lattice, the external field is for some constant and its limit equation produces the viscous Burgers equation (e.g., ). This process corresponds to the case with which we do not treat here.
Let be the space of paths which are right continuous and have left limits for some arbitrary fixed time . For a probability measure on , we denote by the distribution on of the continuous time Markov chain generated by with the initial measure .
The main theorem is stated as follows:
Let be a measurable function. If a sequence of probability measures on satisfies that
for every and for every continuous functions , then for every ,
for every and for every continuous functions , where is the unique weak solution of the following quasi-linear parabolic equation:
Here we define for .
We give examples corresponding to ones in Section 2.2.
The one dimensional standard lattice.
For the embedding in Example 0a., we recover the equation in Theorem 3.1 in :
For the embedding in Example 0b., we have the following equation:
The square lattice.
For the square lattice and its embedding in Example 1a., we have the following equation:
For the square lattice and its embedding in Example 1b., we have the following equation:
The hexagonal lattice, the Kagomé lattice.
For the hexagonal lattice, the Kagomé lattice and their embeddings in Example 2. and 3., we have the following same equation:
4 Replacement theorem
In this section, we formulate the replacement theorem and give its proof. The replacement theorem is given by the form of super exponential estimate and follows from the one-block estimate and the two blocks estimate.
4.1 Local function bundles
For our purpose, we introduce local function bundles which describe the local interactions of particles and the two types of local averages for local function bundles.
A local function bundle on is a function , which satisfies that for each there exists such that depends only on . Here is the graph distance in . We say that a local function bundle is -periodic if it holds that for any , and .
Here we give examples of -periodic local function bundles on . We use the first one and the third one later.
If we define by for and
then is a -periodic local function bundle on .
If we define by for and
then is a -periodic local function bundle on .
Fix . If we define by
then is a -periodic local function bundle on .
Note that a -periodic local function bundle induces a map for large enough in the natural way. To abuse the notation, we indicate the induced map by the same character .
First, for , we define the -ball by
Here is the word metric appearing in Section 6.1. We regard that via the covering map when is large enough for . For a local function bundle , we define the local average of on blocks by for ,
where is a unique element such that and denotes the cardinality of the set, which is equal to . Note that for every . As a special case, we define for and ,
Second, we define the local average of on each -orbit, by for ,
Note that and are -periodic when is -periodic.
If a local function bundle is -periodic and is large enough, then induce the functions on in the natural way. To abuse the notation, we indicate the induced maps by the same characters .
4.2 Super exponential estimate
For a local function bundle , for , let us define , the expectation with respect to the Bernoulli measure .
The following estimate allows us to replace the local averages of the local function bundle by the global averages of the empirical density. We call the following theorem the replacement theorem. We prove it in the form of the super exponential estimate.
(Super exponential estimate of the replacement theorem)
Fix . For any -periodic local function bundles , for every and for every , it holds that
Note that for every , .
We denote by the corresponding distribution on of continuous time Markov chain generated by with the initial measure . Furthermore, we denote by the corresponding distribution on of continuous time Markov chain generated by with the initial measure , i.e., an equilibrium measure. We denote by the expectation with respect to , by the one with respect to and by the one with respect to . For a probability measure on some probability space, we also denote by the expectation with respect to .
By the following proposition, we reduce the super exponential estimate for to the one for .
There exists a constant such that
To simplify the notation, put for . To calculate the Radon-Nikodym derivative: