Hydrodynamic Limit of Condensing Two-Species Zero Range Processes with Sub-critical Initial Profiles

Two-species condensing zero range processes (ZRPs) are interacting particle systems with two species of particles and zero range interaction exhibiting phase separation outside a domain of sub-critical densities. We prove the hydrodynamic limit of nearest neighbour mean zero two-species condensing ZRP with bounded local jump rate for sub-critical initial profiles, i.e., for initial profiles whose image is contained in the region of sub-critical densities. The proof is based on H.T. Yau’s relative entropy method, which relies on the existence of sufficiently regular solutions to the hydrodynamic equation. In the particular case of the species-blind ZRP, we prove that the solutions of the hydrodynamic equation exist globally in time and thus the hydrodynamic limit is valid for all times.


Introduction
In this article, we derive the hydrodynamic limit of a system of two interacting particle systems, specifically two-species zero range processes (ZRPs).The motivation for this study is that hydrodynamic limits provide effective descriptions of large scale interacting particle systems.There is a now good understanding of this limit passage for a range of particle processes leading to one hydrodynamic limit equation.In particular, Kipnis and Landim [19] establish the hydrodynamic behaviour for the one-species zero range process, using the entropy method of Guo, Papanicolaou and Varadhan [17].For systems, however, this limit passage is less well studied, and several tools available for single equations are no longer available, as explained in more detail below.In particular, many systems where a hydrodynamic passage would be of interest both in its own right and as a tool to understand the limiting system of partial differential equations (PDEs) are currently inaccessible to the methods available; the full Patlak Keller-Segel system [18] modelling the evolution of cells or bacteria guided by the concentration of a chemical substance is an example.Yet, there are several recent studies focusing on different models of interacting particle systems.One avenue is to derive equations which incorporate aspects of underlying models, be it by considering the motion of cells only in a stationary, but random, environment mimicking the chemical [14], or by an equation with a singular potential related to a Green's function describing the solution of a second equation [11].The hydrodynamic limit system of an active exclusion process modelling active matter has been recently derived using a two-block estimate and non-gradient estimates [3].
Another approach is to study systems related to underlying zero-range processes (ZRPs) of several species to obtain a limiting system, and this is the approach we pursue here.The focus on ZRPs can be motivated by their nature as a toy model of an interacting particle system.We consider a system of two zero-range processes but the extension to n types is straightforward.Each ZRP is a process on a lattice where particles jump from one site to another according to a jump rate function depending on the number of the two species of particles on this site only (hence the name zero range).
The hydrodynamic limit in the Eulerian scaling t → tN of asymmetric many-species ZRPs with product and translation invariant equilibrium states has been studied in [16].The hydrodynamic limit in the parabolic scaling t → tN 2 for a class of processes not satisfying the assumptions of [16] has also recently been studied [26]; there one type of particles performs a random walk and influences the other type, which is a process of ZRP type.In general, establishing hydrodynamic limits for systems of equations rigorously is a hard problem, with few known results so far.To name a few, the hydrodynamic limit of a two-species simple-exclusion process was first studied in [21], the Leroux system has been derived as a hydrodynamic limit in [24], and hyperbolic systems have also been studied in [25].
Here we consider a system of two ZRPs.We show that the hydrodynamic equation is a quasilinear parabolic system of the form where ∂ t ρ := (∂ t ρ 1 , ∂ t ρ 2 ), ∆Φ(ρ) := (∆Φ 1 (ρ), ∆Φ 2 (ρ)) and Φ is the mean jump rate of the ZRP at a site x ∈ Ì d N under the product and translation invariant equilibrium state of background density ρ ∈ Ê 2 + .Two-species ZRPs, and the phase transition they exhibit, were first studied in [8].In condensing ZRPs, the set of admissible background densities ρ is a strict subset of Ê 2 + .We call such densities sub-critical.One challenge of ZRPs is that they can exhibit condensation phenomena, where particles congregate at the same site [15,7,9].Even for one-species systems, the hydrodynamic limit of ZRPs experiencing condensation is presently unknown.We consider parameter regimes of two-species systems where condensation can occur, but restrict to sub-critical initial profiles, i.e., initial data that take values in the set of sub-critical densities.For one-species ZRPs, the analogous result has been established recently [23] and we extend this argument to the two-species case.Specifically, we apply the Relative Entropy method of H. T. Yau [28], which requires only the one-block estimate proved in Theorem 3.1, not the full replacement lemma [19,Lemma 5.1.10].Thus it does not require the equilibrium states of the ZRP to have full exponential moments, a property not satisfied by condensing ZRPs.This extension of [23] is non-trivial, for two reasons.The first difficulty is that the relative entropy method requires the existence of C 2+θ solutions to (1) for some θ ∈ (0, 1], and that the solution remains in the sub-critical region.By a result of Amann [1] C 1,2+θ solutions exist locally in time, i.e., for small time intervals when starting from C 2+θ initial data.By continuity, there is a non-trivial time interval such that a solution, when starting from the sub-critical region, remains in the sub-critical region.So the general result on the hydrodynamic limit is local in time, being valid for the largest time interval on which we have C 2+θ solutions in the sub-critical region.This result shows that as long as a classical C 1,2+θ solution to the hydrodynamic equation exists, condensation does not occur.The second difficulty to extend the results for one species [19,23] is that the phase space Ê 2 + is now more complicated, and the one-dimensional arguments used in [19,23] do not extend directly.In particular, a novel argument is required to extend [19,Lemma 6.1.10];see Lemma 4.5 and its proof.Specifically, we employ a characterisation of the domain of a convex function via the recession function of its Legendre transform.This characterisation of the domain of convex functions is of interest in its own right in the context of two-species ZRPs.For example, it immediately yields a parametrisation of the boundary of the domain of the partition function via the recession function of the thermodynamic entropy.
Intuitively, condensation means on the level of the governing hydrodynamic limit PDE the formation of singularities where the mass concentrates.For scalar equations, the formation of such singularities can be ruled out by a maximum principle.For systems, however, in general maximum principles do not hold.In this article, we mainly rely on an existence theory for local C 1,2+θ classical solutions established by Amann and focus on proving the local in time hydrodynamic limit.However, for a particular example, the so-called species-blind process, we are able to establish a maximum principle and C 1+θ,2+θ regularity for the hydrodynamic equation.This allows us to obtain that C 1+θ,2+θ solutions exist and remain in the sub-critical region for all times.So in this particular case, the result on the hydrodynamic limit is global in time.
Maximum principles are more complicated for non-linear parabolic systems, since one has to determine the shape of the invariant region in which the solution will have to remain [6,27], while in the scalar case the invariant region is just an interval.For the species-blind process we find that the invariant region of the hydrodynamic equation coincides with the sub-critical region of the ZRP.This is not surprising since the speciesblind process is obtained from a one-species ZRP by colouring particles in two colours, say black and white.The dynamics is the usual ZRP dynamics but at each time of a jump from a site x, we choose the colour of particle to move with the probabilities given by the ratios of the number of black particles and white particles at x to the total number of particles at x, ignoring the colour.It would still be interesting to study the class of parabolic systems arising from 2-species ZRPs in order to determine the largest class of ZRPs that their sub-critical region is an invariant region of the hydrodynamic limit.This would then provide a way to find the invariant region of the associated parabolic systems by calculating the phase diagram of the underlying ZRP.The study of the system of PDEs arising from the ZRP is a different topic and outside of the scope of this article, which mainly focuses on the passage from the microscopic to the macroscopic level by applying the relative entropy method.
Plan of the paper The paper is organised as follows.In Section 2 we collect some preliminary material on two-species ZRPs and describe the particular case of the speciesblind ZRP.Section 3 contains the statements of the main results, and in Section 4 we give the proofs.

The particle model
We briefly give the definition of two-species ZRPs as Markov jump processes (Section A.1.2 in [19]) and their equilibrium states.Main references on this preliminary material are [12,13].We take the discrete d-dimensional N -torus Ì d N as underlying lattice.Each particle interacts only with particles in the same lattice site through a function Here g i (k) is the jump rate of species of type i from any site that contains k ∈ AE 2 0 particles, i.e., k i particles of type i, for i = 1, 2. We impose the natural condition and require where e i = (δ ij ) j=1,2 , i = 1, 2, are the unit vectors in Ê 2 .Note that setting g * := < +∞, we have by ( 2) and (3) that where The state space of a two-species ZRP consists of all configurations η = (η is the number of i-type particles at site x, for i = 1, 2. For any measurable space M we denote by È(M) the set of all probability measures on M .We write p ∈ È( d ) for the nearest neighbour (n.n.) elementary step distribution given by p(x) := 1 2d and by p N ∈ È(Ì d N ) its projection on Ì d N given by p N (x) := p(x + N d ).Also, given a N , we will denote by η i;x;y , i = 1, 2, the configuration resulting from η by moving a type-i particle from x to y. (If η i (x) = 0, then we set η i;x;y = η.)The two-species n.n.symmetric ZRP with jump rate g on the discrete torus Ì d N := {0, 1, . . ., N − 1} d is the unique Markov jump process on the Skorohod space D(Ê + ; Å d;2 N ) of càdlàg paths characterised by the formal generator We will denote by (P N t ) t≥0 the transition semigroup of the n.n.symmetric ZRP.The communication classes of the stochastic dynamics defined by the generator above are the hyperplanes consisting of a fixed number of particles of each species.Since each set Å d;2 N,K is finite, for each (N, K) ∈ AE × AE 2 0 there exists a unique equilibrium distribution ν N,K supported on Å d;2 N,K .The family {ν N,K } (N,K)∈AE×AE 2 0 is the so-called canonical ensemble.However, as proved in [12,Theorem 4.1], in order to have product and translation invariant equilibrium distributions, it is necessary and sufficient that the following compatibility relations for the component functions of two-species jump rates hold, Note that due to the compatibility relations (6) any two-species local jump rate g is uniquely determined by g 1 and the restriction of g 2 to the set {0}×AE 0 , since by induction .
For any increasing path γ to k ∈ AE 2 0 , the factorial of g along γ is defined as for k = 0; we set g!(•) := 1 if k = 0.A two-species local jump rate function g that satisfies (6) yields a well-defined function g! : AE 2 0 → (0, ∞) by the formula g!(k) = g!(k; γ) for some increasing path γ to k.
It is convenient to have a parametrisation of the GCE by the density.This is done via the density function R The proper domain of R is the set + and by differentiation of bivariate power-series, we have that where ϕ • ψ := (ϕ 1 ψ 1 , ϕ 2 ψ 2 ) denotes the pointwise product of two vectors ϕ, ψ ∈ Ê 2 + .Furthermore, this formula extends to the set D Z ∩ ∂D Z if we interpret the directional derivatives ∂ i (log Z) ∈ [0, +∞] as derivatives from the left.With the conventions log 0 = −∞ and e −∞ = 0 the densities ρ ∈ R(D R ) can also be parametrised via the chemical potential by the function , where Z = Z • exp.The density function R : R(D R ) → D R is invertible.Indeed, it is straightforward to check (e.g., see [13, (4.10)]) that for all ϕ ∈ D o R ∩ (0, +∞) 2 , DR(ϕ) = DR(log ϕ) , where Cov(ν 1 ϕ ) denotes the covariance matrix This implies that DR(ϕ) is diagonisable with strictly positive eigenvalues for all ϕ ∈ D o R ∩ (0, +∞) 2 .Furthermore, and for ϕ ∈ D o R such that ϕ 1 ϕ 2 = 0, the matrix DR(ϕ) is triangular, and thus invertible.Therefore the density function R : The fact that R is invertible on all of its domain follows by [13,Proposition 2.3], according to which for every ρ ∈ (0, ∞) 2 there exists a unique maximiser Φ(ρ) ∈ D R ∩ (0, ∞) 2 for the thermodynamic entropy Obviously, for ρ = 0 the supremum is attained at ϕ = 0 (with the convention 0 • (−∞) = 0).Furthermore, since Z is non-decreasing with respect to each variable separately, for any ρ ∈ Ê 2 + \ {0} with ρ 1 ρ 2 = 0, say ρ 2 = 0, the maximisation problem ( 11) is reduced to the corresponding maximisation problem for one of the 1-species jump rate ĝ1 (k) := g 1 (k, 0), k ∈ AE 0 and the supremum is attained at Φ(ρ 1 , 0) = ( Φ1 (ρ 1 ∧ ρc,1 ), 0), where Φ1 , ρc,1 are the mean jump rate and critical density of the 1-species jump rate ĝ1 (see [12,Section 5.2.1] for the 1-species case).Thus for any ρ ∈ Ê 2 + there exists a unique maximiser Φ(ρ) ∈ D R for the thermodynamic entropy S(ρ).As in [13,Proposition 2.3] the function ).Note that the thermodynamic entropy coincides with the Legendre transform of the convex thermodynamic pressure log Z : Ê 2 → [0, +∞], that is, Since ∇(log Z) = R = R • exp, it follows by the formula for the derivative of the Legendre transforms that for all ρ ∈ (0, ∞) Since S is convex the matrix D 2 S(ρ) = D(log Φ)(ρ) is symmetric and strictly positive definite for all 13) can be seen as the macroscopic analogue of the compatibility relations (6).
Using the inverse Φ of R on R(D R ), we can parametrise the grand canonical measures νN ϕ , ϕ ∈ D R , that have finite density via so that they are parametrised by their density.We will denote by ), the product measures on the configuration space Å d;2 Consequently, the product and translation invariant equilibrium states have some exponential moments for all ρ ∈ R(D o R ).They have full exponential moments iff + .It is easy to verify that Φ(ρ) has a probabilistic interpretation as the one-site mean jump rate with respect to the product and translation invariant equilibrium state of density ρ ∈ R(D R ), that is One says that the 2-species ZRP is condensing when R(D R ) = Ê 2 + , in which case there exist densities ρ ∈ Ê 2 + for which there is no grand canonical equilibrium state of

and thus condensation occurs precisely when
One says that condensation of the i-th species, i = 1, 2, occurs at the density ρ ∈ Ê 2 + if R c,i (ρ) < ρ i .All cases are possible, that is, at a given density ρ ∈ Ê 2 + no condensation, condensation of exactly one species and condensation of both species simultaneously can occur.These cases induce an obvious partition of the phase space Ê 2 + .As proved in [13], the extension Φ is the correct one for the equivalence of ensembles in the sense that R c gives the correct limiting background density in the thermodynamic limit.In the case of condensation, i.e., when R(D R ) = Ê 2 + , some additional assumption must be imposed on the jump rate g to ensure that for each ϕ ∈ D R ∩ ∂D R , the onesite marginal ν1 ϕ has heavy tails in the direction normal to the set log In case ∂D R is not differentiable at ϕ, (17) is required to hold for the two limiting normal vectors n + ϕ , n − ϕ at log ϕ.As has been proven in [13,Lemma 3.5], a condition on the jump rate g that guarantees the critical equilibrium states have heavy tails in the direction normal to the logarithm of the boundary is the regularity of its tails, in the sense that for any direction υ ∈ S 1 exists as limit and ϕ c;2 : Here H(µ|ν) denotes the relative entropy between two probability measures µ, ν, The translation invariance of canonical and grand canonical ensembles and the superadditivity of the relative entropy imply convergence for any finite set F ⊆ d , i.e., lim that they depend on a finite number of coordinates.Finally, we briefly recall the notions of local equilibrium and hydrodynamic limits and refer to [19] for more details.We say that a sequence of probability measures Here ) is the product measure with slowly varying parameter as- for all H ∈ C(Ì d ) and all bounded cylinder functions f : The hydrodynamic limit (in the diffusive timescale t → tN 2 ) of the n.n.two-species ZRP is an evolutionary PDE, such that entropy-local equilibria are conserved along its solutions (in the diffusive time-scale) in the following sense: If we start the process from an entropy local equilibrium µ N 0 ∈ È(Å d;2 N ), N ∈ AE, of some sufficiently regular initial profile ρ 0 : Ì d → Ê 2 + at time t = 0 and if there exists a sufficiently regular solution ρ of the hydrodynamic equation on [0, T ) × Ì d starting from ρ 0 , then The main goal of this article is to apply the relative entropy method of H.T. Yau in order to prove the hydrodynamic limit of condensing 2-species ZRPs that start from an initial entropy-local equilibrium {µ N 0 } of sub-critical and strictly positive profile ), which is stated as Theorem 3.2 below.A main ingredient in the proof of the hydrodynamic limit is the one-block estimate which is stated as Theorem 3.1.The relative entropy method also requires the existence of a C 1,2+θ loc classical solution 2 for the hydrodynamic limit and applies the Taylor expansion for C 2+θ functions to the function Φ(ρ t ) of the solution ρ t at each time t > 0 (see (45)) in order to estimate the entropy production 2 and to obtain the bound (52), which is essential in the application of Lemma 4.5.The sub-criticality of the solution ρ is also required for the application of the Large Deviations Lemma 4.4.Together with the C 2+θ regularity of ρ t for each t ≥ 0 it is the main assumption on the solution ρ.Furthermore, in the Taylor expansion the quantities Φ i (ρ t ), i = 1, 2, appear in the denominator, so we have to assume that the solution ρ is coordinate-wise strictly positive.
As already mentioned in the introduction the expected hydrodynamic limit of the 2-species ZRP with product measures is a quasilinear parabolic system of the form (1), which in divergence form is given by Here the divergence with respect to the spatial parameter is applied coordinate-wise, and ∇ρ(t, u) := (∇ρ 1 (t, u), ∇ρ 2 (t, u)) ∈ Ê 2×d is the gradient of ρ with respect to the spatial variable u ∈ Ì d .Furthermore, where D 2 S(ρ) = D(log Φ)(ρ) is a strictly positive definite matrix, the second derivative of the thermodynamic entropy, and for all ρ ∈ R(D o R ), the relations (13) hold and In particular, DΦ(ρ) has positive eigenvalues for all with ρ 1 ρ 2 = 0, the matrix DΦ(ρ) is triangular.It follows that although DΦ(ρ) is not necessarily symmetric, it is uniformly parabolic away from the critical densities, that is, for any compact By the work [1] of Amann, which covers uniformly parabolic systems in general form, it is known that for initial data 22), which by continuity will remain in D o R ∩ (0, +∞) 2 on a possibly even smaller time interval.This establishes the local in time existence of C 1,2+θ sub-critical solutions ρ.On the other hand, by the regularity theory of quasilinear uniformly parabolic systems of the form (22), see [5, Theorem 1.2] and the references therein, it is known that weak solutions to such systems exhibit singularities on a closed subset Q ⊆ [0, T ] × Ì d of zero measure.So we can not simply apply the C 2+θ Taylor expansion on the function Φ(ρ t ) for all times t ≥ 0. Furthermore we do not know whether the sub-critical region R(D R ) is an invariant region for the 2-species ZRP system (22).These are the two main reasons that force us to rely on Amann's local in time existence of regular solutions, and prove a local in time version of the hydrodynamic limit.A further study of the PDE system arising as the hydrodynamic limit of a 2-species ZRP, although interesting, is outside of the scope of this article, which is the passage from the microscopic to the macroscopic description.However, in the example of the species-blind ZRP one can take into advantage its relation with a particular 1-species ZRP to obtain the global in time existence of C 1+θ,2+θ loc solutions and a type of maximum principle, in which the sub-critical region plays the role of the invariant domain.We prove this in Theorem 3.3.

The Species-blind ZRP
We now consider two-species local jump rate functions of the form for some function h : AE 0 → Ê + satisfying the non-degeneracy condition h(k) > 0 for all k ∈ AE.Any jump rate g of this form satisfies (6) since for all k ∈ AE 2 and the factorial of such a jump rate is given by The partition function associated to g is given for ϕ ∈ AE 2 0 with ϕ 2 > 0 by where Ẑ is the partition function associated to the 1-species rate function ĝ(k) := kh(k).
So, in what follows, we assume that h is of the form h(k) = ĝ(k) k , k ≥ 1, for some 1species local jump rate function ĝ with regular tails, i.e., such that the limit inferior φc := lim inf k→+∞ ĝ!(k) 1 k > 0 exists as a limit.In this case the function g defined in ( 23) is a two-species local jump rate.Indeed, the non-degeneracy condition (2) and the Lipschitz condition (3) are easy to verify, as we have seen g satisfies the compatibility condition (6) and obviously is the density function associated to the 1-species jump rate ĝ.In particular D Z = ∅ and thus also (8) holds.We will refer to this nearest neighbour 2-species ZRP as the species-blind ZRP corresponding to the 1-species jump rate ĝ.The density function corresponding to g is given by the formula We set Φ := R−1 and we will compute the inverse We have to solve the system for (ϕ 1 , ϕ 2 ).By adding the two equations we obtain that ρ 1 + ρ 2 = R(ϕ 1 + ϕ 2 ).In particular ρ 1 + ρ 2 ∈ R(D R) for all ρ ∈ R(D R ) and ϕ 1 + ϕ 2 = Φ(ρ 1 + ρ 2 ).Substituting ϕ 1 + ϕ 2 with Φ(ρ 1 + ρ 2 ) in both equations in (24), we can solve for (ϕ 1 , ϕ 2 ) to obtain where the last equality above follows from the identity R(ϕ) = ϕ(log Z) ′ (ϕ) for the 1species density and partition functions, since by this identity we have for all ρ ∈ (0, ρc ) that 1 where ρc is the corresponding critical density of the 1-species jump rate ĝ.Consequently, the inverse Φ := R −1 : R(D R ) → D R is given by the formula Thus the expected hydrodynamic equation of the species-blind ZRP is Since ( 26) is the expected hydrodynamic equation of the species-blind ZRP we will refer to it as the species-blind parabolic system.A classical solution to the species blind 26) with 0 ≤ ρ 1 (t, u)+ρ 2 (t, u) < ρc for all (t, u) ∈ [0, T )×Ì d .Note that for any solution ρ = (ρ 1 , ρ 2 ) of the system (26), the sum ρ 1 +ρ 2 satisfies the parabolic equation ∂ t ρ = Φ(ρ) corresponding to the 1-species ZRP of jump rate ĝ(k) = kh(k).This remark will allows us to prove the global in time existence of solutions to the species-blind system.A similar argument was used for two-species simple exclusion processes in [21].
As an example of the nice properties of the species-blind process, we note that the extended mean jump rate Φ : Ê 2 + → D R of the species-blind process can be computed explicitly and is given by where Φ(ρ) = Φ(ρ ∧ ρc ), ρ ≥ 0, is the extended mean jump rate of the 1-species ZRP with jump rate ĝ.

Main Results
A main probabilistic ingredient in the proof of the hydrodynamic limit of ZRPs is the so-called one-block estimate, which is well known under assumptions that exclude condensing ZRPs (e.g., [19,Section 5.4]).Our first result is a version of the one-block estimate for condensing ZRPs, i.e., R(D R ) = Ê 2 + , under the additional assumptions that the local jump rate g is bounded, has a continuous partition function Z, and has regular tails in the sense of (18).We note that these extra assumptions in the one-block estimate and the hydrodynamic limit below are not required in the non-condensing case, i.e., when R(D R ) = Ê 2 + .In the case that R(D R ) = Ê 2 + , but D Z = Ê 2 + , Theorems 3.1 and 3.2 still hold under the (weaker than boundedness) assumption that g has sub-linear growth at infinity in the sense that lim sup In the case that + , no extra assumption is required on g.Given any (cylinder) function f : Å d;2 N → Ê 2 we set Theorem 3.1 (One-block estimate) Suppose that the ZRP is condensing and that the local jump rate g of the ZRP is bounded, has regular tails in the sense of (18) and its partition function Z is continuous on D Z ∩ ∂D Z .Then for any sequence of initial distributions µ N 0 ∈ È(Å d;2 N ) satisfying the O(N d )-entropy assumption, i.e., C(a) := lim sup for some (and thus for any for all functions F ∈ C([0, T ] × Ì d ; Ê 2 ), T > 0; N denotes the expectation with respect to the diffusively accelerated law of the ZRP starting from µ N 0 ∈ È(Å d;2 N ) and Φ is the extension of Φ given by (11).
We note that the extension Φ of the mean jump rate is required in the statement of the one-block estimate, because η ℓ t can be outside the domain of sub-critical densities.This is the correct extension due to the equivalence of ensembles.The proof of this result is given in Subsection 4.1 below.
Next is the general result regarding the hydrodynamic limit of two-species ZRPs.As noted in the introduction, in order to take into account condensing ZRPs, we apply the relative entropy method of H.T. Yau which requires only the one-block estimate and not the full replacement lemma.But this method relies on the existence of sufficiently regular classical solutions of parabolic systems which are known to exist only locally in time, and thus the result is local in time, valid for the time interval that the unique maximal classical solution of (1) established in [1] exists.We denote by As usual, if I ⊆ Ê is an interval, then we write C 1+a,2+b loc (I × Ì d ) for the space of all functions f such that f ∈ C 1+a,2+b (J × Ì d ) for any compact sub-interval J ⊆ I.This is extended coordinate-wise to vector-valued functions; given a subset A ⊆ Ê 2 , we denote by C 1+a,2+b loc Theorem 3.2 (Hydrodynamic limit) Let (S N t ) t≥0 be the transition semigroup of the two-species symmetric n.n.ZRP on the torus Ì d N , N ∈ AE, with condensing jump rate g satisfying the assumptions of the one-block estimate above, and let Φ be the mean jump rate associated to g.Let be the unique maximal solution of the problem (1).Then any initial entropy local equilibrium with N ∈ AE, is an entropy local equilibrium for all t ∈ [0, T max ).In particular, {µ N t } satisfies (21) for all t ∈ [0, T max ).This theorem is proved in Subsection 4.2.We should note that, although the proof of the hydrodynamic limit relies strongly on the assumption that the classical solution ρ takes values in the set R(D o R ) ∩ (0, ∞) 2 for all times t ≥ 0, and so in particular requires the sequence of initial distributions {µ N 0 } to be an entropy local equilibrium of some sub-critical and strictly positive profile ρ 0 ≡ ρ(0, •), the one-block estimate does not require this assumption.It only requires that {µ N 0 } satisfies the O(N d )-entropy assumption, which can hold even for super-critical profiles, having a Dirac mass of order + .We note also that the assumption that ρ([0, T max )×Ì d ) ⊆ (0, +∞) 2 is a technical one, arising from the fact the Φ(ρ t ) appears in the denominator.If one knew that the region Ê 2 + is strongly invariant for the 2-species system in the sense that ρ 1 ∧ρ 2 becomes strictly positive (and sufficiently fast) for the solution ρ, then one can replace the assumption ρ([0, T max ) × Ì d ) ⊆ (0, +∞) 2 with the assumption ρ 0 ≥ 0 as in [23,Remark 3.3] for the 1-species case.Secondly, if one knew that the region (0, +∞) 2 is invariant for the 2-species ZRP system, starting from C 2+θ non-negative initial data ρ 0 : , use the result for strictly positive data and try to pass to the limit as ε → 0. Since we do not pursue the study of the 2-species ZRP system and its invariant regions at the macroscopic level in this article, we consider only local solutions which are strictly positive and sub-critical and whose existence is established by Amann [1].
The next result states that, when starting from sufficiently regular subcritical initial profiles, the species-blind system (26) has solutions defined globally in time.
The proof of this Theorem can be found in Subsection 4.3 and it is obtained by taking into account the fact that the sum ρ 1 +ρ 2 of the two variables of a solution ρ = (ρ 1 , ρ 2 ) of the 2-species blind system is a solution of the scalar parabolic equation ∂ t (ρ) = ∆ Φ(ρ).Here, by using the strong maximum principle for scalar quasilinear parabolic equations and by proving that classical solutions ρ of the species-blind parabolic system do not become negative, we obtain that the the sub-critical region is an invariant region.We believe that Ê 2 + will be an invariant region of the 2-species ZRP parabolic system in general.Yet, since we do not study this question in this article, in order to be rigorous we prove it in this particular case.We should add that the arguments used strongly rely on the relation to the PDE of the single species ZRP associated to the species-blind ZRP by "ignoring" the species, and thus do not easily extend to the general case.
As a corollary, we find that the hydrodynamic limit for the species-blind process holds globally in time; Subsection 4.4 gives the proof.k ∈ (0, +∞] exists as a limit.Assume further that ĝ is bounded if the critical density ρc of the 1-species ZRP is finite. ) is the unique solution to the species-blind parabolic system (26) starting from ρ 0 .

Proof of Theorem 3.1
The proof of the one-block estimate follows closely the proof for the one-species case found in [19,Section 5.4].The differences are twofold.In [19, Section 5.4], the onespecies case is treated, and we extend this result to two species.However, the main difference is that in [19] the non-condensing case is treated, while we cover the condensing case as well.This is shown by applying the equivalence of ensembles (19) as in [23].
The first step in the proof of the one-block estimate is to replace the jump rate g(η(x)) at the site x with the spatial average g(η(x)) ℓ over a box of size ℓ ∈ AE 0 .This is based on the following lemma which is also useful in the proof of Theorem 3.2.The proof is omitted as it is a simple adaptation of the proof for the one-species case [19, Lemma 6.This lemma, a change of variables and the conservation of the number of particles allow us to replace g(η(x)) with the spatial average g(η(x)) ℓ in the statement of the one-block estimate, and thus the one-block estimate is reduced to proving that lim ℓ→∞ lim sup where μN We establish this identity in a sequence of steps.We first estimate the entropy and the Dirichlet form of the density f N is the density of the law µ N t of the ZRP at time t with respect to the product equilibrium state of density ρ * ∈ A. By [19, Proposition A.9.1], for any initial probability measure µ the entropy H(µ t |ν) of the law µ t := µP t of a Markov semigroup (P t ) t≥0 at time t with respect to an equilibrium state π of (P t ) is a non-increasing function of time.Here the equilibrium π need not be unique or approached by µ t as t → +∞.Therefore, since µ N 0 satisfies the O(N d )-entropy assumption, we have for fixed then by [19,Proposition A.9.2] and the convexity of the functional D N , it follows that ), in order to prove the one-block estimate, it suffices to prove that for some ρ * ∈ A lim sup ℓ→∞ lim sup where the supremum is taken among all densities f ∈ L 1 + (ν N ρ * ).In a second step, following the proof of the one-species case [19,Section 5.4] we cut off large densities.Since Lemma 4.1 requires only the O(N d )-entropy assumption, it follows that lim sup Similarly to the 1-species case, under the assumption that g has sublinear growth at infinity in the sense of ( 27) (which always holds when g is bounded), inequality (32) allows us to cut off large densities, by restricting V ℓ to the set of configurations η which satisfy |η ℓ (0)| 1 ≤ C 1 for some constant C 1 > 0. This way the one-block estimate is reduced to proving that for all constants C 0 , C 1 > 0 lim ℓ→+∞ lim sup In a third step, by adapting the steps 2 to 4 of [19,Sect. 5.4.1] to the two-species case, the one-block-estimate is further reduced to showing that for all constants C 1 > 0, lim sup where the canonical measure ν 2ℓ+1,K is considered as a measure on Å d The final step in the proof of the one-block estimate consists in applying the equivalence of ensembles to prove (34).Since the measure ν 2ℓ+1,K is concentrated on configurations with K particles, the integral appearing in (34) is equal to As in the one-species case, by fixing a positive integer k which will tend to infinity after taking the limit as ℓ → +∞, and decomposing the cube Λ d ℓ in smaller cubes of side-length 2k + 1, the one-block estimate is reduced to showing that where S(m, k) denotes the supremum This is the part of the proof where we need the boundedness and the regularity of the tails (18) of the jump rate g as well as the continuity of the partition function Z on Since the sequence {r m,k n } n∈AE defined by Since we assume that g is bounded, it follows by the equivalence of ensembles that ) and thus we can choose a sequence . By the continuity assumption on Z, the grand canonical ensemble is weakly continuous.By this fact, the continuity of R c and the identity Φ The random variables g η(x) , x ∈ d , are uniformly bounded by g ∞ and i.i.d. with respect to ν ∞ ρ for all ρ ∈ R(D R ) and thus they satisfy the L 2 -weak law of large numbers uniformly over all parameters ρ ∈ R(D R ), which shows that the term in the right hand side above is equal to zero.This completes the proof of the one-block estimate and hence the proof of Theorem 3.1.

Proof of Theorem 3.2
Let A be the interior of the set of all strictly positive sub-critical densities, i.e., and let ρ : [0, T max ) × Ì d → A be the maximal classical solution established in [1] of the initial value problem (1) with ρ(0, •) := ρ 0 ∈ C 2+θ (Ì d ; A).We fix a ∈ A and denote by ψ N t the Radon-Nikodym derivative of ν N ρt(•) with respect to ν N a , ) be the relative entropy of µ N t with respect to ν N ρt(•) .We have the following upper bound on the entropy production, proved in [19,Lemma 6.1.4], for every t ∈ [0, T max ), where Denoting by the limiting entropy density, the main step in the application of the relative entropy method is to use this upper bound on ∂ t H N (t) to get an inequality of the form for some constant γ > 0. Since H(0) = 0 by assumption, this implies by Gronwall's inequality that H(t) = 0 for all t ∈ [0, T max ) as required.Of course, in order for Gronwall's inequality to be applicable, H must belong at least in L 1 loc ([0, T max )).This is the context of the next two lemmas.The first is Remark 6.1.2 in [19] for single-species ZRPs.
Since ν N ρ(•) , ν N a are product measures, the Radon-Nikodym derivative Since Z ≥ 1, we have that 1 Za(ρ) = Z(Φ(a)) Z(Φ(ρ)) ≤ Z(Φ(a)) and therefore where here Z for sufficiently small γ > 0. Then the function u → Z(F a (u, γ)) is well defined and continuous on the torus Ì d , so that its Riemannian sums converge.By (39), (40) and the fact that µ N 0 is an entropy local equilibrium, this yields that for small γ > 0, and the proof of Lemma 4.2 is complete.
Proof: By the relative entropy inequality and [19, Proposition A.1.9.1], according to which the function t → H(µ N t |ν N a ) is non-increasing, for all t ≥ 0 and all γ > 0. Since the proper domain of Z a has interior , the function Z a • ρ is a bounded continuous function on the torus, and therefore, by a computation similar to the one in the proof of Lemma 4.2, we obtain , where For the second term, we have for every u ∈ Ì d that Since Φ(ρ)(Ì d ) ⊆ (0, ∞) 2 and Φ(ρ) is continuous, there exists ϕ 0 ∈ D Z such that and since Φ(a) ∈ D o Z and Φ(a) 1+γ /ϕ γ 0 → Φ(a) as γ → 0, we can choose γ 0 > 0 sufficiently small so that Φ(a) 1+γ /ϕ γ 0 ∈ D o Z and Z Φ(a) 1+γ /ϕ γ 0 ≤ Z Φ(a) + 1 for all γ < γ 0 .Consequently, since by Lemma 4.2 {µ N 0 } satisfies the O(N d )-entropy assumption, by (41) for some constant C ≥ 0 for all γ < γ 0 establishing the claim of Lemma 4.3.
The bound (36) on the entropy production can be estimated explicitly.Since ν N ρt(•) , ν N a are product measures, ψ t can be computed explicitly.Then by differentiating, using the chain rule, the fact that ρ is a solution of the hydrodynamic equation, the relations = R i (ϕ), i = 1, 2 and the relation ( 13) we obtain As already noted in [16], in this computation in the two-species case, one has to use the macroscopic analogue (13) of the compatibility relations (6).
For the other term, by computations of the action of the generator on ψ N t similar to the ones for the single-species case in [19], Since Φ(ρ t ) is C 2+θ for some θ > 0 and the n.n.transition probability has mean zero, the Taylor expansion for C functions yields (with the renormalisation p( d ) = 2d) that Here, for any T ∈ [0, T max ), the remainder r N (t) satisfies the bound where in order to obtain the term t 0 H(s)ds we used Lemma 4.3 to pass the limit inside the integral and lim sup ℓ,N →+∞ denotes the lim sup as N → +∞ and then ℓ → +∞.
To complete the proof of Theorem 3.2, it remains to show that for each t ∈ [0, T max ) we can choose γ > 0 small enough so that the rightmost term in (51) vanishes.We begin by noting that the function G : [0, T max ) × Ì d × Ê 2 + → Ê defined in (50) satisfies + , which for any t ∈ [0, T max ) yields the inequality sup for some constant C t < +∞.Since for any t ∈ [0, T max ) we have ρ( ) is bounded away from the critical densities ϕ c ∈ ∂D Z , and thus there exists ε > 0 such that sup o .It follows that by choosing γ t small enough so that γ t C t < ε, we can pass the limit superior as N → +∞ and then ℓ → +∞ inside the time integral in the rightmost term in (51).Thus in order to complete the proof it remains to show that for each t ∈ [0, T max ) we can choose γ t > 0 small enough so that lim sup The proof of (53) relies on a corollary of the Laplace-Varadhan lemma [4, Section 4.3] for the large deviations principle satisfied by the independent family of the occupation variables {η(x)} x∈ d with respect to the invariant measure ν ∞ ρ on the infinite lattice d for some ρ ∈ A. Since the one-site marginal ν 1 ρ has some exponential moments for ρ ∈ A, by Cramér's theorem, the large deviations functional of the family {η(x)} x∈ d is given by the Legendre transform Λ * ρ of the logarithmic moment-generating functional Λ ρ .Note that (15) implies that modulo an affine function depending on ρ, the rate functional Λ * ρ coincides with the thermodynamic entropy S, that is We omit the proof of this Lemma as it is a simple adaptation of the corresponding result in the one-species case, [19,Lemma 6.1.10].By the bound (52) the function for each fixed t ∈ [0, T max ).Therefore, if we choose γ t > 0 small enough so that 2γ t C t (e 1 + e 2 ) ∈ u∈Ì d Λ ρt(u) o , then for all γ ∈ (0, γ t ) the function γG t satisfies the assumptions of Lemma 4.4, and thus for γ ∈ (0, γ t ) the term in (51) is bounded above by To complete the application of the relative entropy method, it remains to show that by reducing γ t > 0, t ∈ [0, T max ), if necessary, this last term is non-positive.We note that this would follow if we had a bound of the form where and so for γ ∈ (0, γ t ) we would have for all (u, λ) ∈ Ì d × Ê 2 + .Then by choosing γ t > 0 small enough so that in addition , it would follow that (56) is non-positive, and the proof would be complete.The bound (57) is proved in Lemma 4.5.Before we proceed with the proof of Lemma 4.5, we recall some facts on recession functions of convex functions.It is well known [22,Theorem 8.5] that if ψ is a proper lower semi-continuous convex function, then so is its recession function.Using the equivalent definition of recession functions via the recession cone of their epigraphs [22,Section 8], one can express the recession function by the formula

Given a lower semicontinuous proper convex function
(see [10, (12.7.1)]).Particularly useful in the proof of the following lemma is the characterisation of the interior of the proper domain of a convex function ψ via the recession function of its Legendre transform, as stated in [10, (12.7.3)], x ∈ Ê d x, y < (ψ * ) ∞ (y) . (59) Applying (59) to the thermodynamic pressure P := log Z, Z := Z • exp, we get In other words, is a parametrisation of the boundary ∂D Z .This may be compared with [13, (2.14)].Consequently, the part of the boundary ∂D Z on the strictly positive quadrant is given by the parametrisation e S∞(υ)υ , υ ∈ S 1 ∩ (0, ∞) 2 .Along the two axes ϕ 1 = 0 and ϕ 2 = 0, there is only one-species of particles and the critical fugacities in these directions are fugacities of one species ZRPs.
Motivated by the previous sketch, we choose ε > 0 such that K ε := K (ε) ⊆ A, where K (ε) := x∈K D(x, ε), and for any M > 0 we separate the region K , where We prove the claim on each region individually.Obviously the set E M ε is compact and so since the functions Ψ and Λ * are jointly continuous, the claim holds on the region E M ε .
We turn to the region E ε 0 .By its definition, for any (ρ, λ) ∈ E ε 0 we have that λ By the first order Taylor expansion of Φ i , i = 1, 2 around the point ρ ∈ K, Since Φ is smooth on the set A, the matrix D 2 Φ i (ρ) is symmetric for all ρ ∈ A. Denoting by λ i ± (ρ) the real eigenvalues of D 2 Φ i (ρ) we have Furthermore, by the continuity of the eigenvalues λ i ± as functions of ρ ∈ A, So by (62), we have that For the denominator in (61), we note that the rate functional Λ * ρ is C 1 on (0, ∞) 2 and C 2 on A with where S is the thermodynamic entropy.Since Λ * ρ and its derivative vanish at ρ, by Taylor expansion of Λ * ρ around ρ ∈ K Denoting by λ − (ρ) > 0 the minimal eigenvalue of the strictly positive definite matrix D 2 S(ρ), we have by continuity that and yields the bound (61) in the region E ε 0 .It remains to show that the supremum is finite in the region E ∞ M for some M > 0. On one hand, it follows from ( 16) and the compactness of K that Ψ satisfies a bound of the form for some constants C 0 , C 1 ≥ 0. So, to complete the proof, it suffices to show that Λ * has at least linear growth in E ∞ M as |λ| → +∞, i.e., lim where of course the limit as M → +∞ exists as an increasing limit.We begin by noting that lim We choose a sequence {λ n } ⊆ Ê where the equality in the right-hand side holds by (58).Since Φ Then by (63) there exists n 1 ∈ AE such that By (54) and taking into account the fact that Z ≥ 1, we have that for all n ≥ n 1 and all ρ ∈ K, But by the compactness of K, we have that log and then for all n ≥ n 1 ∨ n 2 and all ρ ∈ K This proves that lim inf which completes the proof of Lemma 4.5.Since Lemma 4.5 establishes the missing bound (57), the proof of Theorem 3.2 is complete.

Proof of Theorem 3.3
By [1], it is known that quasilinear parabolic systems have unique maximal classical solutions when starting from initial profiles of class C 2+θ , θ ∈ [0, 1).To show that classical solutions of the blind-species system (26) are global in time, we prove first a maximum principle asserting that the region is invariant under the evolution of the species-blind system.Here ρc (0, +∞] is the critical density of the 1-species ZRP associated with the species-blind ZRP.The proof of this version of the maximum principle for systems of the form (26) relies on the maximum principle for quasilinear PDEs in divergence form found in [2].Since, as we will see, the solution ρ cannot lose regularity, we will obtain the existence of global in time classical solutions.
Since ρ 2 is C 1,2 , it follows that the function A Ψ is C 1 and ∂ υ A Ψ (t, u, r, υ) = ∂ r Ψ(t, u, r)I where I ∈ Ê d×d denotes the identity matrix.By a simple calculation, ∂ r Ψ(t, u, r) = H(r, ρ 2 (t, u)), where H : B ∞ → Ê is given by We have that inf for all δ ∈ [0, +∞] and it is obvious that for some constants c, C ≥ 0. Since the function ρ 1 + ρ 2 is also a solution of the scalar equation ∂ t ρ = ∆ Φ(ρ) with the same initial data ρ 0 , we have by the uniqueness of solutions that a ≡ Φ(ρ 1 + ρ 2 ) We consider the system which is obviously decoupled and can be solved by solving the scalar linear second order parabolic equation twice with initial conditions ρ 01 and ρ 02 .This scalar equation is given in general form by where a ij = aδ ij , b i = ∂ i a and c = ∆a.We note that since a satisfies (72) and a ij = aδ ij , the matrix (a ij ) is uniformly elliptic.Also, since a ∈ C 1+θ,2+θ (Ê + × Ì d ), the coefficients = υ, log υ + µ c , with the convention υ i log υ i = 0 if υ i = 0 since x log x → 0 as x → 0. Finally, points k ∈ AE 2 0 with k i = 0 for some i = 1, 2 contribute to the limit only if υ = e i for some i = 1, 2. For such points k ∈ AE 2 0 , we have k 1 !k 2 != |k| 1 !, and so the first term in the right hand side of (77) vanishes, which agrees with the fact that υ, log υ = 0 if υ = e i , i = 1, 2. This is to be expected, since in the directions υ = e i with i = 1, 2 in the phase space we have only one of the two species of particles, which when on their own perform the underlying 1-species ZRP with critical chemical potential μc = log φc .This completes the proof that g has regular tails.

2 + 2 +
a continuous function of the direction υ ∈ S 1 + .Note that instead of the exponent p = 2, we could have used any p ∈ [1, +∞], replacing the Euclidean sphere S 1 + with the sphere S 1 p,+ := {x ∈ Ê |x| p = 1} with respect to the ℓ p -norm on Ê .According to the equivalence of ensembles [13, Theorem 3.1], if the jump rate has regular tails when R(

2 .
Structural properties of the mobility matrix DΦ : R(D o R ) → Ê 2×2 can be inferred by the properties of DR.For example, for all ρ

Corollary 3 . 1
Let (S N t ) t≥0 be the transition semigroup of the diffusively rescaled speciesblind symmetric n.n.ZRP on the torus Ì d N corresponding to a 1-species jump rate ĝ such that φc := lim inf k→+∞ ĝ!(k) 1