document: 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 kagom lattice ( figure[crystals ] ) 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 @xcite , @xcite 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 ( @xcite ) . 
   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 @xmath0 . 
   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 ( @xcite , @xcite , @xcite , and the expository article @xcite ) . 
   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 @xcite , 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 @xcite 
   , 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 @xmath1 which admits a free action of a free abelian group @xmath2 with a finite quotient graph @xmath3 . 
   we construct a scaling limit of a crystal lattice as follows : let @xmath4 be a positive integer . take a finite index subgroup @xmath5 of @xmath2 , which is isomorphic to @xmath6 when @xmath2 is isomorphic to @xmath0 . 
   then we take the quotient of @xmath1 by @xmath5-action : @xmath7 . 
   we call this finite quotient graph @xmath7 the _ @xmath4-scaling finite graph_. the quotient group @xmath8 acts freely on @xmath7 . here 
   we consider exclusion processes on @xmath7 . to observe these processes in the continuous space 
   , we embed @xmath7 into a torus . 
   we construct an embedding map @xmath9 from @xmath7 into a torus by using a harmonic map @xmath10 in the discrete sense in order that the image @xmath11 converges to a torus as @xmath4 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 @xmath9 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 ( theorem[main ] and examples below ) . 
   we observe that the diffusion coefficient matrices and nonlinear drift terms can be computed by data of a finite quotient graph @xmath3 and a harmonic map @xmath10 . 
   ( see also examples in section [ harmonic ] . ) 
   the hydrodynamic limit for these processes on the crystal lattice is obtained as an extension of the one on @xmath0 . 
   so , first , we review the outline of the proof for @xmath0 , following the method by guo , papanicolaou and varadhan in @xcite . since the lattice @xmath0 is naturally embedded into @xmath12 , 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 @xmath0 and the crystal lattice , that are the convergence of the laplacian and the local ergodic theorem . 
   although the crystal lattice @xmath1 is embedded into an euclidean space by a harmonic map @xmath10 , the combinatorial laplacian on the image of the @xmath4-scaling finite graph @xmath11 does not converge to the laplacian on the torus straightforwardly . 
   it is proved by averaging each fundamental domain by @xmath2-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 _ @xmath2-periodic local function bundles_. a @xmath2-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 @xmath2-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 @xmath2-orbit . 
   the local ergodic theorem in the crystal lattice is formulated by using @xmath2-periodic local function bundles , two types of local averages and the @xmath13-action on the @xmath4-scaling finite graph @xmath7 . 
   in fact , we use only special @xmath2-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 @xmath2-action of a fundamental domain in the first type of the local average and we restrict to a @xmath2-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 @xcite . 
   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 @xcite . 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 ( @xcite ) . 
   as for the crystal lattice , there is another type of the scaling limit . in @xcite , 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. , @xcite , @xcite and @xcite ) . for example , the quenched invariance principle for the random walk on the infinite cluster of supercritical percolation of @xmath0 with @xmath14 is proved by berger and biskup ( @xcite ) . 
   their argument is based on a harmonic embedding of percolation cluster into @xmath12 . our use of the harmonic map @xmath10 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 [ crystal lattices ] , we introduce the crystal lattice and construct the scaling limit by using discrete harmonic maps . in section [ hydrodynamic limit ] , we formulate the weakly asymmetric simple exclusion process on the crystal lattice and state the main theorem ( theorem [ main ] ) . in section 
   [ replacement theorem ] , we introduce @xmath2-periodic local function bundles and show the replacement theorem ( theorem [ super exponential estimate ] ) . 
   we prove the one - block estimate and the two - blocks estimate . in section [ the proof of the main theorem ] , we derive the quasi - linear parabolic equation , applying the replacement theorem and complete the proof of theorem [ main ] . 
   section [ appendixa ] is appendix;a . 
   we prove some lemmas related to approximation by combinatorial metrics to complete the scaling limit argument . 
   section [ appendixb ] 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 @xmath15 to mean that @xmath16 as @xmath17 . 
   we also use the notation @xmath18 to mean that @xmath16 as @xmath19 . 
   in this section , we introduce the crystal lattice as an infinite graph and its realization into the euclidean space . 
   let @xmath20 be a locally finite connected graph , where @xmath21 is a set of vertices and @xmath22 a set of all oriented edges . 
   the graph @xmath1 may have loops and multiple edges . 
   for an oriented edge @xmath23 , we denote by @xmath24 the origin of @xmath25 , by @xmath26 the terminus and by @xmath27 the inverse edge of @xmath25 . here 
   we regard @xmath1 as a weighted graph , whose weight functions on @xmath21 and @xmath22 are all equal to one . 
   we call a locally finite connected graph @xmath20 a _ @xmath2-crystal lattice _ if a free abelian group @xmath2 acts freely on @xmath1 and the quotient graph @xmath28 is a finite graph @xmath29 . 
   more precisely , each @xmath30 defines a graph isomorphism @xmath31 and the graph isomorphism is fixed point - free except for @xmath32 . in other words , 
   a @xmath2-crystal lattice @xmath1 is an abelian covering graph of a finite graph @xmath3 whose covering transformation group is @xmath2 . 
   let us construct an embedding of a @xmath2-crystal lattice @xmath1 into the euclidean space @xmath12 of dimension @xmath33 . 
   given an injective homomorphism @xmath34 such that there exits a basis @xmath35 , @xmath36    then we define a harmonic map associated with @xmath37 . 
   fix an injective homomorphism @xmath37 as above . 
   we call an embedding @xmath38 , a _ 
   @xmath37-periodic harmonic map _ if @xmath10 satisfies the followings : _ 
   @xmath10 is @xmath2-periodic _ 
   , i.e. , for any @xmath39 and any @xmath30 , @xmath40 and _ @xmath10 is harmonic _ , i.e. , for any @xmath39 , @xmath41=0 $ ] , where @xmath42 . 
   we note that a @xmath37-periodic harmonic map @xmath10 depends on @xmath37 and call @xmath10 a _ periodic harmonic map _ in short when we fix some @xmath37 .    for @xmath43 
   , we take a lift @xmath44 of @xmath25 , and define @xmath45 . by the @xmath2-periodicity 
   , @xmath46 does not depend on the choices of lifts . for @xmath47 , 
   let us define a @xmath48-matrix by @xmath49 here the matrix is symmetric and positive definite . 
   we call the matrix @xmath50 the _ diffusion coefficient matrix_.    _ examples _    1 .   the one dimensional standard lattice . 
   the one dimensional standard lattice @xmath1 which we identify the set of vertices @xmath21 with @xmath51 and the set of ( unoriented ) edges with the set of pairs of vertices @xmath52 . 
   now @xmath51 acts freely on @xmath1 by the additive operation in @xmath51 and the quotient finite graph consists of one vertex and one loop as un oriented graph . when we regard @xmath1 as an oriented graph , we add both oriented edges to @xmath1 and the quotient graph consists of one vertex and two oriented loops ( figure [ one - dim ] ) . 
   + let us choose a canonical injective homomorphism @xmath53 . in our formulation 
   , choose a basis @xmath54 in @xmath55 and define @xmath53 by setting @xmath56 for @xmath57 so that @xmath58 . by identifying the set of vertices of @xmath1 with @xmath51 
   , we define an embedding map @xmath59 , @xmath60 . 
   this embedding map @xmath10 is a @xmath51-periodic harmonic map . in this case , 
   +    3 .   let us give another example of periodic harmonic map for the one dimensional standard lattice @xmath1 . 
   now we define a @xmath51-action on @xmath1 in the following way : for @xmath62 , @xmath39 , define @xmath63 . 
   then this induces a free @xmath51-action on @xmath1 and the quotient graph consists of two vertices and two edges between them as an unoriented graph . 
   let @xmath37 be the injective homomorphism as the same as in example 0a . 
   we define an embedding map @xmath64 by setting @xmath65 , @xmath66 . 
   then @xmath10 is a periodic harmonic map . 
   the image of @xmath10 is not isomorphic to the previous one ( figure [ one - dim2 ] ) . in this case , 
   +   in example 0b.,width=487 ] 4 .   the square lattice . 
   the square lattice has the standard embedding in @xmath68 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 @xmath1 with @xmath69 and the set of edges with the set of pairs of vertices @xmath70 . 
   now @xmath69 acts freely on @xmath1 by the additive operation in @xmath69 and the quotient graph is the bouquet graph with one vertex and two unoriented loops . when we regard @xmath1 as an oriented graph , we add both oriented edges to @xmath1 and the quotient finite graph is the bouquet graph with one vertex and four oriented loops . 
   + let us choose a canonical injective homomorphism @xmath71 . 
   that is , choose a basis @xmath72 in @xmath68 and define @xmath71 by setting @xmath73 for @xmath74 so that @xmath75 . by identifying the set of vertices of @xmath1 with @xmath69 
   , we define an embedding map @xmath59 , @xmath76 . 
   this embedding map @xmath10 is a @xmath69-periodic harmonic map . in this case , 
   @xmath77 6 . 
   let us give another example of periodic harmonic map for the square lattice @xmath1 . 
   choose a basis @xmath78 in @xmath68 and define @xmath71 by setting @xmath79 for @xmath74 so that @xmath80 . 
   in the same way as above example 1a , we define an embedding map @xmath59 , @xmath76 . 
   ( figure [ square ] . ) 
   this embedding map @xmath10 is a @xmath69-periodic harmonic map . in this case , @xmath81 +    7 . 
   let us give an example an embedding map @xmath10 which is periodic but not harmonic . 
   we choose an action of @xmath69 on the square lattice @xmath1 in the following way : again , we identify the set of vertices @xmath21 of @xmath1 with @xmath69 . for @xmath82 , @xmath83 , define @xmath84 . then this induces a free @xmath51-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 @xmath37 be the same as in example 1a . 
   we define an embedding map @xmath85 by setting @xmath86 , @xmath87 for @xmath88 . 
   then @xmath89 is periodic but not harmonic since for @xmath90 , @xmath91 = ( 1 , 0)+(-1 , 0 ) + ( 1 , 1/2 ) + ( 1 , -1/2)=(2 , 0 ) \neq ( 0,0)$ ] . 
   the hexagonal lattice . 
   + the hexagonal lattice admits a free @xmath69-action with the quotient graph consisting of two vertices and three edges as an unoriented graph . 
   we define a fundamental subgraph @xmath92 by setting the set of vertices @xmath93 and the set of ( unoriented ) edges @xmath94 . 
   then the hexagonal lattice has a subgraph isomorphic to @xmath92 and is covered by copies of the subgraph translated by the @xmath69-action . 
   choose a basis @xmath95 in @xmath68 and define @xmath71 by setting @xmath79 for @xmath74 so that @xmath80 . 
   we define an embedding map @xmath10 by setting @xmath96 , @xmath97 , @xmath98 and @xmath99 for @xmath88 . 
   ( figure [ hexagonal ] . ) 
   then @xmath10 is a periodic harmonic map . in this case , @xmath100 +    9 .   the kagom lattice . 
   + the kagom lattice admits a free @xmath69-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 @xmath92 by setting the set of vertices @xmath101 and the set of ( unoriented ) edges @xmath102 . 
   then the kagom lattice has a subgraph isomorphic to @xmath92 and is covered by copies of the subgraph translated by the @xmath69-action . 
   choose a basis @xmath95 in @xmath68 and define @xmath71 as the same as in example 2 . 
   we define an embedding map @xmath10 by setting @xmath103 , @xmath104 , @xmath105 , @xmath106 , @xmath107 for @xmath88 . 
   ( figure [ kagome ] . ) 
   then @xmath10 is a periodic harmonic map . in this case , 
   @xmath100 +     _ 
   remark_.    the notion of periodic harmonic map on @xmath2-crystal lattice is studied by kotani and sunada and including the standard realization which they introduced in @xcite 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@xcite ) . 
   the existence of periodic harmonic map for every injective homomorphism producing lattices in @xmath12 and the uniqueness up to translation is proved in theorem 2.3 and theorem 2.4 in @xcite . 
   let us construct the scaling limit of the crystal lattice . 
   suppose that @xmath2 is isomorphic to @xmath0 . 
   let @xmath108 be an arbitrary positive integer and @xmath5 the subgroup isomorphic to @xmath6 . 
   the subgroup @xmath5 acts also freely on @xmath1 and its quotient graph @xmath109 is also a finite graph @xmath110 . 
   then @xmath111 acts freely on @xmath7 . 
   we call @xmath7 the _ @xmath4-scaling finite graph_. the map @xmath112 satisfies that @xmath113 for all @xmath39 and all @xmath30 since @xmath10 is @xmath2-equivariant . 
   we have the torus @xmath114 , equipped with the flat metric induced from the euclidean metric . 
   the torus depends on @xmath115 , however , we do not specify it in the following . 
   then the map @xmath116 induces the map @xmath117 we call @xmath9 the _ @xmath4-scaling map_. ( figure[scaling ] . ) 
   @xmath118    -scaling finite graph by a harmonic map in the covering space , width=487 ]    next , we observe convergence of the combinatorial laplacian on @xmath7 . since the degrees of @xmath119 might be different , depending on each @xmath120 , we consider  average " of the combinatorial laplacian on a fundamental domain . 
   let @xmath121 be the degree of a vertex @xmath119 , i.e. , the cardinality of the set @xmath122 . define the combinatorial laplacian @xmath123 associated with @xmath124 acting on the space of continuous functions @xmath125 by @xmath126,\ ] ] for @xmath127 and @xmath119 . we show that the combinatorial laplacian converges to the laplacian on @xmath128 in the following sense : for every twice continuous derivative functions @xmath129 , for every @xmath130 , for each @xmath131 , take an arbitrary sequence of vertices @xmath132 such that @xmath133 is a lift of @xmath120 and @xmath134 as @xmath17 , then by the taylor formula , @xmath135 since @xmath10 is harmonic , @xmath136 since @xmath137 , the last term is equal to @xmath138 , where @xmath50 is a diffusion coefficient matrix and @xmath139 and @xmath140 . 
   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 @xmath110 be the @xmath4-scaling finite graph of @xmath1 . 
   we denote the configuration space by @xmath141 . 
   we denote the configuration space for the whole crystal lattice @xmath142 by @xmath143 . 
   each configuration is defined by @xmath144 with @xmath145 or @xmath146 and by @xmath147 in the same way . 
   we consider the bernoulli measure @xmath148 and @xmath149 on @xmath150 , @xmath151 , respectively , for @xmath152 . 
   they are defined as the product measures of the bernoulli measure @xmath153 on @xmath154 , where @xmath155 . 
   let @xmath156 be the @xmath157-space of @xmath55-valued functions on @xmath150 . 
   the action of @xmath13 on @xmath7 lifts on @xmath150 by setting @xmath158 for @xmath159 and @xmath119 . 
   the group @xmath13 also acts on @xmath156 by @xmath160 for @xmath161 . for @xmath162 and @xmath163 , we denote by @xmath164 the configuration defined by exchanging the values of @xmath165 and @xmath166 , i.e. , @xmath167 for each @xmath162 , we define the operator @xmath168 by setting @xmath169 . 
   we see that @xmath170 and @xmath171 for @xmath172 . 
   the above notations also indicate corresponding ones for @xmath173 the configuration space on the whole crystal lattice . 
   the symmetric simple exclusion process is defined by the generator @xmath174 acting on @xmath175 as @xmath176    the weakly asymmetric simple exclusion process is defined as a perturbation of the symmetric simple exclusion process . 
   we denote by @xmath177\times \mathbb{t}^{d})$ ] the space of continuous functions with continuous derivatives in @xmath178 $ ] and the twice continuous derivatives in @xmath128 . for each function 
   @xmath179\times \mathbb{t}^{d})$ ] , the weakly asymmetric simple exclusion process on @xmath7 is defined by the generator @xmath180 acting on @xmath175 as @xmath181 where @xmath182.\ ] ] here @xmath183 is the @xmath4-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 @xmath184 from @xmath24 to @xmath26 ( @xmath25 is an edge ) at time @xmath185 , while in the perturbed process , a particle jumps approximately with rate @xmath186 therefore , the external field which is now @xmath187 gives a small asymmetry of the order @xmath188 in the jump rate . 
   notice that we obtain the symmetric simple exclusion process when @xmath189 is constant . 
   _ remark _    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 @xmath190 for some constant @xmath191 and its limit equation produces the viscous burgers equation ( e.g. , @xcite ) . 
   this process corresponds to the case with @xmath192 which we do not treat here . 
   let @xmath193 , z_{n})$ ] be the space of paths which are right continuous and have left limits for some arbitrary fixed time @xmath194 . for a probability measure @xmath195 on @xmath150 , 
   we denote by @xmath196 the distribution on @xmath193 , z_{n})$ ] of the continuous time markov chain @xmath197 generated by @xmath198 with the initial measure @xmath195 . 
   the main theorem is stated as follows :    [ main ]    let @xmath199 $ ] be a measurable function . if a sequence of probability measures @xmath195 on @xmath150 satisfies that @xmath200=0,\ ] ] for every @xmath201 and for every continuous functions @xmath202 , then for every @xmath203 , @xmath204=0,\ ] ] for every @xmath201 and for every continuous functions @xmath202 , where @xmath205 is the unique weak solution of the following quasi - linear parabolic equation : @xmath206    here we define @xmath207 for @xmath43 . 
   we give examples corresponding to ones in section [ harmonic ] . 
   _ examples _    1 .   the one dimensional standard lattice . + for the embedding in example 0a 
   , we recover the equation in theorem 3.1 in @xcite : @xmath208 + for the embedding in example 0b . 
   , we have the following equation : @xmath209 2 .   the square lattice . 
   + for the square lattice and its embedding in example 1a . 
   , we have the following equation : @xmath210 + for the square lattice and its embedding in example 1b . 
   , we have the following equation : + @xmath211 3 .   the hexagonal lattice , the kagom lattice . + for the hexagonal lattice , the kagom lattice and their embeddings in example 2 . and 3 
   , we have the following same equation : @xmath212 
   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 .      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 @xmath213 on @xmath214 is a function @xmath215 , which satisfies that for each @xmath216 there exists @xmath217 such that @xmath218 depends only on @xmath219 . 
   here @xmath220 is the graph distance in @xmath1 . 
   we say that a local function bundle @xmath221 is @xmath2-periodic if it holds that @xmath222 for any @xmath30 , @xmath39 and @xmath147 .    here 
   we give examples of @xmath2-periodic local function bundles on @xmath223 . 
   we use the first one and the third one later .    _ examples _    * if we define @xmath221 by for @xmath39 and @xmath147 @xmath224 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . * 
   if we define @xmath221 by for @xmath39 and @xmath147 @xmath225 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . 
   * fix @xmath226 . 
   if we define @xmath227 by @xmath228 then @xmath213 is a @xmath2-periodic local function bundle on @xmath223 . 
   note that a @xmath2-periodic local function bundle @xmath221 induces a map @xmath229 for large enough @xmath4 in the natural way . to abuse the notation , 
   we indicate the induced map by the same character @xmath213 .    first , for @xmath230 
   , we define the @xmath231-ball by @xmath232 here @xmath233 is the word metric appearing in section [ approximation ] . 
   we regard that @xmath234 via the covering map when @xmath4 is large enough for @xmath231 . for a local function bundle @xmath215 
   , we define the local average of @xmath213 on blocks @xmath235 by for @xmath39 , @xmath236b(d_{x_{0}},r)\right|}\sum_{z \in [ x]b(d_{x_{0}},r ) } f_{z } : z \to \mathbb{r},\ ] ] where @xmath237 $ ] is a unique element @xmath30 such that @xmath238 and @xmath239b(d_{x_{0}},r)\right|$ ] denotes the cardinality of the set , which is equal to @xmath240 . 
   note that @xmath241 for every @xmath242 . as a special case , 
   we define for @xmath147 and @xmath39 , @xmath243b(d_{x_{0}},r)\right|}\sum_{z \in [ x]b(d_{x_{0}},r ) } \eta_{z } : z \to \mathbb{r}.\ ] ]    second , we define the local average of @xmath213 on each @xmath2-orbit , @xmath244 by for @xmath39 , @xmath245 note that @xmath246 and @xmath247 are @xmath2-periodic when @xmath213 is @xmath2-periodic . 
   if a local function bundle @xmath213 is @xmath2-periodic and @xmath4 is large enough , then @xmath248 induce the functions on @xmath150 in the natural way . to abuse the notation , we indicate the induced maps by the same characters @xmath248 .      for a local function bundle @xmath215 , for @xmath39 , 
   let us define @xmath249 $ ] , the expectation with respect to the bernoulli measure @xmath149 . 
   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](super exponential estimate of the replacement theorem )    fix @xmath194 . for any @xmath2-periodic local function bundles 
   @xmath215 , for every @xmath242 and for every @xmath201 , it holds that @xmath250 where @xmath251    note that for every @xmath242 , @xmath252 @xmath253 . 
   we denote by @xmath254 the corresponding distribution on @xmath193 , z_{n})$ ] of continuous time markov chain @xmath197 generated by @xmath255 with the initial measure @xmath195 . 
   furthermore , we denote by @xmath256 the corresponding distribution on @xmath193 , z_{n})$ ] of continuous time markov chain @xmath197 generated by @xmath255 with the initial measure @xmath257 , i.e. , an equilibrium measure . 
   we denote by @xmath258 the expectation with respect to @xmath196 , by @xmath259 the one with respect to @xmath254 and by @xmath260 the one with respect to @xmath256 . for a probability measure @xmath261 on some probability space , 
   we also denote by @xmath262 the expectation with respect to @xmath261 .    by the following proposition , 
   we reduce the super exponential estimate for @xmath196 to the one for @xmath254 . 
   [ radon - nikodym ] there exists a constant @xmath263 such that @xmath264    to simplify the notation , put @xmath265 for @xmath119 . to calculate the radon - nikodym derivative : @xmath266    \\ & = \exp\bigg[\sum_{x \in v_{n}}h(t , x)\eta_{x}(t)-\sum_{x \in v_{n}}h(0 , x)\eta_{x}(0 ) -\int_{0}^{t}\bigg\{\sum_{x \in v_{n}}\frac{\partial h}{\partial t}(t , x)\eta_{x}(t ) \\ & \ \ \ \ \ \ \ \ \ \ \ 
   \ \ \ + \frac{n^{2}}{4}\sum_{e \in e_{n}}\big(\eta_{oe}(1-\eta_{te})e^{h(t , te)-h(t , oe ) } + \eta_{te}(1-\eta_{oe})e^{h(t , oe)-h(t , te ) }   \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\eta_{oe}(1-\eta_{te } ) - \eta_{te}(1-\eta_{oe})\big)\bigg\}dt\bigg ]    \\ & = \exp\bigg[\sum_{x \in v_{n}}h(t , x)\eta_{x}(t)-\sum_{x \in v_{n}}h(0 , x)\eta_{x}(0 )    -\int_{0}^{t}\big\{\sum_{x \in v_{n}}\frac{\partial h}{\partial t}(t , x)\eta_{x}(t ) \\ & + \frac{n^{2}}{2}\sum_{x \in v_{n}}\sum_{e \in e_{n , x}}\eta_{x}\left(e^{h(t , te)-h(t , oe)}-1\right ) - \frac{n^{2}}{4}\sum_{e \in e_{n}}\left(e^{h(t , te)-h(t , oe)}+e^{h(t , oe)-h(t , te)}-2\right)\eta_{oe}\eta_{te}\big\}dt\bigg ] 
   .    \\\end{aligned}\ ] ]    by the inequality @xmath267 for @xmath268 , we have that @xmath269   \\ & -n^{2}\sum_{e \in e_{n , x}}\left(h(t , te)- h(t , oe)\right ) - 
   \frac{1}{2}n^{2}\sum_{e \in e_{n , x}}\left(h(t , te)- h(t , oe)\right)^{2}\big|   \\ & \le \frac{1}{6}n^{2}\sum_{e \in e_{n , x}}\left|h(t , te)- h(t , oe)\right|^{3}e^{\left|h(t , te)- h(t , oe)\right|},\end{aligned}\ ] ] and thus , since @xmath10 is harmonic , @xmath270 \\ & = \frac{1}{2}\sum_{e \in e_{n , x}}\sum_{i , j=1}^{d}\frac{\partial^{2}h}{\partial x_{i } \partial x_{j}}(t , x)v_{i}(e)v_{j}(e ) + \frac{1}{2}\sum_{e \in e_{n , x}}\left(\nabla_{{\bf v}(e)}h(t , x)\right)^{2 } + o_{n}.\end{aligned}\ ] ]    furthermore , @xmath271    hence , there exists a constant @xmath263 depending only on @xmath189 and @xmath272 such that @xmath273 is bounded from above by @xmath274 . 
   it completes the proof . 
   the super exponential estimate for @xmath254 induces the one for @xmath196 since it holds that for any borel sets @xmath275 , z_{n})$ ] , @xmath276 . 
   furthermore , it is enough to prove the super exponential estimate for @xmath256 since for any borel sets @xmath275 , z_{n})$ ] , @xmath277 . 
   let @xmath278 be the spaces of probability measures on @xmath279 , respectively . 
   define @xmath280 , @xmath281 the @xmath282-bernoulli measure on @xmath150 , @xmath151 , respectively . here 
   we introduce a functional on @xmath283 , which is the dirichlet form for the density function .    for @xmath284 , 
   put the density @xmath285 . 
   the dirichlet form of @xmath286 is defined by @xmath287    note that @xmath288 . 
   _    the functional @xmath289 is also called the @xmath290-functional in the different literatures . 
   let us define the subset of the space of probability measures on @xmath150 by for @xmath291 , @xmath292    the proof of the super exponential estimate for @xmath256 is reduced to the following :    [ local ergodic theorem ] for every @xmath291 and every @xmath242 , @xmath293    first , we prove theorem[super exponential estimate ] by using therem[local ergodic theorem ] . 
   fix any @xmath294 . by the above argument 
   , it is enough to show that for any @xmath2-periodic local function bundles @xmath215 , for every @xmath242 and for every @xmath201 , @xmath295 where @xmath251    by the chebychev inequality , for every @xmath296 and every @xmath201 , @xmath297    now , the operator @xmath298 acting on @xmath299 is self - adjoint for all @xmath296 and all @xmath242 . 
   let @xmath300 be the largest eigenvalue of @xmath298 . by using the feynman - kac formula , @xmath301    therefore , 
   it is suffice to show that for every @xmath296 , @xmath302 since ( [ eigenvalue ] ) implies that @xmath303 and we obtain the theorem by taking @xmath304 to the infinity .    by the variational principle , 
   the largest eigenvalue @xmath300 is represented by the following : @xmath305 see @xcite appendix 3 for more details . 
   denote the average of @xmath261 by @xmath13-action by @xmath306 then @xmath307 is @xmath13-invariant so that @xmath308=\mathbb{e}_{\widetilde \mu}\left|\widetilde f_{x , k } - \langle f_{x}\rangle(\overline \eta_{x_{0 } , \epsilon n})\right|.\ ] ] the functional @xmath309 is also @xmath13-invariant , i.e. , @xmath310 for @xmath284 , @xmath311 . 
   thus , it is suffice to consider @xmath13-invariant measures @xmath261 to estimate the largest eigenvalue @xmath300 . 
   furthermore , it is suffice to consider the case where @xmath261 satisfies @xmath312 there exists a constant @xmath313 depending on @xmath213 such that @xmath314 , thus we reduce @xmath284 to every @xmath13-invariant measure satisfying that for every @xmath315 , @xmath316 this shows that we can reduce to @xmath317 for every @xmath318 , and thus it is enough to show for every @xmath315 and every @xmath242 , @xmath319 to obtain ( [ eigenvalue ] ) . 
   this follows from theorem[local ergodic theorem ] . 
   it completes the proof .      in this section , we prove the one - block estimate . 
   we regard a probability measure @xmath261 on @xmath150 as one on @xmath151 by periodic extension . 
   let @xmath320 be the covering map by @xmath5-action , and define the periodic inclusion @xmath321 by @xmath322 . 
   we identify @xmath261 on @xmath150 with its push forward by @xmath323 . on the other hand , we identify an @xmath5-invariant probability measure on @xmath151 with a probabiltiy measure on @xmath150 . 
   first , for a finite subgraph @xmath324 of x , we define the restricted state space @xmath325 and the @xmath282-bernoulli measure by @xmath326 on @xmath327 . 
   let us define the operator acting on @xmath328 by @xmath329 . for @xmath330 
   , @xmath331 stands for the restriction of @xmath261 on @xmath327 and @xmath332 its density . 
   we also define the corresponding dirichlet form of @xmath333 by @xmath334 for large enough @xmath4 , we regard @xmath335 as a subgraph of @xmath7 by taking a suitable fundamental domain in @xmath21 for @xmath5-action .    by the convexity of the dirichlet form , @xmath336 by putting @xmath337 . 
   the one - block estimate is stated as follows :    [ the one - block estimate](the one - block estimate . ) for every @xmath2-periodic local function bundles , @xmath215 , every @xmath242 and every @xmath315 , @xmath338    for any probability measure @xmath339 , we apply the above argument by setting @xmath335 as @xmath340 and @xmath341 . since @xmath342 and @xmath343 are @xmath13-invariant , @xmath344    since @xmath339 and @xmath345 , it holds that @xmath346    we note that @xmath347 is compact with respect to the weak topology , and thus @xmath348 has a subsequence which convergences to some @xmath261 in @xmath347 . 
   let @xmath349 be the set of all limit points of @xmath350 in @xmath347 . by the above argument , @xmath351 for all @xmath352 . 
   therefore , since @xmath353 for every @xmath352 by the definition of @xmath354 , we obtain that @xmath355 for every @xmath356 and every @xmath357 . 
   this shows that random variables @xmath358 are exchangeable under @xmath261 . by the de finetti theorem 
   , there exists a probability measure @xmath359 on @xmath360 $ ] such that @xmath361 . 
   since @xmath362 it is enough to show that @xmath363}\mathbb{e}_{\nu_{\rho}}\left| \widetilde f_{x , k } - \langle f_{x } \rangle \left(\overline \eta_{x_{0 } , k}\right ) \right|=0,\ ] ] for every @xmath2-invariant local function bundles @xmath213 .    by the definition of the @xmath2-periodic local function bundle , there exists a constant @xmath364 such that for every @xmath39 , @xmath365 depends on at most @xmath366b(d_{x_{0}},l)\}$ ] . 
   therefore we obtain that there exists a constant @xmath313 depending only on @xmath213 such that @xmath367\right]^{2}\le c(f)\cdot \frac{l^{d}}{k^{d } } \to 0 \ \ \ \ \ \text{as $ k \to \infty$}.\ ] ]    note that @xmath368 $ ] for every @xmath242 since the bernoulli measure @xmath149 is @xmath13-invariant . 
   in addition , we also obtain that there exists a constant @xmath369 not depending on @xmath370 , @xmath371 finally , since @xmath372 is a polynomial with respect to @xmath370 , in particular , uniformly continuous on @xmath360 $ ] , we obtain that @xmath373 } \mathbb{e}_{\nu_{\rho}}\left| \widetilde f_{x , k } - \langle f_{x } \rangle(\overline \eta_{x_{0 } , k } ) \right|=0 \ \ \ \ \ \text{for every $ x \in d_{x_{0}}$}.\ ] ] this concludes the theorem @xmath374      in this section , we prove the two - blocks estimate . 
   we identify a probability measure on @xmath150 with its periodic extension on @xmath151 in the same manner as section[section of the one - block estimate ] . 
   ( the two - blocks estimate)[the two - blocks estimate ] for every @xmath315 , @xmath375    let us denote by @xmath376 the space of probability measures on @xmath377 . 
   we define the map for @xmath30 , @xmath378 by @xmath379 , @xmath147 . for @xmath330 , we define the push forward of @xmath261 by @xmath380 via @xmath381 . 
   let us denote by @xmath382 the set of all limit points of @xmath383 in @xmath384 as @xmath17 and by @xmath385 the set of all limit points of @xmath386 as @xmath387 . 
   we put @xmath388 , where @xmath389 stands for a copy of @xmath390 . 
   then , it holds that @xmath391 note that @xmath392    we introduce two types of generators acting on @xmath393 and the corresponding dirichlet forms . the first one is used for treating two different states at the same time independently . 
   the second one is used for treating exchanges of particles between two different states .    as in section[section of the one - block estimate ] , we define a subgraph @xmath394 of @xmath1 by setting @xmath395 and the operator acting on @xmath328 by @xmath396    for @xmath397 , we denote by @xmath398 the restriction of @xmath261 on @xmath399 . 
   define the dirichlet form of @xmath400 by @xmath401    let us introduce the notation which describes exchanges of states for @xmath402 . for @xmath403 , @xmath404 is the configuration obtained by exchanging values @xmath405 and @xmath406 , i.e. , @xmath407 is defined by setting    @xmath408    @xmath409    moreover , for @xmath410 , we define @xmath411 . 
   we define the operator acting on @xmath412 by @xmath413    the corresponding dirichlet form of @xmath400 is defined by @xmath414    we prove two lemmas needed later . 
   the first one is easy to show , so we omit the proof .    [ 
   quasi - iso ] there exist constants @xmath415 such that for all @xmath30 , @xmath416 where @xmath220 is the graph distance of @xmath1 .    for @xmath417 and for @xmath163 
   , @xmath418 is the configuration obtained by exchanging two values @xmath405 and @xmath419 , i.e. , @xmath420 and moreover for @xmath417 , we define the operator @xmath421 for @xmath422 . these notations also indicates ones for @xmath151 . 
   the second one is the following :    [ path ]    for every @xmath13-periodic functions @xmath422 and every @xmath423 , @xmath424    for @xmath425 , there exists a path @xmath426 such that @xmath427 and @xmath428 . 
   define a sequence of edges @xmath429 by setting @xmath430 . 
   for @xmath163 , let us define @xmath431 , inductively . 
   we note that @xmath432 . 
   then we have that @xmath433 if @xmath434 is @xmath13-periodic , then for each @xmath435 , @xmath436 therefore @xmath437 it completes the proof . 
   let us define the subset of @xmath384 by for a constant @xmath438 , @xmath439    then we have the following lemma . 
   [ inclusion ] there exists a constant @xmath438 such that @xmath440    as in section[section of the one - block estimate ] , we define a subgraph @xmath441 of @xmath1 by setting @xmath442 for @xmath443 . 
   we take large enough @xmath444 for the diameter of @xmath335 , @xmath443 , so that @xmath445 for @xmath446 and @xmath447 for @xmath448 by taking a suitable fundamental domain in @xmath21 by @xmath5-action . 
   take @xmath449 . for @xmath450 
   , if we define @xmath451 by @xmath452 and @xmath453 then @xmath454 . let us consider the operator acting on @xmath328 by @xmath455 for @xmath456 , we denote the density of @xmath457 by @xmath458 
   . then @xmath459 for any @xmath456 , we put @xmath337 . by the convexity of the dirichlet form and the @xmath13-invariance of @xmath37 , @xmath460 since @xmath461 , @xmath462 therefore @xmath463 for any @xmath464 . 
   furthermore @xmath465 for any @xmath466 by the continuity of the functional @xmath467 . 
   for @xmath456 , by the convexity of the dirichlet form , @xmath468    by the @xmath13-invariance of @xmath456 and by lemma[path ] , @xmath469    by lemma[quasi - iso ] and the definition of @xmath456 , for all @xmath30 such that @xmath448 , @xmath470 by setting @xmath471 and the continuity of the @xmath467 , for every @xmath466 , we have that @xmath472 . 
   it concludes that @xmath473 . 
   let us prove theorem[the two - blocks estimate ] .    denote by @xmath474 the set of all limit points of @xmath475 as @xmath19 . for every @xmath476 , it holds that @xmath477 and @xmath478 by the continuity of the functionals @xmath467 and @xmath479 . 
   these show that for any @xmath480 , @xmath481 and thus for any @xmath482 and for any @xmath483 , @xmath484 , @xmath485 and @xmath486 , i.e. , @xmath487 is exchangeable on @xmath377 . by the de finetti theorem there exists a probability measure @xmath359 on @xmath488 $ ] such that @xmath489}\nu_{\rho}\otimes \nu_{\rho}\lambda(d\rho)$ ] .    as in the proof of theorem[the one - block estimate ] , @xmath490}\mathbb{e}_{\nu_{\rho}}\left|\overline \eta_{x_{0 } , k } - \rho \right|^{2}=0 $ ] , therefore , by the triangular inequality , @xmath491}\mathbb{e}_{\nu_{\rho}\otimes \nu_{\rho}}\left|\overline \eta_{x_{0 } , k}-\overline \eta_{x'_{0 } , k}\right|   \\ & \le 2 \sup_{\rho \in [ 0 , 1]}\mathbb{e}_{\nu_{\rho}}\left|\overline \eta_{x_{0 } , k}- \rho \right| \to 0 \ \ \ \ \ \text{as $ k \to \infty$}.\end{aligned}\ ] ] finally , by lemma[inclusion ] , @xmath473 for some @xmath438 and thus , @xmath492 this completes the theorem @xmath493      let us prove theorem[local ergodic theorem ] by using the one - block estimate theorem[the one - block estimate ] and the two - blocks estimate theorem[the two - blocks estimate ] . 
   first , we note that there exist positive constants @xmath494 and @xmath495 such that for any @xmath147 , @xmath496 and thus there exists a constant @xmath497 depending on @xmath498 and @xmath443 such that for any @xmath147 , @xmath499 uniformly . then , since @xmath261 is @xmath2-invariant as a probability measure on @xmath151 , @xmath500 where the last inequality comes from the fact that for any @xmath501 it holds that @xmath502 .    applying theorem[the two - blocks estimate ] 
   , we have that @xmath503 for every @xmath2-periodic local function bundles @xmath213 , @xmath504 is uniformly continuous on @xmath488 $ ] . 
   therefore , @xmath505 furthermore , applying the one - block estimate theorem[the one - block estimate ] , for every @xmath242 , @xmath506 it completes the proof of theorem[local ergodic theorem ] . 
   in this section , we prove theorem[main ] . 
   let @xmath507 be the @xmath4-scaling map . 
   we define the empirical density by @xmath508 where @xmath509 is the delta measure at @xmath510 . 
   the empirical density is the measure valued process . 
   we denote by @xmath511 \times \mathbb{t}^{d})$ ] the space of continuous functions with continuous derivatives in @xmath178 $ ] and twice continuous derivatives in @xmath128 . for every @xmath512 
   \times \mathbb{t}^{d})$ ] , we define @xmath513 to abuse the notation , we denote the inner product in @xmath514 by @xmath515    let us define the process as follows : @xmath516 where @xmath517 and @xmath518 where @xmath519 here @xmath520 are martingales with respect to the filtration @xmath521 , where @xmath522 and it holds that @xmath523 then we have the following lemma by applying the doob inequality . 
   [ doob ] @xmath524=0.\ ] ]    for @xmath525 \times \mathbb{t}^{d})$ ] and for each @xmath162 , we have that @xmath526\left[j_{s}\left(\phi_{n}(te)\right)-j_{s}\left(\phi_{n}(oe)\right)\right].\end{aligned}\ ] ]    by the regularity of @xmath527 and the compactness of @xmath128 , we note that there exists a constant @xmath528 depending only on @xmath527 , such that uniformly , @xmath529 thus @xmath530^{2 }     \le n^{2}|e_{n}|c'\frac{c(j)^{2}}{|v_{n}|^{2}n^{2 } } = \frac{c''}{|v_{n}|},\ ] ] where @xmath531 is a constant such that @xmath532 and @xmath533 . 
   then we obtain that @xmath534 applying the doob inequality for the right continuous martingale @xmath535 , @xmath536 \le 4 \mathbb{e}_{n}^{h}\left|m_{n}(t)\right|^{2},\ ] ] we conclude that @xmath537=0 $ ] . 
   we denote by @xmath538 the space of nonnegative borel measures with the total measure less than or equal to one on @xmath128 , endowed with the weak topology . since @xmath128 is a compact metric space , the space of continuous functions @xmath125 with the supremum norm is separable . 
   fix a dense countable subset @xmath539 of @xmath125 , then the weak topology of @xmath540 is given by the distance @xmath541 by @xmath542 for @xmath543 where @xmath544 . 
   we note that @xmath540 with the weak topology is compact . 
   define the space of paths in @xmath540 by @xmath545 , \mathcal{m}):=\left\{\xi_{\cdot } : [ 0 , t ] \to \mathcal{m } \ 
   \text{$\xi$ is right continuous with left limits.}\right\},\ ] ] equipped with the skorohod topology . for a given process @xmath546 
   \to \mathcal{m}$ ] such that @xmath547 , \mathcal{m})\right)=1 $ ] , we denote by @xmath548 the distribution of @xmath549 on @xmath550 , \mathcal{m})$ ] . 
   then we show that the sequence @xmath551 has a subsequential limit . 
   the following proposition gives a sufficient condition for this . 
   see @xcite section 4 , theorem 1.3 for the proof . 
   [ relative compactness ] if for every @xmath552 , @xmath553 , and every @xmath201 , @xmath554 then there exists a subsequence @xmath555 and a probability measure @xmath556 on @xmath550 , \mathcal{m})$ ] such that @xmath557 weakly converges to @xmath556 as @xmath558 . 
   the next proposition claims that each subsequential limit @xmath556 in proposition[relative compactness ] is absolutely continuous with respect to lebesgue measure on the torus for each time @xmath185 , and its density has the value in @xmath488 $ ] a.e . 
   the proof is the same as in @xcite section 4 , pp.57 , so we omit the proof . 
   [ absolute continuity ] all limit points @xmath556 of @xmath551 are concentrated on trajectories of absolutely continuous measures with respect to the lebesgue measure for each time @xmath185 , i.e. , there exists a borel set @xmath559 , \mathcal{m})$ ] such that @xmath560 and for every @xmath561 and every @xmath562 $ ] , @xmath563 is absolutely continuous with respect to the lebesgue measure @xmath564 . 
   moreover , the density @xmath565 satisfies that @xmath566 , @xmath564-a.e .    to simplify the notation 
   , we put @xmath567 for @xmath568 , \mathbb{t}^{d})$ ] , respectively . 
   then we have :    @xmath569 \cdot ( -\eta_{oe}\eta_{te}+\eta_{oe})(j(t , te)- j(t , oe ) )    \\ & = \frac{n^{2}}{2|v_{n}|}\sum_{x \in v_{n}}\sum_{e \in e_{n , x}}\left[\left\{\exp\left(h(t , te ) - h(t , oe)\right)-1\right\}\cdot \left(j(t , te ) - j(t , oe)\right)\eta_{oe}+\left(j(t , te)- j(t ,   oe)\right)\eta_{oe}\right ] \\ & -\frac{n^{2}}{4|v_{n}|}\sum_{e \in e_{n}}\left\{\exp\left(h(t , te ) - h(t , oe)\right)- \exp\left(h(t , oe)-h(t , te)\right)\right\}\cdot \eta_{oe}\eta_{te}(j(t , te)- j(t , oe)).\end{aligned}\ ] ]    for @xmath23 , we denote the directional derivative along @xmath46 by @xmath570\times \mathbb{t}^{d}$}.\ ] ]    applying the inequality @xmath571 for @xmath268 , by the regularity of @xmath189 and by the compactness @xmath128 , there exists a constant @xmath318 not depending on each point of @xmath178\times \mathbb{t}^{d}$ ] such that for every @xmath4 and for every @xmath162 , @xmath572 @xmath573 @xmath574    by the convergence of the combinatorial laplacian in section[scaling limit ] , we have that @xmath575 hence ,    @xmath576    we replace @xmath196 by @xmath548 regarding the empirical density @xmath549 as the measure on @xmath550,\mathcal{m})$ ] . 
   let us prove the following lemma :    [ equi - continuity ] for every @xmath127 and for every @xmath201 , @xmath577=0.\ ] ]    for every continuous functions @xmath578 , it holds that @xmath579 since by ( [ n2l ] ) there exists a constant @xmath580 such that for large enough @xmath4 , @xmath581 uniformly , we obtain that by the chebychev inequality and by the triangular inequality , for every @xmath201 , for every @xmath582 and for large enough @xmath4 , @xmath583 \le ( 1/\delta)\mathbb{e}_{n}^{h}\left[c \gamma + 2 \sup_{0 \le t \le t } \left|m_{n}(t)\right|\right].\ ] ] then by lemma[doob ] , @xmath584\le c\frac{\gamma}{\delta}.\ ] ] therefore for every @xmath201 , @xmath585=0.\ ] ] it completes the proof . 
   we have the following estimate : there exists a constant @xmath586 depending only on @xmath189 and @xmath527 such that for every @xmath162 and for every @xmath587 , @xmath588    thus , by the regularity of @xmath189 and @xmath527 , putting @xmath589 for @xmath162 , @xmath590 here we regard @xmath591 as a local function bundle independent of states and denote by @xmath592 the local average . by the uniform continuity of the twice derivative of @xmath527 , putting @xmath593 it holds that @xmath594 here 
   we also regard @xmath595 as a local function bundle .    by the above argument 
   , we obtain that    @xmath596    here @xmath597 is the local function bundle appearing in the third example in section[local function bundles ] and @xmath598 its local average .    by applying theorem[super exponential estimate ] and by the continuity of @xmath599 , @xmath600 and @xmath601 on the compact space @xmath178 \times \mathbb{t}^{d}$ ] 
   , it holds that for every @xmath562 $ ] and for every @xmath201 ,    @xmath602    @xmath603    and    @xmath604    here we use @xmath605 for every @xmath119 in the third estimate above . by the triangular inequality , it holds that for every @xmath562 $ ] and for every @xmath201 ,    @xmath606    applying the convergence of the combinatorial laplacian in section [ scaling limit ] , we have that @xmath607    recall that @xmath608    by lemma[doob ] and by the chebychev inequality , for every @xmath201 , @xmath609 furthermore , by the triangular inequality , we have that for every @xmath610 and for every @xmath562 $ ] ,    @xmath611    by lemma[characteristic function ] , we replace @xmath612 by @xmath613 and the summation for @xmath311 by the integral . since by lemma[equi - continuity ] and by proposition[relative compactness ] , the sequence @xmath614 is relatively compact in the weak topology , for a limit point @xmath556 there exists a subsequence @xmath615 weakly converging to @xmath556 . by proposition[absolute continuity ] , the empirical density @xmath616 concentrates on an absolutely continuous trajectory @xmath617 as @xmath558 . by the assumption of theorem[main ] , we replace @xmath618 by @xmath619 , and then we have that for every @xmath201 and for every @xmath620 $ ] ,    @xmath621    here we replace @xmath46 for @xmath226 by @xmath46 for @xmath43 by the @xmath2-invariance of @xmath622 .    by the lebesgue dominated convergence theorem as @xmath19 and by the triangular inequality 
   , we have that for every @xmath201 and for every @xmath620 $ ] ,    @xmath623    this shows @xmath624 concentrates on @xmath370 , which is a weak solution of the quasi - linear parabolic equation ( [ pde1 ] ) . 
   furthermore , @xmath370 has finite energy by lemma[energy ] . by the uniqueness result of the weak solution lemma[uniqueness ] in section[appendixb ] , we conclude that the limit point @xmath556 of @xmath551 is unique and @xmath549 concentrates on @xmath625 as @xmath4 goes to the infinity . 
   that is , for every @xmath201 , @xmath626 where @xmath627 is the skorohod distance in @xmath193 , \mathcal{m})$ ] . in particular , since @xmath194 is arbitrary , it follows that for every @xmath203 , for every @xmath201 and for every continuous functions @xmath127 , @xmath628=0.\ ] ] it completes the proof of theorem[main ] . 
   take a @xmath51-basis @xmath629 of @xmath2 and identify @xmath2 with @xmath0 . 
   define the standard generator system of @xmath2 by @xmath630 . 
   we introduce the length function associated with @xmath631 , @xmath632 by @xmath633 for @xmath30 . 
   then the map @xmath634 induces the metric in @xmath2 , which is called the word metric associated with @xmath631 . 
   let @xmath635 be the image of @xmath629 by the natural homomorphism @xmath636 
   . then @xmath637 generates @xmath13 . the length function associated with @xmath638 , 
   @xmath639 is also defined in the same way . to abuse the notation 
   , we denote the word metric in @xmath13 associated with @xmath638 by the same symbol @xmath233 . 
   we define an @xmath640-norm in @xmath641 by @xmath642 for @xmath643 and the distance @xmath644 in @xmath12 by @xmath645 for @xmath646 . 
   denote by @xmath647 the induced metric in @xmath128 from @xmath648 . 
   fix @xmath649 and a fundamental domain @xmath650 such that @xmath651 and @xmath652 is connected in the following sense : for any @xmath653 there exist a path @xmath654 in @xmath22 such that @xmath655 and @xmath656 are all in @xmath652 . 
   this kind of set @xmath652 always exists if we take a spanning tree in @xmath3 and its lift in @xmath1 . to abuse the notation , 
   we denote by @xmath657 the images of @xmath658 by the covering map , respectively . 
   we also fix a fundamental domain @xmath659 which is identified with @xmath660 . to abuse the notation , we denote by @xmath661 the image of @xmath662 by the covering map . 
   we define the map @xmath663:v \to \gamma$ ] as follows : for @xmath39 , there exists a unique element @xmath30 such that @xmath238 since @xmath2 acts on @xmath1 freely . 
   define @xmath237:=\sigma$ ] . since there exists a constant @xmath664x_{0})-\phi(x)\|_{1}$ ] by the @xmath2-perodicity , we have that @xmath665x_{0})-\phi([z]x_{0})\|_{1 } -2c_{0 } \le \|\phi(x)-\phi(z)\|_{1 } \le \|\phi([x]x_{0})-\phi([z]x_{0})\|_{1 } + 2c_{0}\ ] ] for any @xmath666 . furthermore , since @xmath667x_{0})- \phi([z]x_{0})\|_{1}=\left|[x]-[z]\right|$ ] 
   , we have that @xmath668-[z]\right| -2c_{0 } \le \|\phi(x)-\phi(z)\|_{1 } \le \left|[x ] -[z]\right| + 2c_{0}.\ ] ]    as in section[harmonic ] , suppose that we have an injective homomorphism @xmath669 such that @xmath670 take a fundamental parallelotope @xmath671 . in the similar way to the above 
   , we define the map @xmath663:\gamma \otimes \mathbb{r } \to \gamma$ ] as follows : for @xmath672 , there exists a unique element @xmath30 such that @xmath673 . 
   define @xmath674:=\sigma$ ] . 
   since there exists a constant @xmath675x_{0})- { \bf x}\|_{1}$ ] by the @xmath2-periodicity , we have that @xmath676x_{0})-\phi([{\bf z}]x_{0})\|_{1 } -2c_{1 } \le \|{\bf x } - { \bf z}\|_{1 } \le \|\phi([{\bf x}]x_{0})-\phi([{\bf z}]x_{0})\|_{1 } + 2c_{1}\ ] ] for any @xmath677 . furthermore , we have that @xmath678-[{\bf z}]\right| -2c_{1 } \le \|{\bf x}-{\bf z}\|_{1 } \le \left|[{\bf x } ] -[{\bf z}]\right| + 2c_{1}.\ ] ]    let us define an @xmath679-ball in @xmath128 of the center @xmath680 by @xmath681 . 
   we show the following :    [ ball approximation ] there exists a constant @xmath682 depending only on @xmath679 such that for any @xmath680 and for any @xmath108 , @xmath683 here @xmath684 stands for the volume of a borel set @xmath272 and @xmath685 the cardinality of a set @xmath686 .    for any @xmath680 , take a lift @xmath687 then @xmath688x_{0})- \tilde { \bf z}\|_{1 } \le c_{1}$ ] . for sufficiently small @xmath689 , 
   take a lift @xmath690 of @xmath691 . 
   again , from the above argument , it holds that @xmath692p \subset \widetilde b^{1}_{\tilde{\bf z}}(n\epsilon ) \subset \cup_{|\sigma| \le \epsilon n + 2c_{1}}\sigma [ \tilde{\bf z}]p$ ] . 
   note that @xmath693 , and thus @xmath694p\right)\right| \le vol\left(p \right)(2^{d}/d!)\left((\epsilon 
   n + 2c_{1})^{d } -(\epsilon n -2c_{1})^{d}\right)$ ] . 
   since @xmath695p\right ) / vol\left(p\right)= \left| \cup_{|\sigma| \le \epsilon n}\sigma [ \tilde{\bf z}]d_{x_{0}}\right| / |v_{0}|$ ] and @xmath696 , it concludes that there exists a constant @xmath697 depending only on @xmath679 such that @xmath698d_{x_{0 } } \right|}{n^{d}|v_{0}|}\right| \le \frac{c(\epsilon)}{n}.\ ] ] the cardinality of the set @xmath699 is invariant under translation . 
   it completes the proof . 
   let us define a measure @xmath261 on @xmath128 by @xmath700 . 
   let @xmath701 be a characteristic function defined by @xmath702 for the empirical density @xmath703 on @xmath128 , @xmath163 , then we have the following lemma . 
   [ characteristic function ] there exists a constant @xmath704 depending only on @xmath689 , such that for any @xmath163 and any @xmath705 , @xmath706d_{x_{0}}\right|}\sum_{x \in \cup_{|\sigma|\le \epsilon n}\sigma [ z]d_{x_{0 } } } \eta_{x}\right|   \le   \frac{c(\epsilon)}{n},\ ] ] where @xmath707    take a lift @xmath708 of @xmath705 and a lift @xmath709 of @xmath710 . in the similar way to the proof of lemma[ball approximation ] , we obtain that @xmath711d_{x_{0 } }   \subset   \left\{x \in v \ \bigg| \ 
   \left\|\frac{1}{n}\phi(x ) - \frac{1}{n}\phi(\tilde z)\right\|_{1 } \le \epsilon \right\ } \subset   \bigcup_{|\sigma|\le \epsilon 
   n + 2c_{0}}\sigma [ \tilde z]d_{x_{0}}.\ ] ] furthermore , we take a lift @xmath712 of @xmath163 , then it holds that @xmath713 and @xmath714d_{x_{0 } } } \widetilde \eta_{x}\right| \le   \frac{1}{|v_{n}|}\sum_{x \in \cup_{\epsilon n - 2c_{0 } \le |\sigma|\le \epsilon n + 2c_{0}}\sigma [ \tilde z]d_{x_{0}}}\widetilde \eta_{x}.\ ] ] the last term is bounded by @xmath715 , and thus there exists a constant @xmath716 depending only on @xmath679 such that @xmath714d_{x_{0 } } } \widetilde \eta_{x}\right| \le   \frac{c_{1}(\epsilon)}{n}.\ ] ] by lemma[ball approximation ] and @xmath717d_{x_{0 } } \right|=\left|\cup_{|\sigma| \le \epsilon n}\sigma d_{x_{0 } } \right|=\left|\cup_{|\sigma| \le \epsilon n}\sigma[z]d_{x_{0 } } \right|$ ] , @xmath718d_{x_{0 } } \right|}\right|   & \le \frac{1}{|v_{n}|\mu\left(b^{1}_{\phi_{n}(z)}(\epsilon)\right)}\frac{1}{\left|\cup_{|\sigma| 
   \le \epsilon n}\sigma[z]d_{x_{0 } } \right|}\frac{c(\epsilon)}{n } |v_{n}|      \\ & \le   \frac{c_{2}(\epsilon)}{n^{d+1}},\end{aligned}\ ] ] where @xmath719 is a constant depending only on @xmath679 . finally , @xmath720d_{x_{0}}\right|}\sum_{x 
   \in \cup_{|\sigma|\le \epsilon n}\sigma [ z]d_{x_{0 } } } \eta_{x}\right| & \le \frac{1}{\mu\left(b^{1}_{\phi_{n}(z)}(\epsilon)\right)}\frac{c_{1}(\epsilon)}{n } + \frac{c_{2}(\epsilon)}{n^{d+1}}|v_{0}|(\epsilon n)^{d } \\ & \le \frac{c_{3}(\epsilon)}{n},\end{aligned}\ ] ] where @xmath721 is a constant depending only on @xmath679 
   . it completes the proof . 
   [ energy ] suppose that @xmath722 is a sequence of probability measures on @xmath193,\mathcal{m})$ ] . 
   for any limit point @xmath723 of @xmath722 , @xmath723-a.s . 
   there exists a measurable function @xmath205 such that @xmath724 , @xmath370 has @xmath725\times \mathbb{t}^{d}),\ ] ] for @xmath726 , and satisfies @xmath727 for every @xmath728\times \mathbb{t}^{d})$ ] and @xmath726 .    for fixed @xmath729 , 
   we define a lattice @xmath730 whose vertex set @xmath731 is the subset of @xmath732 in the following : @xmath731 is the orbit of @xmath390 by @xmath13 , i.e. , @xmath733 . define @xmath734 the set of oriented edges @xmath213 such that @xmath735 for some @xmath736 . 
   then @xmath13 acts on @xmath737 naturally . 
   a configuration @xmath738 on @xmath7 induces the one on @xmath737 by restriction . 
   we use the same symbol @xmath738 for this restriction . for @xmath739\times 
   \mathbb{t}^{d})$ ] and for @xmath740 , we define @xmath741,\mathbb{r})$ ] , @xmath742 and @xmath743 for any @xmath284 , we have that @xmath744 we use the cauchy - schwarz inequality in the last inequality . 
   by using the argument in the proof of lemma[path ] , there exists a constant @xmath580 such that @xmath745    since @xmath746 and @xmath747 , we get @xmath748 note that @xmath749 for large enough @xmath4 . consider for @xmath296 the self - adjoint operator @xmath750 and suppose @xmath751 to be the largest eigenvalue of this operator . by the variational formula @xmath752    by the simple inequality for @xmath753 , @xmath754 , the last formula is bounded by @xmath755 . 
   we put @xmath756 . on the other hand , @xmath757 where @xmath758 is the directional derivative along @xmath759 and @xmath760 .    by the entropy inequality , @xmath761 since @xmath762 for some constant @xmath531 , by the feynman - kac formula 
   , we obtain @xmath763 \times \mathbb{t}^{d})}^{2 } + |v_{0}|c'.\ ] ] for a limit point of @xmath764 , @xmath723 , we get @xmath765 \le 16ct \|j\|_{l^{2}([0,t ] \times \mathbb{t}^{d})}^{2}+|v_{0}|c'.\ ] ] denote a countable dense subset of @xmath766\times \mathbb{t}^{d})$ ] by @xmath767 , we also get the following estimate : @xmath768 \le |v_{0}|c'.\ ] ] see @xmath769 pp.107 , section 5.7 for details 
   . therefore for almost all @xmath370 , there exists @xmath770 such that for every @xmath728\times \mathbb{t}^{d})$ ] , @xmath771 that is , @xmath772 this implies the linear functional @xmath773\times \mathbb{t}^{d } ) \to \mathbb{r}$ ] defined by @xmath774 is extended on @xmath775\times\mathbb{t}^{d})$ ] . by the riesz representation theorem 
   , there exists @xmath776\times\mathbb{t}^{d})$ ] such that @xmath777 for every @xmath728\times \mathbb{t}^{d})$ ] and every @xmath778 
   . this yields lemma[energy ] . 
   [ uniqueness ] for any @xmath779\times \mathbb{t}^{d})$ ] , a weak solution of the quasi - linear partial differential equation @xmath780 with the measurable initial value @xmath199 $ ] , of bounded energy , i.e , @xmath781 is unique .      first , the author would like to thank professor tsuyoshi kato for his constant encouragement and helpful suggestions . 
   he wishes to express his gratitude to professors motoko kotani , nobuaki sugimine , satoshi ishiwata and makiko sasada for their valuable advice and helpful discussions . 
   second , he would like to thank professor yukio nagahata for his helpful comments on this subject and the background of the hydrodynamic limit . 
   third , he would like to thank doctors sei - ichiro kusuoka and makoto nakashima for their advices on the presentation of this paper . 
   fourth , he would like to thank professor kazumasa kuwada for his valuable comments on the earlier version of the manuscript and his encouragement . the author could not write up this paper without a great deal of his advice . 
   fifth , the author partially carried out this work at riken center for developmental biology in kobe . 
   he would like to thank professors hiroki r. ueda and yohei koyama for their interest in this topic and their encouragement . 
   sixth , the author partially carried out this work at max planck institute for mathematics in the sciences in leipzig . 
   he would like to thank professors jrgen jost and nihat ay for helpful discussions . 
   the author is supported by the research fellowships of the japan society for the promotion of science for young scientists . 
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summary: 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 .