Hydrodynamic limits for long-range asymmetric interacting particle systems

We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${\mathbb Z}^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+\alpha)}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<\alpha<1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $\alpha\geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.


Introduction
In this paper, we consider hydrodynamic limits in a class of mass conserving particle systems in several dimensions n ≥ 1 on Z n with certain asymmetric long-range interactions. These limits, when they exist, capture the space-time scaling limit of the microscopic empirical mass density field of the particles as the solution of a 'hydrodynamic' equation governing a macroscopic flow. When the interactions are symmetric and finite-range, such limits have been shown in a variety of stochastic particle systems (cf. [9], [20], [28]). Also, when the interactions are asymmetric and finite-range, for systems, such as 'simple exclusion' and 'zero-range', as well as other processes, hydrodynamics has been proved (cf. [2], [3], [4], [14], [16], [17], Chapter 8 in [20], [24], and reference therein).
However, less is known about hydrodynamics when the dynamics is of long-range type, although such processes are natural in applications, for instance with respect to wireless communications. The only works, to our knowledge, which considers 'longrange' limits are [5] and [19], where hydrodynamics of types of symmetric, long-range exclusion and zero-range processes was shown.
In this context, our main purpose is to derive the hydrodynamic equation in a general class of asymmetric long-range particle models, which includes simple exclusion and zero-range systems. Another motivation was to understand if there is a 'mode coupling' basis for certain 'stationary' fluctuation results in asymmetric long-range models seen in [6], [25]. There, the fluctuations of the empirical mass density field, translated by characteristic speeds, was shown to obey in a sense either a stochastic heat or Burgers equation, depending on the strength of the long-range interactions. One may ask whether such fluctuations could be inferred from associated hydrodynamics through mode coupling analysis (cf. [28]), as is the case with respect to asymmetric finite-range systems.
Informally, the particle systems studied follow a collection of dependent random walks which interact in various ways. For instance, in the exclusion and zero-range particle systems, the random walks interact infinitesimally in time and space respectively. In the exclusion process, particles move freely except in that jumps, according to a jump probability p(·), to already occupied locations are suppressed. Whereas, in the zero-range process, the jump rate of a particle at a site depends on the number of particles at that site, but the location of the jump is freely chosen according to p(·).
In this article, we will consider a general form of the 'misanthrope' process, for which features of exclusion and zero-range interactions are combined, so that both the jump rate and location of jump may depend infinitesimally on the other particles.
In such dynamics, as mass is preserved, that is no birth or death allowed, there is a family of product invariant measures ν ρ indexed by density ρ. Let η t (x) denote the number of particles at location x at time t. By 'long-range', to be concrete, we mean, for α > 0 and d ∈ Z n , that p(·) takes the form where d > 0 means d i ≥ 0 for 1 ≤ i ≤ n and d = 0. The form we have chosen may be generalized as discussed in Subsection 3.1.
We will start the process in certain 'local equilibrium' nonstationary states µ N , that is when initially particles are put independently on lattice sites, according to a varying mass density ρ 0 , where the marginal at vertex x has mean ρ 0 (x/N ), and N is a scaling parameter. We will restrict attention to initial densities ρ 0 such that the relative entropy of µ N with respect to an invariant measure ν ρ * for ρ * > 0 is of order N n . In effect, this means ρ 0 = ρ 0 (u) is a function which equals a constant ρ * for all u large. This restriction is further discussed in Subsection 3.1.
The choice of θ is usually determined by the time needed in order for a single particle to travel a microscopic distance of order N n , or a nonzero macroscopic distance. When α > 1, as p(·) has a mean, the travel time is of the same order as in the finite-range asymmetric case, namely of order N , indicating θ = 1, the 'Euler' scale. While, when 0 < α < 1, because of the heavier tail in p(·), the travel time is of shorter duration, and it turns out θ should be taken as θ = α, an anomalous scale, interestingly the same as in [19] when the jumps are symmetric. However, in the case α = 1, time should be speeded up by N/ log N Our main results are as follows. When 0 < α < 1 (Theorem 3.1), we derive that the hydrodynamic equation is a weak form of ∂ t ρ(t, u) = [0,∞) n F (ρ(t, u − v), ρ(t, u)) − F (ρ(t, u), ρ(t, u + v)) v n+α dv, Hydrodynamics for long-range asymmetric systems Finally, we comment, although we have chosen the 'entropy' method of proof of hydrodynamics, that there are other techniques, such as 'compensated compactness' (as in [3]) and 'relative entropy' (cf. Chapter 6, [20]), which might be explored with profit to treat related and different scenarios.
The structure of the article is as follows. In Section 2, we introduce the processes studied and, in Section 3, we state our main results, Theorems 3.1 and 3.2, and related remarks. After some preliminaries in Section 4, we prove Theorem 3.1 in Section 5, relying on 1 and 2-block estimates shown in Section 6. In Section 7, we prove Theorem 3.2, stating key inputs, Theorems 7.2, 7.3, 7.4, and 7.5, which are then proved in Sections 8,10,11,12, with the aid of estimates in Section 9 and the Appendix.

Models
Let N 0 = N ∪ {0}. We will consider a class of n ≥ 1 dimensional 'misanthrope' particle systems evolving on the state space X = N Z n 0 , which includes simple exclusion and zero-range systems. The configuration η t = {η t (x) : x ∈ Z n } gives the number of particles η t (x) at locations x ∈ Z n at time t. Let p : Z n → [0, ∞) be a single particle transition rate such that d p(d) < ∞. We say that a function f : X → R is local if it depends only on a finite number of occupation variables {η(x) : x ∈ Z n }.
In the simple exclusion process, at most one particle may occupy each site, η(x) = 0 or 1 for all x ∈ Z n . Informally, each particle carries an exponential rate 1 clock. When a clock rings, the particle may displace by d with probability proportional to p(d). If the destination site is empty, this jump is made, however, if the proposed destination is occupied, the jump is suppressed and the clock resets, hence the name 'simple exclusion'. Formally, the system is the Markov process on {0, 1} Z ⊂ X with generator action on local functions given by where η x,y is the configuration obtained from η by moving a particle from x to y: See [23] for the construction and further details of the simple exclusion process.
In the zero-range process, however, any number of particles may occupy a site.
Informally, each site x holds an exponential clock with rate g(η(x)), where g : N 0 → R + is a fixed function, such that g(0) = 0 and g(k) > 0 for k ≥ 1. When a clock rings, from that site a particle at random displaces by d with chance proportional to p(d). The name 'zero-range' comes from the observation that, infinitesimally, particles interact only with those on the same site. Formally, the zero range process is a Markov process on X with generator action on local functions given by To construct the zero range process, we require that g be Lipschitz. See [1] for the construction and further details of the zero range process.
In this article, we concentrate on 'decomposable' misanthrope systems, where b(l, m) = g(l)h(m), in terms of functions g and h, satisfying the restrictions on b(·, ·) above. Such a class is large enough to include exclusion and zero-range processes, yet concrete enough to streamline later proofs. We comment, although not pursued here, that more general misanthrope systems might also be considered with more involved estimates and notation.
The associated generator action reduces to the form To aid in construction of the process and for other estimates, we will impose that (i) g is Lipschitz: |g(k + 1) − g(k)| ≤ κ for k ≥ 0, (ii) h be bounded, in which case, h is also Lipschitz, |h(k + 1) − h(k)| ≤ κ 1 := 2 h ∞ for k ≥ 0, and (iii) |g(a)h(b) − g(u)h(v)| ≤ κ 2 |a − u| + |b − v| for a, b, u, v ≥ 0. The last condition (iii) is a sufficient ingredient to construct the process, and forces g to be bounded if h is nontrivial (cf. equation (7.1) in [18]). However, it is not a necessary condition, and will not be used in the main body of the paper.
Since g(0) = 0, we have g(l) ≤ κl. We also have h(0) > 0. If h has a zero, and M 0 < ∞ is the first root, then h(m) = 0 for m ≥ M 0 ; in this case, the process, starting with less than M 0 particles per site, remains so in the evolution.

Long range asymmetric transitions
In this article, we concentrate on 'long range' totally asymmetric processes, where p(·) is in form Here, · is the Euclidean norm, and d = (d 1 , ..., d n ) > 0 means d i ≥ 0 for all i, but d = 0. We require α > 0 so that d p(d) < ∞. Although more general transition rates can be treated, as discussed in Subsection 3.1, the form of p chosen allows for simplified notation and encapsulates the complexity of the more general situation.
We will distinguish three cases α < 1, α = 1, and α > 1. The transition rate p(·) has a finite mean exactly when α > 1, and the corresponding model shares some of the properties of the finite-range situation where p is compactly supported. However, when α < 1, the behavior of the associated process does reflect that long jumps are more likely. The α = 1 case, although borderline, turns out in some ways to be similar to the α > 1 case.
In particular, a random walk with transition rate p(·) will take an order γ N steps to travel an order N distance on Z n where These orders will be relevant when discussing hydrodynamic space-time scaling of the process.

Invariant measures
As the decomposable misanthrope system is mass conservative, one expects a family of invariant probability measures ν ρ indexed by particle density ρ ≥ 0. In fact, there is a family of translation-invariant product stationary measures, for a general class of misanthrope processes, including the long-range asymmetric decomposable models, when and i, j ≥ 0 otherwise, which we will also assume (cf. [18]).
In the case h is nontrivial, this implies a linear relation between g and h.
To specify the marginal Θ ρ of the measure ν ρ = x∈Z n Θ ρ , consider the probability measureΘ λ on N 0 given byΘ where Z(λ) is the normalizing constant. Let ρ(λ) = k≥0 kΘ λ (k) be the mean ofΘ λ . Both Z(λ) and ρ(λ) are well defined for 0 ≤ λ < λ c , where λ c = ∞ if M 0 < ∞ and λ c = lim inf k↑∞ g(k)/h(k) otherwise. One can see they are strictly increasing on this range, and so invertible. Let ρ c = lim λ→λc ρ(λ). Now, for density ρ The functions, will play important roles in the sequel. One can observe that Φ, Ψ are C ∞ on their domains.
Moreover, as g and h are Lipschitz, both Φ and Ψ will be Lipschitz. (following the method of Corollary 2.3.6 [20]). Also, note by boundedness of h that Ψ ≤ h ∞ . Hence, we have the following inequalities, In later calculations, we will need finite exponential moments of η(x) and g(η(x)) with respect to ν ρ for ρ ∈ [0, ρ c ). Since g(η(x)) ≤ κη(x) for some constant κ > 0, we note g(η(x)) will have a finite exponential moment if η(x) does. We say that the FEM satisfied for all γ ≥ 0 and ρ ∈ [0, ρ c ). FEM is a condition on the rates g and h, which we will assume holds throughout. For instance, if lim k→∞ h(k)/g(k + 1) = 0 or M 0 < ∞, then FEM holds.
We will also assume that ρ c = M 0 , a condition on the rates g and h that ensures a stationary measure at each possible density.
To relate with zero-range and simple exclusion, if we set h ≡ 1, the measure ν ρ reduces to the well known family of invariant probability measures for the zero-range process. However, when h(1) = 0, we recover that ν ρ is the Bernoulli product measure with parameter ρ ∈ [0, 1]. Finally, we remark, with respect to ν ρ * , one may construct an L 2 (ν ρ * ) Markov process as in [26]. The associated adjoint L * may be computed as the generator of the process with reversed jump rates p * (d) = p(−d) for d ∈ Z n .

Initial, empirical and process measures
We will examine the scaling behavior of the process as seen when time is speeded up by γ N and space is scaled by parameter N ≥ 1. Let L N := γ N L and process η N t := η γ N t for t ≥ 0. Let T N We will focus on the cases α = 1, discussing the case α = 1 in Subsection 3.1. Then, for α = 1, we have γ N = N α∧1 . The space-time scaling is the 'Euler' scaling when α > 1, but is an anomalous scale when α < 1.
Define, for t ≥ 0, the empirical measure We will use the following notation for spatial integration against test functions G: x N .
For T > 0 fixed, the measure-valued trajectories {π N t : 0 ≤ t ≤ T } are in the Skorohod space D([0, T ], M + (R n )), where M + (R n ) is the set of positive Radon measures on R n endowed with the vague topology.
Suppose that we start the process at level N according to an initial measure µ N . We denote the distribution at times t ≥ 0 by µ N t := µ N T N t . The initial measures that we will use are such that the law of large numbers holds in probability with respect to an initial density profile ρ 0 as N → ∞: We will assume that ρ 0 : R n → R is continuous, and the range of ρ 0 lies in [0, ρ c ). We will also suppose that µ N is a product measure, whose marginal at x ∈ Z n is Θ ρ0(x/N ) with mean ρ 0 (x/N ). Moreover, we will assume that the relative entropy of µ N with respect to an invariant measure ν ρ * for 0 < ρ * < ρ c is of order N n . Then, for x large, the marginals of µ N , in this case would be very close to those of ν ρ * and, in particular, ρ 0 (x) ∼ ρ * . For convenience, throughout, we will assume that ρ 0 is such that it equals ρ * outside a compact set. We also remark the measures {µ N }, by their definition, are stochastically bounded by Define also {P N } N ≥1 to be the sequence of probability measures on the Skorohod space D([0, T ], M + (R n )), governing π N · when the process η · starts from µ N . Expectation with respect to P N will be denoted as E N .

Additional assumption when α > 1
We will assume further the following condition, in force only when α > 1. Namely, we will assume the misanthrope process is 'attractive', that is b(n, m) is increasing in n and decreasing in m. In other words, the 'decomposable' process is 'attractive' when g is increasing and h is decreasing.

Model assumptions summary
To summarize, we gather here some of the assumptions on the rates and initial measures discussed earlier.
2. The following ensure product stationary measures indexed by all possible densities with finite exponential moments.
3. The rate p is long-range and is in form (2.1).
4. The initial measures µ N satisfy the following.
is a continuous function such that ρ 0 (u) = ρ * for u outside a compact set, and 0 < ρ * < ρ c ; • the relative entropy H(µ N |ν ρ * ) = O(N n ). 5. The parameter α is such that α = 1 in the main body of the article. Remarks about α = 1 are in Subsection 3.1.
6. When α > 1, the rates are 'attractive', that is g is increasing and h is decreasing.

Results
We will split the main results according to the settings α < 1 and α > 1. The case α = 1 is discussed in the remarks in Subsection 3.1.
Suppose α < 1. and consider the operator L acting on smooth, compactly supported test functions G : R n → R by Note that the integral in (3.1) is well-defined as the integrand is O( v −(n+α−1) ) for v near the origin. Let also Theorem 3.1. Suppose α < 1. Then, the sequence {P N } N ≥1 is tight, and every limit point P * is supported on absolutely continuous measures π t = ρ(t, u)du whose densities are weak solutions of the hydrodynamic equation ∂ t ρ = L(ρ) with initial condition ρ(0, u) = ρ 0 (u). That is, for test functions G with compact support in [0, T ) × R n , so that G(T, ·) ≡ 0, We now assume α > 1, and state the hydrodynamic limit in this setting.
Theorem 3.2. Suppose α > 1, and in addition that the process is 'attractive'. Then, {P N } N ≥1 converges weakly to the point mass supported on the absolutely continuous measure π t = ρ(t, u)du whose density is the weak entropy solution (cf. (3.3)) of the hydrodynamic equation with initial condition ρ(0, u) = ρ 0 (u). Here, 1(n) is the unit vector in the direction 1, ..., 1 , and γ α is the constant defined by γ α = We comment, as is well known, the scalar conservation law (3.2) may not have a classical solution for all times. However, a weak solution ρ(t, u) exists (cf. [10], [15]): We say that a weak solution ρ(t, u) is a 'weak entropy' solution if in the weak sense, with respect to nonnegative test functions G with compact support in [0, T ) × R n , that Kruzkov proved that there is a unique bounded weak entropy solution if ρ 0 is bounded, which is implied by our assumptions [21]. See [10], [15], and [21] for further discussion about weak entropy solutions.

Remarks
We now make several remarks about Theorems 3.1 and 3.2.

1.
Uniqueness of solution. When α < 1, an open question is to understand in what sense a weak solution is unique. If there is an unique weak solution ρ(t, u) of the hydrodynamic equation, then P N would converge weakly to δ ρ(t,u)du . However, it is not clear what additional criteria, if at all, as in the finite-range or α > 1 setting, needs to be imposed to ensure an unique weak solution.
In this context, we note, in [19], for certain attractive long-range symmetric zerorange evolutions, with symmetric jump rate p sym (d) = p(d) allows variation in all directions, as opposed to (3.1). Uniqueness of weak solution, under an 'energy' condition, is shown there. Symmetric long-range exclusion processes are also considered in [19]. However in such models, as is well known, the hydrodynamic equation is linear, and so uniqueness of solution is more immediate.

2.
General jump rates. The jump rate p(·) may be generalized to a larger class, in which jumps are allowed in all directions. When α < 1, the jump rate can be in form, say is the standard basis. We note, in this case, p gen may even be symmetric as in [19]. However, when α > 1, the same generalization is allowed, except the jump rate must have a drift, dp gen (d) = 0.
Under these generalizations, the form of Theorems 3.1 and 3.2 remain the same except that the hydrodynamic equation now involves straightforwardly the constants and when α > 1, in (3.2) and (3.3), γ α ∂ 1(n) is replaced by dp gen (d) · ∇. The proofs are the same, albeit with more notation.

3.
Case α = 1. Although we assume throughout that α = 1 and do not consider the case α = 1 in the sequel, we remark, when α = 1, a log correction is needed in the definition of the empirical measure since the jump rate p does not have mean, but just 'barely' so in that d ≤N dp(d) = O(log(N )). In this case, instead of π N t , we should use the rescaled measures 1 N n x∈Z n η (N/ log(N ))t (x)δ x N . The arguments, when α > 1, are straightforwardly adapted to yield the equation ∂ t ρ + ∂ 1(n) [Φ(ρ)Ψ(ρ)] = 0, here γ α being replaced by 1.

4.
Long-range communication α > 2 versus 1 ≤ α ≤ 2. In the Euler scale, when α > 2, as opposed to when 1 < α ≤ 2, the influence from long distances to the origin, say, is minimal. From considering the single particle displacement rates, the chance a particle displaces by order N is of order N 1−α . So, the likelihood of a particle a distance of order N or more away from the origin to pass by is minimal when α > 2, but this chance it appears is nontrivial when 1 < α ≤ 2.
In this case, it seems not possible to overestimate the chance of travel of particles located at sites x where |x| ≥ cN to an N neighborhood of the origin by a convergent sum as in [24]. In particular, it is not clear how to use the method in [24] to approximate the process starting from L 1 initial densities by those starting from arbitrary initial states. Please see Lemma 5.7 in [24] for details about this approximation method.
Hence, rather than start in an L 1 density ρ 0 , under which the system would have only a finite number of particles at each scaling level N as in [24], we have tried to understand infinite volume effects, using the 'entropy' method, by starting in a non-integrable density ρ 0 . That ρ 0 (u) = ρ * for large u is a consequence of this method.

5.
Use of 'attractiveness' when α > 1. Only for the proof of Theorem 3.2 is 'attractiveness' used. This condition allows to show in Step 1 of Section 7 that solutions ρ are in L ∞ when M 0 = ∞. However, when M 0 < ∞, we have a priori that ρ ∈ L ∞ and 'attractiveness' is not needed for this point.
On the other hand, 'attractiveness' is used to rewrite the generator of a coupled process in (9.2), and then to bound it in Lemma 9.1. These are important ingredients for the 'ordering' Lemma 9.3, which is used to show a 'measure weak' formulation of the entropy condition in Theorem 7.3, proved in Section 10.
6. Initial conditions. Only in the proof of Theorem 3.2 is the full description of the initial measures µ N used. In particular, the full structure is employed in Step 3a in Subsection 10.1, for the proof of the entropy condition inequality. However, with respect to the proof of Theorem 3.1, we note only the fact that the marginals of µ N at x ∈ Z n have mean ρ 0 (x/N ) is used.

Preliminaries
Throughout this paper, a test function will be a smooth C 1,2 function G : [0, T ) × R n → R with compact support. Typically, given a test function G, we will denote, in terms of the letter R, that its support lies in [0, T ) × [−R, R] n . Define G = sup t,u |G(t, u)|, and similarly ∇G , ∇ 2 G and ∂ s G . Often, we will write G t (x) for G(t, x) in the sequel.
Define also |y| = max{y 1 , ..., y n } for y = (y 1 , ..., y n ) ∈ Z n . In later calculations, we will use the notion of an 'l-block' average of a function f = f (η): That is, define In particular, η l (x) = 1 (2l+1) n |y|≤l η(x + y). Form now the mean-zero martingale with respect to G t , π N t : Also, with respect to its quadratic variation, t is a mean-zero martingale. Explicitly, we may compute Here, and in the body of the paper, our convention will be that the sums over d implicitly contain the restriction that d = (d 1 , . . . , d n ) > 0, that is d i > 0 for 1 ≤ i ≤ n, as p is supported on such d, to reduce notation.

Entropy and Dirichlet forms
Recall the distribution of the process at the N th level at time t ≥ 0, Consider the relative entropy H(µ N t |ν ρ * ) of µ N respect to ν ρ * . In terms of the Radon-Nikodym derivative Recall the adjoint L * defined in Subsection 2.2. Define now the Dirichlet form of a density f by D(f ) = − √ f L sym √ f dν ρ * , where we define L sym = (L + L * )/2 as the symmetric part of L. We will on occasion define new Dirichlet forms in terms of pieces of the above Dirichlet form. For x, y ∈ Z n , define the bond Dirichlet form as where p sym (d) = (p(d) + p(−d))/2. By properties of ν ρ * , one can calculate D x,y (f ) = D y,x (f ). Roughly speaking, D x,y (f ) is a measure of how much f (η) can vary as one particle is moved from x to y or vice versa. In particular, if D x,y (f ) = 0, then f (η) = f (η x,y ) when p(y − x)g(η(x))h(η(x + y)) = 0. In terms of these bond forms, the 'full' Dirichlet form may be written as D(f ) = (1/2) x,y D x,y (f ).
One may relate the entropy and Dirichlet form as follows, justification below: t t 0 f N s ds. Moreover, by our entropy assumption on the initial distributions {µ N }, and with C 0 = C/(2t), we have In the finite volume, (4.2) and (4.3) are well-known (cf. Chapter 5 in [20]). In the infinite volume, to obtain finiteness of the relative entropy, In particular, by the construction estimates in Sections 8 of [18], for Lipschitz functions u on the complete, separable metric space X 0 , we have T N,R t u → T N t u as R ↑ ∞; also, as |T N,R t u(η)|, |T N t u(η)| ≤ c u e crt η X0 + |u|(0) ∈ L 1 (µ N ), where c u is the Lipschitz constant with respect to u, 0 is the empty configuration, and c r is a constant depending on process parameters, we have the 'convergence', , as R ↑ ∞. Therefore, µ N,R t converges weakly to µ N t by the Portmanteau theorem (cf. Section 3.9 in [13]).
Now, note that the localized Dirichlet form is greater than the Dirichlet form D K involving only bonds in a fixed box with width K for all large R, and that such fixed forms increase as K grows to the full one. We claim that the form D K is lower semi- is Lipschitz in X 0 by use of the construction assumption (iii; see also 1. in Subsection 2.5), and E ν ρ * [f (η x,x+y )g(η(x))h(η(x + y))] = E ν ρ * [f (η)g(η(x + y))h(η(x))] (applied with f = dµ N,R s /dν ρ * and dµ N s /dν ρ * ). With these ingredients, it is straightforward to conclude (4.2) and (4.3). See also [22] and references therein for related approaches.
Recall now the 'entropy inequality' (cf. Appendix 1 in [20]): For γ > 0, and bounded or nonnegative f , A common application of the entropy inequality is to bound the numbers of particles in various sets.
Proof. By the entropy inequality (4.4), and finite exponential moments FEM, the left-side of the display, for γ > 0, is bounded by For later reference, we state the following 'truncation' bounds, which holds under FEM, using also the entropy inequality; see p 90-91 in [20].

Generator and martingale bounds
We now collect a few useful bounds. Let σ n be the surface area of the part of an unit radius n-sphere, centered at the origin, contained in the first orthant. In this subsection, to make expressions compact, we will adopt the convention that A(x) orB(x) is a sum Before going to the proof, we remark that we have made precise the constant C G , especially its dependence on R, as it will be of use in a later estimate (cf. Lemma 11.1).
Proof. First, as h is bounded and g is Lipschitz, by (2.3), we have The sum over d can be divided into a sums over 1 ≤ d ≤ N and d > N . We may bound G s Both sums over x add over at most 2((R + 1)N ) n sized regions. Hence, by Lemma 4.1, the expected value of both sums are less than 2K(R + 1) n N n . Also, the sums over d can be bounded as follows: We note also, an alternate bound, Then, We state here straightforward corollaries of the proof of Lemma 4.3, adjusting the values of a and b in the sums over d near (4.6).

Lemma 4.4.
We have, when α < 1, that The difference of quadratic variations defined in (4.1) can be bounded as follows: However, we have already bounded this expression in the proof of Lemma 4.3 by C G .

Tightness of {P N }
We now show, when α = 1, that the sequence {P N } is tight and therefore weakly relatively compact. For smooth G with compact support, let P N G be the induced distribution [20]). We will in fact show sufficient tightness estimates with respect to the uniform topology, stronger than the Skorohod topology.
Proposition 4.7. The sequence {P N G } is tight with respect to the uniform topology: For smooth G with compact support in R n , the following holds.

For every
But, by Lemma 4.1, To prove the second condition, for t > s, we may write The second term on the right-side of (4.8) is bounded through the triangle inequality, Doob's inequality, and the quadratic variation estimate Lemma 4.6: For the first term on the right-side of (4.8), as is done in the proof of Lemma 4.3, we bound the integrand by (4.5). We now analyze the first term in (4.5); the other term is similarly handled. Write the first term as I 1 + I 2 , in terms of a parameter A, where We may bound I 1 , as in the proof of Lemma 4.3, by For the term I 2 , we use the following approach. For each δ, = 0, to finish the proof.

Proof outline: hydrodynamic limits when α < 1
We outline the proof of Theorem 3.1, refering to '1 and 2-block' estimates later proved in Section 6.
Step 1. First, by Doob's inequality and the quadratic variation bound Lemma 4.6, for As G has compact support, we may choose t < T large enough so that G t , and hence π N t , G t vanishes. Therefore, for such t, Step 2. Next, in order for π N 0 , G 0 + t 0 π N s , ∂ s G s ds + t 0 N α L N π N s , G s ds to look like the weak formulation of a hydrodynamic equation, we will replace t 0 N α L N π N s , G s ds by appropriate terms. Noting the generator expression near (4.1), We now truncate the sum over d to when d is at least N and at most DN . By Lemma 3)), the excess vanishes, where ↓ 0 and D ↑ ∞, after N ↑ ∞. Therefore, after limits on N , and D are taken in order, vanishes in probability. Here, and elsewhere, we write N and DN for N and DN . We remark that one may link D to by specifying D = −1 in what follows. We have chosen however to separate the parameters to highlight their roles. We also comment that the truncations on d are of use to bound quantities such as ∇ α,d G s (x/N ) in Step 3c, and others in the proofs of the 1 and 2-block estimates later quoted in Step 3.
Step 3a. We will now like to replace the nonlinear terms 'gh d (η s (x))' by functions of the empirical measure π N s .
The first replacement involves substituting gh d (η s (x)), with its average over l-blocks: (gh d ) l (η s (x)), where l diverges after N diverges, but before the limits on and then D. By a discrete integration-by-parts, smoothness and compact support of G, the error introduced is of the expected order which vanishes, noting Lemma 4.1.
Therefore, we have, as N ↑ ∞, l ↑ ∞, ↓ 0 and D ↑ ∞, that Step 3b. Next, we perform what is usually called the '1-block' replacement. Recall the the 'averaged' function of the local mass density. That is, we wish to show . By discrete integration-by-parts and bounding G(x/N ) by 1(|x| ≤ RN ) G , it will be enough to show that both vanish as N ↑ ∞ and then l ↑ ∞ for fixed , D. This is proved as a consequence of Proposition 6.1 in Subsection 6.1.
After this 1-block replacement, we have Step 3c. The final estimate is the so-called '2-blocks' replacement, where η l s (x) is replaced by η N s (x) in terms of a parameter . We will write N instead of N throughout.
Hydrodynamics for long-range asymmetric systems That is, we will like to show for fixed and D, as in order N ↑ ∞, ↓ 0 and l ↑ ∞, that |. Then, to show the 2-blocks replacement, it will enough to show, for fixed , D that This is a consequence of Proposition 6.3 in Subsection 6.2.
We now observe that an N -block is macroscopically small, and may written in terms of π N s as follows: vanishes in probability as N ↑ ∞ and ↓ 0, for fixed , D.
Step 4. We may replace the Riemann sums with integrals limited by and D. As Φ, Ψ are Lipschitz and Ψ is bounded (cf. (2.3)), and as ∇ α,d G s is smooth, the error accrued is Further, we may then replace the limits in the integrals by 0 and ∞, respectively.
The error of this replacement, comparing to Riemann sums, vanishes by Lemma 4.4, as converges to zero in probability as N ↑ ∞ and ↓ 0.
Step 5. Now, according to Proposition 4.7, the measures {P N G } are tight, with respect to uniform topology. Let {N k } be a subsequence where the measures converge to a limit point P * . The function of π, Step 6. Now, we claim that P * is supported on on measures π s that are absolutely continuous with respect to Lebesgue measure, and so π s = ρ(s, u)du for an L 1 loc function ρ(s, u). Indeed, this follows, under condition FEM, with the same proof given for zerorange processes on pages 73-75 of [20]. We also have π 0 , G 0 = ρ 0 , G 0 from our initial conditions. Hence, π s , ι (· As Ψ is bounded, the second term on the right-side is bounded by 2κ ψ ∞ ρ(s, u). Note also that sup w∈R n E P * t 0 | π s , 1(| · −w| ≤ R) ds = sup w∈R n E P * t 0 |u−w|≤R ρ(s, u)duds < ∞ by Lemma 4.1 and lower semi-continuity in π of the associated mapping. Then, as G has compact support, by the L 1 loc convergence, and use of dominated convergence, we have, with respect to each limit point P * , a.s.
Since G has compact support in [0, T ) with respect to time, we may replace t by T . In other words, every limit point P * is supported on absolutely continuous measures, π s = ρ(s, u)du, whose densities ρ(s, u) are weak solutions of the hydrodynamic equation.
This concludes the proof of Theorem 3.1.

1-block and 2-block estimates
We discuss the 1 and 2 block estimates when α < 1, and also a 1-block estimate when α > 1 in the next three subsections.
6.1 1-block estimate: α < 1 We now prove the 1-block replacement used in Section 5. As a comment, in Step 3, due to the long range setting, we use a somewhat nonstandard estimate.
Proof. The proof goes through a few steps.
Step 1. We first introduce a truncation. As both |h|, |Ψ| ≤ h ∞ , and both g, Φ are Lipschitz, we can bound H 1 Once again, by Lemma 4.2, as |H 1 It will be enough to show, for each A, as N ↑ ∞ and l ↑ ∞, that the following vanishes, Step 2. Recall the densityf N t in Subsection 4.1. The expected value above equals Given the Dirichlet bound onf N t in (4.3) of order N n /N α , we need only show that Step 3. At this stage, there is a trick that is not part of the standard 1-block argument because, in H 1 0,d,l , we in fact have 2 l-blocks, about 0 and d. Let ξ and ζ be configurations on [−l, l] n that equal η and τ d η, respectively, on [−l, l] n . Define Let ν 1 ρ * (dξ, dζ) be the product measure on pairs of configurations (ξ, ζ) induced by ν ρ * , and letf l,d (ξ, ζ) be the conditional expectation of f R,N (η) given configurations η that equal ξ on [−l, l] n and ζ on [−l + d, l + d] n . Define now Given (2RN + 1) n /N n ≤ (2R + 1) n and DN d = N N α / d n+α is bounded in terms of and D, it will be sufficient to show that be the bond Dirichlet forms with respect to configurations ξ and ζ respectively. Define now a new Dirichlet form, In Lemma 6.2 below, we prove the following bound D 2 Because of the truncation, 1(ξ l (0) ∨ ζ l (0) ≤ A), we may restrict the supremum to sub-probability densities f supported on a finite set of configurations (ξ, ζ) satisfying ξ l (0) ∨ ζ l (0) ≤ A. As the mass ν 1 ρ * (ξ, ζ) is bounded below uniformly by a constant C(n, l) > 0 for these finite number of configurations, we have the uniform bound for the sub-probability density, f (ξ, ζ) ≤ C −1 (l), on its domain. Hence, from any sequence of such densities, one can extract a subsequence which converges pointwise. See Chapter 5 in [20] for another approach.
The supremum in (6.1), for each N and l, is attained at some density denoted f N,l .
In particular, it will be enough to show that lim sup Step 6. We try to make the integrand independent of l. To simplify expressions, we assume now that l is such that (2l + 1) n = q(2k + 1) n , that is, an l-block is partitioned into k-blocks. When l is not in this form, the following argument may be straightforwardly adapted with more notation. Let B 1 , .., B q denote the k-blocks. Then, We can then take the limit as l ↑ ∞, that is, as q ↑ ∞ to obtain, by a local central limit theorem or equivalence of ensembles estimate as in Corollary 1.7 in Appendix 2 [20], the expression sup 0≤ρ1,ρ2≤A But, this quantity, say using a Chebychev bound, vanishes uniformly for 0 ≤ ρ ≤ A as k ↑ ∞ by the law of large numbers, since Φ(ρ 1 )Ψ(ρ 2 ) = E νρ 1 ×νρ 2 [g(ξ(0))h(ζ(0))].
As a remark, this last step is rather interesting. Normally, the usual 1-block estimate ends by showing that an average of a function of the ξ(y) converges to its expected value. Here, in the α < 1 case, we end up with term that looks like a covariance. Proof. By the convexity of the Dirichlet form, for i = 1, 2, is less than Note, for each l, N and R, that bond is counted at most four times. Hence, we may bound the last display further by The proof of the 2-blocks estimate is similar to the preceding 1-block estimate, so we will give only a brief overview of the key differences.
Proof. The proof uses several steps.
Step 1. Analogous to the 1-block proof, we introduce a truncation. We can bound the second term of H 2 x,d,l, N (η) by Lemma 4.2. Since Ψ is Lipschitz, the truncated second term is less than The proposition will follow if we show, as N ↑ ∞, ↓ 0 and l ↑ ∞, that for both x * = x and x * = x + d.
As in the standard 2-blocks estimate, we will replace an N block, η N s , by an average of l-blocks, η l s . Specifically, we will replace η N s (x * ) by The expected error introduced is of order E N t 0 N −2n |x|≤R η s (x)ds, for some R , which vanishes by say Lemma 4.1.
By bounding the 'average' over y by a supremum, it will be enough to show that Step 2. From here, the proof of the 2-blocks estimates proceeds in the same way as for the 1-block estimate. We can write the expected value in terms off N t and then majorize by a factor t times .
which looks like the standard 2-blocks estimate, say in Chapter 5 of [20].
Step 3. We may introduce the indicator function 1(η l (x) ∨ η l (x + y) ≤ A) to the integrand by Lemma 4.2. By translation-invariance of ν ρ * , we can shift the summand by τ −x . Recall the averaged density f R+D,N , introduced in Step 2 in Subsection 6.1. Multiplying and dividing by (2(R + D)N + 1) n and noting that the factor (2(R + D)N + 1) n /N n is bounded, by convexity of the Dirichlet form, it will be enough to show the following vanishes, as N ↑ ∞ and ↓ 0: Step 4. Let ξ 1 and ξ 2 be configurations on [−l, l] n , equal to η and τ y η, respectively. Let ν 2 ρ * (dξ 1 , dξ 2 ) be the associated induced measure with respect to ν ρ * . Let alsof l,y (ξ 1 , ξ 2 ) be the conditional expectation of f R+D,N (η) given configurations η that equal ξ 1 on [−l, l] n and ξ 2 on [−l + y, l + y] n . The last display in Step 3 equals With D w,z 1 (f ) = D w,z (f ) and D w,z 2 (f ) = D w,z (τ y f ), we now introduce a Dirichlet form, is a Dirichlet form on the bond between the centers of the l-blocks involved. Note, with the convention 0/0 = 1 when p sym (y) = 0, we have D * (f )/p sym (y) = D 0,y (f )/p sym (y) = D y,0 (f )/p sym (y). Importantly, a zero form D * l (f ) = 0 implies that f is invariant to particle motion within each l-block and also motion between the centers. In this case, f takes a constant value along each of the hyperplanes H 2 j = (ξ 1 , ξ 2 ) : |y|≤l (ξ 1 (y) + ξ 2 (y)) = j for j = 0, 1, ..., 2(2l + 1) n A.
In Lemma 6.4 at the end of the Subsection, for 2l < |y| ≤ N , we prove the bound D * l (f l,y ) ≤ C 2 N −α + C 3 ( ) α . Therefore, it will be enough to show the following vanishes: As in the 1-block proof, as particle numbers are bounded, we may take limits, as N ↑ ∞ and ↓ 0, to restrict the supremum above to densities f such that D * l (f ) = 0.
Step 5. Hence, at this stage, f equals a constant C j along each hyperplane H 2 j for j ≤ 2(2l + 1) n A. Because f is a probability density, these constants C j are non-negative and j C j ν 2 ρ * (H 2 j ) = 1. Therefore, we need only show vanishes, where ν 2,l,j is the canonical measure on configurations (ξ 1 , ξ 2 ) which distributes j particles among the two l-blocks. However, both the expectations under ν 2,l,j of ξ l 1 (0) and ξ l 2 (0) equal j 2(2l+1) n . Hence, adding and subtracting j 2(2l+1) n inside the absolute value, it will be enough to control (1, 0, . . . , 0). By the equivalence of ensembles as used in Step 6 of Proposition 6.1, noting ν j/[2(2l+1) n ] is a product measure with identical marginals, the variance vanishes as l ↑ ∞.

Moving particle lemma
We now prove the following bound on D * l,y (f l,y ). Part of the strategy is inspired by [19] where a similar 'moving particle' estimate was proved. The development here is simpler and more general than that which was used in [19].
Recall that p is supported on y such that y > 0, that is when y = 0 and y i ≥ 0 for 1 ≤ i ≤ n (cf. (2.1)). Then, p sym (y) = p(y) + p(−y) /2 is supported on y such that y > 0 or y < 0. We note if the dimension n = 1, then p sym is supported on all y = 0. Lemma 6.4. Suppose 2l < |y| ≤ N and D(f ) ≤ C0N n N α . Then, with respect to constants C 2 = C 2 (R, D, n) and C 3 = (R, D, n), we have Proof. Recall the definition of D * l in (6.2). First, by the same argument as in Lemma 6.2, the sum Therefore, we need to control the form D * (f l,y )/p sym (y), which reflects motion from 0 to y = (y 1 , . . . , y n ) = 0. If y is such that neither y > 0 or y < 0, we may split y into its positive and negative parts, y = y + − y − , where both y + , y − > 0; note also that y − y + = −y − < 0. Straightforwardly, for such a y, noting the definition of D * in (6.3), by properties of the invariant measure ν ρ * and the inequality (u + v) 2 ≤ 2(u 2 + v 2 ), we have D * (f l,y ) p sym (y) ≤ 2 D 0,y + (f l,y ) p sym (y + ) + 2 D y + ,y (f l,y ) p sym (−y − ) .
In the following, analysis of the Dirichlet forms on the right-side of the above display are similar and lead to the same bound. Without loss of generality, we will assume now that y is positive, y > 0.
By convexity of the Dirichlet form, We now split the term D z,z+y (f )/p sym (y), reflecting a displacement by y, into jumps, one displacing by k = (k 1 , . . . , k n ) where 0 ≤ k i , |k| ≤ |y|, and 0 = k = y, and one displacing by y − k. If p sym were supported on all y = 0 (the case when n = 1), or if y − k > 0 for all k (the case when y = (|y|, . . . , |y|)), then these two jumps would suffice. If y − k is not positive, then we split y − k into its positive and negative parts, making three jumps.
When a k is summed over all k such that 0 ≤ k i , |k| ≤ |y|, and 0 = k = y, each bond is counted at most three times. Denoting k the sum over such k, we have k a k ≤ 3D(f ). (6.8) In particular, from (6.7) and (6.8), we have Hence, we have As |y| n scales like |y|, and |y| ≤ N , we have |y| n n+α /[(|y| + 1) n − 2] = O(( N ) α ). Therefore, in terms of a constant C 3 = C 3 (R, D, n).

1-block estimate: α > 1
The proof of the 1-block estimate, when α > 1, is similar to that when α < 1, but with fewer complications. The argument is also similar to that in the standard finite-range setting in [20]. For completeness, we summarize the proof. Proof. Following the proof of Proposition 6.1, for α < 1, we may introduce the indicator function 1(η l s (x) ∨ η l s (x + d) ≤ A), and bound the expectation in the display by In Lemma 6.6 below, when D(f ) = O(N n−1 ), we show that D l,d (f l,d ) ≤ C 1 /N . Therefore, we can replace the supremum in (6.9) by that over densities f such that D l,d (f ) ≤ C 1 /N . As the truncation enforces a finite configuration space, after N ↑ ∞, the supremum may be further replaced by D l,d (f ) = 0. In this case, f will be a constant C j ≥ 0 on hyperplanes of the form H j = ξ : l] n |. Moreover, as j C j ν l,d ρ * (H j ) ≤ 1, we may bound (6.9) by a supremum over hyperplanes: where ν l,d,j is the canonical measure supported on the hyperplane H j . As before, in the proof of Proposition 6.1, we can partition [−l, l] n into k-blocks assuming (2l + 1) n = q(2k + 1) n for simplicity. Let B 1 , .., B q be the q number of k-blocks. Then, Under the measure ν l,d,j , the distributions of y∈Bi g(ξ(y))h(ξ(y + d))) − ΦΨ(ξ l (0)) do not depend on i. Therefore, it is enough to show lim sup k→∞ lim sup l→∞ sup j E ν l,d,j (gh d ) k (ξ(0)) − ΦΨ(ξ l (0)) . Now, we would like to replace ΦΨ(ξ l (0)) by ΦΨ(ρ), where ρ = j/|[−l, l] n ∪ d + [−l, l] n |, for each j ≤ 2A as l ↑ ∞. This holds because [−l, l] n and d + [−l, l] n will have sufficient overlap for large l. To make this precise, bound |ΦΨ(ξ l (0)) − ΦΨ(ρ)| ≤ C(A)|ξ l (0) − ρ| since Φ, Ψ are Lipschitz and ξ l (0) ≤ 2A. As d is fixed, the number of sites outside the overlap is of order O(l n−1 ). Then, because of the truncation of particle numbers, for each A, we have ξ l (0) = ρ + O(l −1 ).
We now prove the bound on D l,d (f l,d ). Although the argument is similar to a finiterange setting estimate in [20], as it is short, we include it for convenience of the reader. When α > 1, as the expected jump size d dp(d) is finite, one expects in Euler scale to recover a similar hydrodynamic equation as when the jumps have finite range. The strategy employed here is to follow the scheme of arguments in [24] and Chapter 8 in [20] for finite-range processes.
However, in the long-range setting, several important steps are different. In particular, we have worked to remove reliance on 'attractiveness', a monotonicity condition on the rates, although it is still used in two, albeit, important places, namely to bound the hydrodynamic density as an L ∞ object in Step 1 below, and to show the 'Ordering' Lemma 9.3, which is used to prove a so-called measure weak entropy formulation. On the other hand, the proof includes new arguments to bound uniformly the 'mass difference from ρ * ' in the system (Theorem 7.4), and to handle the 'initial boundary layer' estimate (Theorem 7.5), needed to apply a form of DiPerna's uniqueness characterization.
The first step in the argument is to use a 1-block replacement estimate. Here, we do not rely on 'attractiveness' as in [24], but the 'entropy' method. Part of the reason for this choice, as discussed in Subsection 3.1, is that, when 1 < α < 2, it is not clear how to use the 'L 1 -initial density' method in [24]. However, an artifact of using the 'entropy' method is that we need to start from initial profiles ρ 0 , which are close to ρ * at large distances.
Since a '2-blocks' estimate is not available in the general asymmetric model, as also in [24] and Chapter 8 of [20], we use the concept of Young measures and DiPerna's characterization of measure-valued weak entropy solutions of the hydrodynamic equation to finish.
In terms of the process η t , define a collection of Young measures as Integration with respect to π N,l t against test functions is as follows: )) to be the space of functions π : t ∈ [0, T ] → π t ∈ M + (R n × [0, ∞)) such that π t , F is essentially bounded in time for every continuous function F with compact support in R n × [0, ∞). The topology on L ∞ ([0, T ], M + (R n × [0, ∞))) is such that elements π andπ are close if they give similar values upon integrating against a dense collection of test functions over space, λ, and time, that is if More precisely, the distance between π andπ is where {F k } k≥1 is a dense sequence in the space of compactly supported functions in R n × [0, ∞), with respect to the uniform topology. Here, where {h k } k≥1 is a dense sequence of functions in L 1 [0, T ] (cf. p. 200 in [20]). Note now that π N,l t ∈ L ∞ ([0, T ], M + (R n × [0, ∞))), and accordingly {Q N,l } are measures on L ∞ ([0, T ], M + (R n × [0, ∞))). The general strategy, as in [24], is to characterize limit points Q * of {Q N,l } in terms of unique 'measure weak' solutions to the hydrodynamic equation.
At this point, we remark that functions F (s, u, λ) = G(s, u)f (λ) where f is not compactly supported, but bounded |f (λ)| ≤ Cλ for all large λ, will have use in later development. Although such functions are not part of the topology on L ∞ ([0, T ], M + (R n ×[0, ∞))), we establish in Subsection 7.1, for a subsequence {Q N ,l } converging to Q * , that  We now define the notion of 'measure weak' solution. Consider the weak formulation of the differential equation in terms of a weak solution ρ(s, u). The measure weak formulation is obtained by replacing ρ(s, u) where ever it appears by λ and then integrating against the measure ρ(s, u, dλ) with respect to λ. So, f (ρ(s, u)) becomes f (λ)ρ(s, u, dλ). If ρ(s, u, dλ) is a solution of the resulting equation, it is called a measure weak solution.
For example, It is known that there is a unique bounded weak solution of the hydrodynamic equation which satisfies Kruzkov's entropy condition, with bounded initial data w 0 (cf. [21], [10], [15]). The corresponding measure weak formulation is given by where q(λ, c) = sgn(λ − c)(ΦΨ(λ) − ΦΨ(c)) and G is a nonnegative test function. We will say that ρ(t, u, dλ) is a measure weak entropy solution, or satisfies the entropy condition 'measure weakly', if it is a measure weak solution of the hydrodynamic equation that measure weakly satisfies the entropy condition.
We are now ready to state DiPerna's uniqueness theorem (cf. Theorem 4.2 in [11]) for such measure weak solutions. Theorem 7.1. Suppose w(t, u, dλ)du is a measure weak entropy solution of ∂ t w + υ · ∇Q(w) = 0.
Here, Q ∈ C 1 , υ ∈ R n , and initial condition w(0, u, dλ) = δ w0(u) , where w 0 is bounded and integrable. Suppose also that the following three conditions are satisfied: 1. Bounded support and probability measure: The support of w(t, u, dλ) is bounded in the interval A = [a, b], for some a, b ∈ R, uniformly in (t, u) ∈ [0, T ] × R n . Also, for each (t, u), w(t, u, dλ) is a probability measure.
Given this preamble, we now begin the main part of the proof of Theorem 3.2.
Step 1. First, we claim that the measures {Q N,l } are tight. This follows the same proof as given in Lemma 1.2, Chapter 8 in [20]. Next, as N ↑ ∞ subsequentially, we may obtain a weak limit Q l , and as l ↑ ∞ subsequentially, we obtain a limit point Q * . We claim that Q * is supported on measures in the form π(s, du, dλ) = ρ(s, u, dλ)du, which are absolutely continuous in u. This also follows the same proof as given for item 1, p. 201 of [20].
In addition, ρ(s, u, dλ) is supported in a bounded interval, uniformly in s, u: If M 0 < ∞, that is h(m) = 0 for some m, then there can be at most M 0 particles per site in the process. In particular, η l s (x) ≤ M 0 for all x, s, l, and so 0 ≤ ρ(s, u, dλ) ≤ M 0 for all s, u, without using 'attractiveness'. On the other hand, if h(m) > 0 for all m, by the 'basic coupling' proof, using 'attractiveness', and that the measures {µ N } are 'stochastically bounded' by ν ρ # where ρ # = ρ 0 ∞ , as given for item (ii) in the proof of Theorem 1.1 of Chapter 8, p. 201-203 in [20], we obtain ρ(s, u, dλ) is supported in [0, ρ # ] (cf. related comments, on the 'basic coupling', at the beginning of Section 9).
Theorem 7.3. The entropy condition holds measure weakly for any c ∈ R: Theorem 7.5. The initial condition holds, lim inf We prove Theorems 7.2, 7.3, 7.4, 7.5, in Subsections 8, 10, 11, and 12, respectively Step 3. Although our initial condition, as ρ 0 (u) = ρ * for |u| large, is not integrable, the functionρ 0 (u) = ρ 0 (u) − ρ * , is also bounded, and belongs to L 1 (R n ). By considering ρ * -shifted solutions, we will see that the items in Steps 1 and 2 allow to use DiPerna's Theorem 7.1 to characterize the limit points Q * . First, we note the following equivalences.
Hence, all limit points Q * of {Q N,l } are the same, uniquely characterized in terms of the weak entropy solution of the hydrodynamic equation, Q * = δ ρ(t,u) .
Step 5. We now relate the limit points {Q * } to the limit points of {P N }, and thereby prove Theorem 3.2. We have shown, for test functions f (s)G(u) that for all 0 > 0. Then, as Q N,l on a subsequence converges to Q * , by (7.1), Now, by the assumption FEM, limit points of {P N } are supported on absolutely continuous measures π s =ρ(s, u)du; this observation, made in Step 6 in Section 5 for the case α < 1, also directly applies when α > 1. Then, as π → t 0 f (s) π N s , G ds is continuous, for every limit point P * , we have But, as tightness of {P N G } was shown with respect to the uniform topology (Proposition 4.7), the limit R n G(u)ρ(s, u)du is continuous function in time s. One also has that R n G(u)ρ(s, u)du is continuous in s (cf. Theorem 2.1 [7]). Therefore, R n G(u)ρ(s, u)du = R n G(u)ρ(s, u)du for all times s.
We conclude all limit points P * are the same, that is, supported on absolutely continuous measures π t = ρ(t, u)du whose density is the unique weak entropy solution of the hydrodynamic equation, and so Theorem 3.2 follows.

Proof of (7.1)
We first note, for all large A and λ, by the bound |f (λ)| ≤ Cλ and compact support of G, Then, by Lemma 4.2 and that π → t 0 π s , λ1(λ ≥ A) ds is a lower semi-continuous function, we have In particular, as π → t 0 π s , |G s (u)||f (λ)|1(λ ≥ A) ds is also lower semi-continuous, We now argue the left to right equivalence. In the left-side of (7.1), by ( The right-side of (7.1) follows now by (7.3) applied again. The right to left equivalence in (7.1) follows by similar steps in reverse, given now Here, without loss of generality we have replaced '>' by '≥' to maintain the correct bounds implied by weak convergence.

Measure weak solutions: Proof of Theorem 7.2
The argument follows some of the initial reasoning given for the proof of Theorem 3.1, in the α < 1 case, relying however on the 1-block estimate Lemma 6.5.
Step 1. The same estimate as in Step 1 in Section 5, with respect to the martingale Here, we recall from (4.1), Step 2. We would like to replace G s x+d N − G s x N by ∇G s ( x N ) · d/N . To this aim, noting gh d (η s (x)) ≤ κ h ∞ η s (x), by Lemma 4.5, we may truncate the sum on d to d ≤ N , in terms of a parameter which will vanish after N diverges. Next, as |G s which vanishes in expected value, noting Lemma 4.1, as N ↑ ∞ and ↓ 0. Therefore, converges to zero in probability as N ↑ ∞ and ↓ 0. Moreover, with similar reasoning, we may further replace the sum on d to a truncated sum over d ≤ D, where D will diverge after N . Indeed, as gh d (η) ≤ κ h ∞ η(x), the error in such a replacement is of order Since the sum on d is of order D 1−α , the expected error, by Lemma 4.1, as N ↑ ∞ and D ↑ ∞ vanishes.
Step 3a. Now, by the method of Step 3a in Section 5 for the α < 1 case, we substitute gh d (η s (x)) with (gh d ) l (η s (x)) where l will go to infinity after N but before D. We will also replace η s (x) by η l s (x) in the first and second terms in (8.1). Hence, converges to zero in probability as these limits in order are taken.

A coupled process
We introduce the basic coupling for misanthrope processes. LetP N denote the distribution of the coupled process (η t , ξ t ) with generatorL, given by its action on test where min x,y = min{b(η(x), η(y)), b(ξ(x), ξ(y))}. From the form of the generator, it follows that the marginals are themselves misanthrope processes.
Suppose now that the process is 'attractive', that is when b(n, m) = g(n)h(m), with g increasing and h decreasing in particle numbers. Then, if η s (x) ≤ ξ s (x) for all x ∈ Z n , at any later time t ≥ s, we still have the same ordering. This observation is the crux of the proof of the 'L ∞ ' bound in [20], referred to in Step 1 in Section 7. This is the first of the two ways where 'attractiveness' is used in the proof of Theorem 3.2.
We will use the following teminology. For any set Λ ⊆ Z n , we write η ≥ ξ on Λ if η(x) ≥ ξ(x) for all x ∈ Λ, and we write η > ξ on Λ if η ≥ ξ on Λ and η(x) > ξ(x) for at least one x ∈ Λ. If η ≥ ξ or ξ ≥ η on Λ, we say that η and ξ are ordered on Λ. Otherwise, we say that η and ξ are unordered on Λ.
Let U Λ (η, ξ) = 1(η and ξ are not ordered on Λ). Let U x,d (η, ξ) = U {x,x+d} (η, ξ). We also Define the coupled empirical measure bỹ We now introduce martingales which will be useful in the sequel. The first two are the coupled versions of M N,G t and the associated 'variance' martingale: For test functions G on the coupled space, define the martingale, With respect to the quadratic variation, t is also a martingale. We may compute Note that U x,d (η s , ξ s ) = |U ± x,d (η s , ξ s )|. That there is a sum of G's in the last line of the computation is because η s and ξ s are not ordered.
When the process is 'attractive', we have In this case, the second line of the generator computation (9.1) simplifies to We remark that this is the second of two places where the 'attractiveness' condition is explicitly used, featuring in the proof of the 'Ordering Lemma', stated later.
Lemma 9.1. When α > 1 and G is nonnegative, Proof. The bound follows because in (9.2) all the terms are nonnegative.
In the next two results, we will start the coupled process (η s , ξ s ) from an arbitrary initial distributionμ N whose marginals are µ N and ν c , for a 0 ≤ c ≤ M 0 if M 0 < ∞, and c ≥ 0 if otherwise. The coupled process measure is denoted byP N and the associated expectation is given byẼ N .
For the quadratic variation, M N,G t , a straightforward computation gives that Lemma 9.2. When α > 1, we havẽ Proof. In the expression for the quadratic variation, we may bound factors (gh which we split as A 1 + A 2 , the term A 1 involving G s x+d N − G s x N 2 and A 2 involving the other squared quantity. Since, gh d (η(x)) ≤ κ h ∞ η(x) by (2.3), we observe that |gh d (η s (x)) − gh d (ξ s (x))| ≤ 2 h ∞ κ(η s (x) + ξ s (x)). Hence, A 1 ≤ A 11 + A 12 , where A 11 and A 12 involve each only the η · and ξ · process respectively. By the proof of Lemma 4.6, starting from (4.7), E N A 11 ≤ K G t/N n . A similar bound and argument holds when ξ s (x) is present as ν c is invariant, and therefore ξ s ∼ ν c andẼ N a≤|x|≤b ξ s (x) = c(b n − a n ). Hence, The remaining partẼ N A 2 , since the the sum of the G's squared is bounded by This finishes the proof.
We now state an 'Ordering Lemma' which, in essence, tells us that η t and ξ t are ordered on average, even if they are not initially ordered. This result is analogous to those in the finite-range setting, Lemma 3.3 in [24] and Lemma 2.2 on p. 209 of [20].  [aN, bN ] and [a, b] = n j=1 [a j , b j ] denotes the n-dimensional hyperrectangle with diagonal extending from a to b.
We also have, for all d with d ≥ 1, that lim sup We postpone the proof the 'Ordering Lemma' to the Appendix. In this proof, the second statement will be seen to follow from the first, along with an induction argument.

Entropy condition: Proof of Theorem 7.3
We note, as specified in the definition of the measure weak entropy condition, the test functions G in this section are nonnegative.
Step 1. Since ρ ≥ 0 a.e. (cf. Step 1 of Section 7), it is enough to prove Theorem 7.3 when c ≥ 0. When the max occupation number M 0 < ∞, it is enough to consider 0 ≤ c ≤ M 0 .
Suppose we may show, for 0 > 0 and t ≤ T , that In terms of Young measures and Q N,l , (10.1) is written In this case, the desired measure weak formulation of the entropy condition would follow: By tightness of {Q N,l }, let Q * be a limit point. Such a Q * is supported on absolutely continuous measures π s = ρ(s, u, dλ)du and ρ(0, u, dλ) = δ ρ0(u) (cf. Step 1 of Section 7). Then, noting the form of q, as ΦΨ(λ) ≤ κ h ∞ λ (cf. (2.3)), by the weak convergence statement (7.1), we would have Q * a.s. that Step 2. To begin to establish (10.1), consider a coupled process (η t , ξ t ) where the initial distribution is such that ξ 0 is the invariant measure ν c with density c. We will specify the form of the coupled initial distribution at the beginning of Subsection 10.1, and show there a coupled version of the microscopic entropy inequality: For > 0, and t ≤ T , we have Step 3. We now show how the microscopic entropy inequality (10.1) can be deduced from the coupled microscopic entropy inequality (10.2). It is enough to show that the following terms vanish as N and then l go to infinity: To analyze the second term, we note, as Ψ is bounded by h ∞ and Φ is Lipschitz, that , the error vanishing, as N ↑ ∞, l ↑ ∞ and A ↑ ∞, by Lemma 4.2 and that ξ · ∼ ν c . Also, now note that q(z, w) is uniformly continuous on On the other hand, for the first term, by the triangle inequality, ||η l But, since the state ξ s has distribution ν c , it follows that which vanishes by the law of large numbers as l ↑ ∞.

Proof of (10.2)
We proceed in some steps, recalling estimates in Section 9. First, we specify the initial coupled distribution in Step 2 above: We will takeμ N as a product measure over x ∈ Z n with x-marginal given byμ Such a coupled initial measure may be constructed (cf. Section II.2 in [23]) as the x-marginals of µ N and ν c are stochastically ordered, that is the marginal of µ N , Θ ρ0(x/N ) , is stochasically more or less than then the marginal of ν c , Θ c , if ρ 0 (x/N ) is more or less than c respectively. Then,P N is the coupled process measure starting from µ N .
Step Since G has compact support in [0, T ) × R n , we have π N t , G t = 0 for t ≥ T , and so It follows, as G is nonnegative, from the bound in Lemma 9.1, that We now replace the second integral in the last display by one with a nicer form. We make substitutions following the same reasoning as in Step 2 of Section 8, the estimates for the ξ · process easier as ξ · ∼ ν c . First, we limit the sum over d to when d is at most N , where N ↑ ∞ and then ↓ 0. Next, G s Finally, the sum over d is replaced by that when d is at most D, which tends to infinity after N diverges. After this replacement, we have with probability tending to 1, as N ↑ ∞ and D ↑ ∞, that Step 2. As in Step 3a in Section 5, we may substitute l-averages for |η s (x) − ξ s (x)| and (gh d (η s (x)) − gh d (ξ s (x)))O x,d (η s , ξ s ), where l diverges after N but before D, through a discrete integration-by-parts, the smoothness and compact support of G, as well as the particle bound Lemma 4.1, and with respect to the ξ · process that ξ · ∼ ν c . Then, we have with high probability as N, l, and D go to infinity.
Step 3a. We now begin to perform a '1-block' replacement in the last display, which will allow us to access the Young measure formulation.
It is only here that we leverage the full form of the initial coupled distribution in order to treat the first term on the left-side of (10.4). Let A 1 and A 2 be the set of sites x in Z n where ρ 0 (x/N ) ≥ c and ρ 0 (x/N ) < c respectively. Write, using the coupling, noting that G is nonnegative, that We now add and subtract ρ 0 (x/N ) − c inside the square bracket. Noting the compact support of G, we observẽ A similar argument, using that ξ 0 has distribution ν c , works for the difference between ξ 0 (x)−c. Hence, with high probability as N ↑ ∞, we may bound above 1 Step 3b. Now, we will replace |η s ( in the second integral of (10.4).
Indeed, by the compact support of ∂ s G and ∇G, and Therefore, the sum S 1 1 vanishes. But, by the 1-block estimate Proposition 6.5, we have lim sup and its counterpart with η · replaced by ξ · ∼ ν c also vanishes. Hence, the expectation of the time integral of S 1 2 vanishes in the limit as N ↑ ∞ and l ↑ ∞.
Step 3c. When η s and ξ s are not ordered on x + [−(l + D), l + D] n , as h, Ψ ≤ h ∞ are bounded, and g, Φ are Lipschitz, we have are both bounded by a constant times (η l s (x) + ξ l s (x)). Therefore, we may introduce the indicator function 1(η l s (x) ∨ ξ l s (x) ≤ A) when taking expectations, the error vanishing as N ↑ ∞ and l ↑ ∞ by Lemma 4.2, and that ξ · ∼ ν c .
Once this indicator is introduced, both terms L 1 , L 2 are bounded by a constant C = C(A), which allows further to introduce the indicator function 1(η l s (x + d) ∨ ξ l s (x + d) ≤ A), say by Lemma 4.2 and that ξ · ∼ ν c , the error vanishing as N ↑ ∞, l ↑ ∞ and A ↑ ∞. Now, for k = 1, 2, we have Therefore, to complete the 1-block replacement, it is enough to show, for fixed l, that  Step 3d. Recall that U Λ (η, ξ) indicates when η and ξ are not ordered on Λ, and also that U x,d (η, ξ) = U {x,x+d} (η, ξ). We then have the bound, for large enough N where R + > R. However, by the Ordering Lemma 9.3, we have for each l and d thatẼ N t 0 1 N n |x|≤R + N U x,d (η s , ξ s )ds vanishes as N ↑ ∞. Hence, (10.5) holds. and the 1-block replacement follows.
In particular, we have with high probability as N, l, and D go to infinity. To recover (10.2) from (10.6), we may group together the terms involving d, and remove the bound d ≤ D, by appealing to the argument in Step 3a in Subsection 8. Then, the sum on d is replaced by γ α ∂ 1(n) G s ( x N ).
We leverage the weak formulation of the entropy condition (7.2).
Step 1. Consider a test function in form G(s, u) = H(s)G(u) for G nonnegative, and c = ρ * . Define V G (s) = R n ∞ 0 G(u)|λ − ρ * |ρ(s, u, dλ)du. By the 'Mass Bounding' Lemma 11.1 shown below, V G is finite. Moreover, by the measure weak entropy condition inequality (Theorem 7.3), where we recall q(λ, c) = sgn(λ − c)(ΦΨ(λ) − ΦΨ(c)). Since Φ, Ψ are Lipschitz, and also The idea now will be to choose G and H, approximating the constant 1 and an indicator of a time point t, so that the right-hand side is well bounded. This will be done through an iteration scheme in the next step.
With respect to H = H i and G = G i , we have Step 3. Iterating the above inequality k times, starting with i = 0, gives (11.1) Step 4. Choose now k = n + 2. Note that V G0 ↑ V G∞ as R ↑ ∞. Then, the supremum over 0 < γ < δ/2, 0 < δ ≤ T − t and 0 ≤ t < T of the left-side of (11.1), Q * a.s., increases To capture the limit of the right-side, note Then, the first term on the right-side of (11.1) converges to V G∞ (0) as R ↑ ∞.
However, by the 'Mass Bounding' Lemma 11.1, we have E Q * ess sup 0≤t≤T V Gn+2 (t) = O(R n ), and so Hence, Q * a.s., by Borel-Cantelli lemma, as R ↑ ∞, the second term on the right-side of (11.1) vanishes.
Step 5. Therefore, we have with respect to a Q * probability 1 set. Moreover, on this set, as −∂ s H 0 is positive on (t − δ, t + δ), we have that V G∞ is locally integrable on [0, T ]. Also, by Fatou's lemma, for each t and small enough δ > 0, we have In fact, for each Lebesgue point t of V G∞ , as δ ↓ 0, we have V G∞ (t) ≤ V G∞ (0). We conclude, as Lebesgue points are dense, that Q * a.s.
finishing the argument.

Mass bounding lemma
The following result bounds the mass in finite regions.
Proof. First we bound |λ − c| by λ + |c|. Since ρ(t, u, dλ) is a probability measure, we have R n G(u) ∞ 0 |c|ρ(t, u, dλ)du = O(R n ). Therefore, we only need to prove To this end, for RN ≥ l, note By our initial conditions, E N π N 0 , G 1 ≤ (4R + 1) n ρ 0 ∞ , and by Lemma 4.3, we have E N T 0 |N L N π N s , G 1 |ds = O(R n ), independent of N and l. Also, by Doob's inequality and Lemma 4.6 . Therefore, for all large N , we have E N,l ess sup 0≤t≤T π N,l t , G(u)λ = O(R n ). Finally, as ess sup 0≤t≤T π N,l t , G(u)λ is a lower semi-continuous function of π N,l , we may take subsequential limits as N, l ↑ ∞, for which Q N,l ⇒ Q * , to obtain (11.2).

Initial conditions: proof of Theorem 7.5
The strategy is to approximate the initial density ρ 0 in compact sets via the weak form of the entropy inequality.
Step 1. Since ρ 0 is a continuous function that equals a constant ρ * outside of a compact set [−R, R] n , it is uniformly continuous. Fix a δ = (δ 0 , . . . , δ 0 ) with 0 < δ 0 < 1. Consider a regular division of R n into countably many overlapping hyper-rectangles Finitely many of these hyper-rectangles cover [−R, R] n . The parameter δ may be chosen so that ρ 0 varies at most > 0 on each hyper-rectangle.
Note that (2R) n is the volume of [−R, R] n , ρ(s, u, dλ) is a probability measure (cf. Step 1 in Section 7), and |λ − c i | − |ρ 0 − c i | = |λ − c i | ≥ 0 for all but finitely many hyper-rectangles. Then, a Fubini-Tonelli theorem may be applied, so that Step 2. Suppose, for all i, that lim sup Then, by Fatou-Lebesgue lemma , we would have from which Theorem 7.5 would follow as > 0 is arbitrary.
By taking a supremum over time, noting |H(s)| ≤ 1, the right-side of the inequality (12.2) is bounded by γ α C 2 t ess sup 0≤s≤T R n ∞ 0 |∂ 1(n) G (u) ||λ − c|ρ(s, u, dλ)du. However, the left-side, by dominated convergence, converges, as δ ↓ 0, to We obtain As |∂ 1(n) G| is compactly supported, noting the 'Mass Bounding Lemma' 11.1 again, the expected value of the right-side of the above display vanishes as t goes to zero. Therefore, the first line of (12.1) holds.
To obtain the second line of (12.1), instead of bounding the right-side of (12.2) by a For only finitely many i does c i differ from ρ * and |ρ 0 (u) − c i | > 0. Also, by the comment in the previous paragraph, for each i, the expected value of the right-side of the above display is bounded. Note now, by the regular division, that the support of each ∂ 1(n) G i is overlapped by the support of at most an uniformly bounded number, in terms of the covering, of other {∂ 1(n) G j }. Note also, from construction, that ∂ 1(n) G j is uniformly bounded in j. Also, from Theorem 7.4, we have that E Q * T 0 R n ∞ 0 |λ − ρ * |ρ(s, u, dλ)duds ≤ T R n |ρ 0 (u) − ρ * |du < ∞. Hence, summability in (12.1) follows, and the proof of Theorem 7.5 is complete.
In passing, we remark that this proof, making use of the weak formulation of the entropy condition, seems new and more direct than proofs in [24] and [20] which introduce types of particle couplings in the finite-range setting, without going to the continuum equation. We note, in the PhD thesis [27], an alternate argument for the first line of (12.1) through a simpler and different particle coupling will be found.

A Proof of the ordering Lemma 9.3
Step 1. We now show the first part of the lemma. Let G s (u) be a nonnegative smooth function that is 1 on hyper-rectangle [a, b] = n j=1 [a j , b j ] and decreases to 0 outside of [a − δ, b + δ] = n j=1 [a j − δ j , b j + δ j ] where δ = (δ 1 , . . . , δ n ) with δ i > 0 and δ < 1. Then, noting the computation of NL N π N s , G s in (9.1) and (9.2), we have −NL N π N s , G s =: J 1 − NL N π N s , G s .
As the expectation is of order O(N n ) by Lemma 4.1 and that ξ · ∼ ν c , the last display vanishes as N ↑ ∞. This completes the proof of the first part of Lemma 9.3.
Step 4. We now show the second part of Lemma 9.3. In general, gh d (η s (x))−gh d (ξ s (x)) may not vanish, and so the first part is not coercive. To work around this issue, we would like to introduce the indicator function 1(η s (x) ∨ ξ s (x) ∨ η s (x + d) ∨ ξ s (x + d) < A) into the associated expectation. This is justified if we show that The expectation above is bounded by the sum of E N t 0 1 N n |x|≤RN 1(η s (x) > A)ds, and three other expectations containing the indicator functions 1(ξ s (x) > A), 1(η s (x+d) > A), and 1(ξ s (x + d) > A)).
Thus, the induction step and therefore the second part of Lemma 9.3 would be proved.
We thus obtain, moving the negative terms to the other side of the inequality,  Hence, the expected value S 1 ≤ C(R)/N → 0, as N ↑ ∞, completing the proof of (A.3).