Relaxation Schemes for Interacting Exclusions

We investigate the interaction of one-dimensional asymmetric exclusion processes of opposite speeds, where the exchange dynamics is combined with a creation-annihilation mechanism, and this asymmetric law is regularized by a nearest neighbor stirring of large intensity. The model admits hyperbolic (Euler) scaling, and we are interested in the hydrodynamic behavior of the system in a regime of shocks on the inﬁnite line. This work is a continuation of a previous paper by Fritz and Nagy [ FN06 ] , where this question has been left open because of the lack of a suitable logarithmic Sobolev inequality. The problem is solved by extending the method of relaxation schemes to this stochastic model, the resulting a priory bound allows us to verify compensated compactness.


Introduction and Main Result
A rigorous treatment of hyperbolic (Euler) scaling problems requires specific conditions because a direct compactness argument is not available; the celebrated Two Blocks Lemma, see Theorem 4.6 and Theorem 4.7 of [GPV88], can not be extended to macroscopic block averages. Just as in the case of parabolic energy inequalities, the diffusive (elliptic) component of the microscopic evolution does vanish in a hyperbolic scaling limit, thus we can not control spatial fluctuations. This problem is closely related to the formation of shock waves resulting in a breakdown of existence of classical solutions to the macroscopic equations. Assuming smoothness of the macroscopic solution, this difficulty can be avoided by means of the relative entropy method of [Yau91], hydrodynamic limit (HDL) for a large class of models can be derived in this way. In the case of attractive systems coupling and other specific techniques can be applied even in a regime of shocks, therefore advanced methods of PDE theory, as the entropy condition of S. Kruzkov play a crucial role in the proofs, see e.g. [Rez91], [KL99] and [Bah04] with some further references. Since the models of [FT04] and [FN06] are not attractive, a new tool, the stochastic theory of compensated compactness is used there to pass to the hydrodynamic limit; the microscopic entropy flux is evaluated by means of a logarithmic Sobolev inequality (LSI). Our present model is more difficult, these techniques alone are not sufficient to control the mechanism of creation and annihilation. The main purpose of this paper is to extend the PDE method of relaxation schemes to microscopic systems with a hyperbolic scaling, we are going to expose several versions of the argument.

Relaxation schemes:
In case of non -attractive systems with general initial conditions the method of compensated compactness, cf. [Tar79] or [Ser00] and [Daf05] is an effective tool, its applications to microscopic systems are discussed in the papers [Fri01], [Fri04], [FT04], [FN06] and [Fri09]. However, compensated compactness alone is not sufficient in the present situation, it should be supplemented by another tool called the method of relaxation schemes in the PDE literature, see [Liu87], [CL93] and [CLL94] for the first results, [GT00] or [Daf05] for explanation and further discussions. The basic idea is not difficult: the single conservation law ∂ t u + ∂ x f (u) = 0 in one space dimension can be obtained as the zero relaxation limit of the linear system (1.1) with a nonlinear source (relaxation) term on its right hand side. We have B > 0, so unless ( f (u ) − v ) → 0 , this term might explode if 0 < → 0 ; but the negative sign of v / is encouraging. To see a successful relaxation, we have to find a clever Liapunov function h = h(u, v) such that whence under suitable conditions on f and the initial data follows, where C 0 does not depend on . Let us remark that the equation above is easily controlled if we have a function J = J(u, v) such that h u (u, v)∂ x v+h v (u, v)∂ x u = ∂ x J , that is h is a Lax entropy for (1.1) if B = 0 , but there are other possibilities, too. Therefore it is reasonable to expect v ≈ f (u ) in a mean square sense as → 0 , consequently the limit u of u satisfies ∂ t u + ∂ x f (u) = 0 in a weak sense, see e.g. Sections 6.7 and 16.5 of [Daf05] for complete proofs which are technically much more complex than the presentation here.
The hyperbolic scaling limit of our model of interacting exclusions with creation and annihilation shall be understood as a microscopic version of the zero relaxation limit for the LeRoux system where x ∈ , u ∈ [−1, 1] , ρ ∈ [0, 1], F (u) := (1/3)(4−(4−3u 2 ) 1/2 and B ≥ 1/2 , consequently the limiting equation for u reads as ∂ t u + ∂ x (F (u) − u 2 ) = 0 . The first proof uses h = (1/2)(F (u) − ρ) 2 as our Liapunov function, the result obtained in this way can be improved by choosing h as a Lax entropy of the Leroux system.

The model:
In view of our naive physical picture of electrophoresis, we consider ±1 charges moving in an electric field on such that positive charges are jumping to the right at rate 1 if allowed (i.e. there is no particle on the next site), negative charges are jumping to the left at unit rates. The exclusion rule is in force: two or more particles (charges) can not coexist at the same site. However, when two opposite charges meet, then they either jump over each other at rate 2 , or they are both annihilated at rate β > 0 . To compensate annihilation, charges of opposite sign can be created at neighboring empty sites, again at rate β . Because of technical reasons, the process is regularized by a nearest neighbor stirring of intensity σ > 0 , all elementary actions are independent of each other. The mathematical formulation of the model is summarized as follows.
The configuration space, Ω of our system is the set of sequences ω := (ω k ∈ {0, 1, −1} : k ∈ ) , i.e. ω k is interpreted as the charge of the particle at site k ∈ , ω k = 0 indicates an empty site, and η k := ω 2 k denotes the occupation number. The process is composed of the following local operations. If b = (k, k + 1) is a bond of , i.e. b ∈ * , then stirring ω ↔ ω b means that ω k and ω k+1 are exchanged, the rest of the configuration is not altered. The action ω ↔ ω b+ creates a couple of particles on the bond b := (k, k + 1) if it is empty: (ω b+ ) k = +1 and (ω b+ ) k+1 = −1 if ω k = ω k+1 = 0 , other coordinates are not changed. Annihilation of a couple, ω ↔ ω b× means that (ω b× ) k = (ω b× ) k+1 = 0 if ω k = +1 , ω k+1 = −1 ; (ω b× ) j = ω j otherwise. The stochastic dynamics is then defined by the following formal generators, see [Lig85] on the construction of interacting particle systems. These operators are certainly defined for finite functions, i.e. for ϕ : Ω → depending only on a finite number of variables, and the set of finite functions is a core of the full generator. The totally asymmetric process of interacting exclusions (INTASEP) is generated by . This generator lets ⊕ particles jump to the right, particles jump to the left at rate 1 , if allowed, while a collision ⊕ → ⊕ occurs at rate 2 . Both particle number η k and total charge ω k are preserved by INTASEP. The two-parameter family, {λ ρ,u : 0 < ρ < 1 , 0 ≤ |u| < ρ} of translation invariant stationary product measures is characterized by λ ρ,u (η k ) = ρ and λ ρ,u (ω k ) = u ; here and later on we use the short hand notation λ(ϕ) ≡ ϕ dλ . The degenerated stationary states λ ρ,u with ρ = 0 , ρ = 1 or |u| = ρ play no role in our calculations. The study of interacting exclusions and some related models goes back to the paper [TV03] by B. Tóth and B. Valkó, where HDL of INTASEP with hyperbolic scaling is derived in a smooth regime with periodic boundary conditions.
The creation -annihilation process (CRANNI) is generated by * : Only total charge ω k is preserved by CRANNI, and within the class λ ρ,u : 0 ≤ |u| < ρ its stationary measures are characterized by the principle of microscopic balance: The smaller root, F (u) is between 2/3 and 1 , while the second one is not allowed because F * (u) ≥ 5/3 for all u ∈ [−1, 1] . Consequently λ * u := λ F (u),u is a stationary measure of the process generated by * if |u| < 1 , and λ * u (ω k ) = u , while λ * u (η k ) = ρ = F (u) . Note that these measures are reversible with respect to * , this fact shall be exploited several times in our computations.
Since we want to pass to HDL in a regime of shocks by means of the theory of compensated compactness, our process has to be regularized, e.g. by an overall stirring of large intensity, cf. [FT04] and [FN06]. The generator of the stirring process reads as (1.7) This process is reversible with respect to any λ ρ,u , and both η k and ω k are preserved. HDL of the process generated by σ := o + σ in a regime of shocks was determined in [FT04], here we are interested in the hyperbolic scaling limit of the creation -annihilation process generated by := o + β * + σ , see (1.3), (1.4) and (1.7) for definitions, where β and σ are positive parameters to be specified later. The main goal of our paper is to develop a microscopic theory of relaxation schemes, by means of which the macroscopic behavior of this creation -annihilation process can be described.
Let us remark that in the paper [FN06] HDL of the process generated by κ := o + α κ + σ was investigated. The spin -flip generator κ reads as Since κ violates conservation of total charge, and Therefore the family of stationary product measures is just {λ κ ρ := λ ρ,κρ : 0 < ρ < 1} such that λ κ ρ (η k ) = ρ and λ κ ρ (ω k ) = κρ . Although we can not improve results of [FN06] in that way, it might be interesting to see that this model also exhibits relaxation.

Currents:
To understand the microscopic structure of our model, let us summarize some more information on the generators; j π,? below denotes the current of a conservative quantity π , which is induced by a generator ? . By direct computations we . The case of is trivial: ω k = ∆ 1 ω k and η k = ∆ 1 η k , where ∆ 1 ξ k := ξ k+1 + ξ k−1 − 2ξ k for any sequence, ξ indexed by , thus j ωs k (ω) = ω k − ω k+1 and j ηs k (ω) = η k − η k+1 are the associated currents. The spin -flip dynamics does not induce any current of η because κ η k = 0 . Moreover, the identity κ ω k = κ (ω k − κη k ) = 2(κη k − ω k ) indicates relaxation in the sense that ω k ≈ κη k in the scaling limit, see the heuristic explanation in Section 1.4 below.
The action of the creation -annihilation process is less transparent. We have (1.5). Since λ * u is reversible with respect to * , the effect of j ω * vanishes in the hydrodynamic limit; λ * u (j ω * k ) = 0 . On the other hand, where b = (k, k + 1) and b− := (k − 1, k) , whence λ ρ,u ( * η k ) = 2C(ρ, u) = (3/2)(ρ − F (u))(ρ − F * (u)) . Observe now that the second factor is negative because F * (u) ≥ 5/3 , thus we have a good reason to suspect that the terms * η k give rise to relaxation: ρ ≈ F (u) in the scaling limit. This problem, however, is more difficult than the previous one because * η is not a linear function of ω and η , see the end of Section 1.4, and also Section 1.5 for some hints.

Macroscopic equations:
Under hyperbolic scaling of space and time, at a level ∈ (0, 1] of scaling the scaled densities read as ρ (t, x) := η k (t/ ) and u (t, x) := ω k (t/ ) if |x − k| < /2 ; later on we shall redefine these empirical processes in terms of block averages. A formal application of the principle of local equilibrium suggests that in the case of interacting exclusions ρ and u converge in some sense to weak solutions of the following version 2 of the LeRoux system: for a correct derivation see [TV03] or [FT04] concerning o or σ = 0 + σ , respectively. In the second case we had to assume that σ = σ( ) → +∞ such that σ( ) → 0 and σ 2 ( ) → +∞ as → 0 ; the uniqueness of the limit is not known in a regime of shocks. Under the same assumptions on σ , in the paper [FN06] it was shown that HDL of the spin -flip dynamics generated by κ results in a Burgers equation: (1.14) and the limit is unique even in the regime of shocks. As we suggest in the next subsection, and demonstrate at the end of the paper, (1.14) can also be derived as the zero relaxation limit of the modified LeRoux system allowing us to do the replacementũ ≈ κρ . Since κ η k = 0 and κ ω k = 2(κη k − ω k ) , this heuristic picture reveals quite well the microscopic structure of the spin -flip dynamics, note that the vanishing terms, σ( ) ∂ 2 xρ and σ( ) ∂ 2 xũ have been omitted. In view of the previous subsection, see (1.9), (1.11) and (1.12) in particular, a formal calculation yields (1.16) as the macroscopic equation for the creation -annihilation process generated by := o + β * . Assuming smoothness of the macroscopic solution, it would not be difficult to formulate and prove the statement in a rigorous manner. Let us remark that the flux F (u) − u 2 is neither convex nor concave, thus the structure of shocks developed by this equation is rather complex. It is important that the graph of F (u) − u 2 does not contain any linear segment, thus the uniqueness theorem of [CR00] applies. The relaxation scheme for (1.16) is less convincing than that of (1.15), it can be written as As it has been stressed already in Section 1.1, in order to demonstrate relaxationρ ≈ F (ũ ) , we have to find an effective Liapunov function, possibly a Lax entropy for (1.13). Since (ρ − F (u))C(ρ, u) ≤ −(1/2)(ρ − F (u)) 2 , even H := (1/2)(ρ − F (u)) 2 seems to be an effective Liapunov function, see Section 1.1, and (1.2) in particular.

Thermodynamic entropy:
Conservation laws play a fundamental role in the study of nonlinear hyperbolic systems, see [Ser00] or [Daf05] with references on the original work by Peter D. Lax. The function S = S(ρ, u) of the state variables ρ, u ∈ is a Lax entropy with flux Φ = Φ(ρ, u) for a system of two conservation laws, as the LeRoux system, if ∂ t S + ∂ x Φ = 0 along classical solutions. Such a couple (S, Φ) is called a Lax entropy -flux pair ; in case of (1.13) these are characterized by (1.18) Differentiating the right hand sides above with respect to u and v , a linear wave equation is obtained for S , which admits a rich class of solutions, cf. [Ser00].
An interesting example, the relative entropy of λ ρ,u and λ v,π at one site is defined as By a direct computation (1.21) therefore the thermodynamic entropy satisfies (1.19), consequently it is really a Lax entropy for (1.13).
Let us demonstrate at an intuitive level that thermodynamic entropy might be an effective Liapunov function for our relaxation schemes. For this purpose the parameters v and π should be specified such that S u (ρ, κρ) = 0 in case of spin -flips, while S ρ (F (u), u) = 0 for creation and annihilation.
Choosing v = 1/2 and π = κ/2 in the definition of S , we see that consequently the spin -flip dynamics exhibits relaxation to the Burgers equation. Although this problem has been solved already in [FN06] with another method, some remarks on this approach are added at the end of the paper.
To understand relaxation of creation and annihilation, set e.g. π = 0 and v = F (0) = 2/3 in the definition of S , then whence by a direct computation which is the necessary bound for relaxation to (1.16).
Unfortunately, our probabilistic calculations presuppose that the underlying entropy has bounded first and second derivatives, therefore we have to look for something else. In Section 3 we show that, although it is not a Lax entropy, H(ρ, u) := (1/2) ρ − F (u) 2 is an effective Liapunov function even for the microscopic system of creation and annihilation, while H κ (ρ, u) := (1/2)(u − κρ) 2 applies in the case of spin flips. Computations are quite simple in both cases, but the results are not as good as possible. In case of the creation -annihilation process generated by = * + σ , this statement can be improved a bit by means of a clever Lax entropy -flux pair, see Sections 1.6 and 5.6. The optimal result on the spin -flip dynamics is that of [FN06], cf. Section 5.7.

Main result:
It is well known that (1.16) develops shocks in a finite time, and uniqueness of its weak solutions breaks down at the same time, thus we must be careful with definitions. A measurable u : 2 . Here and below a subscript "c" refers to compactly supported functions, + := [0, +∞) , 2 + := + × , and C 1 co ( 2 + ) is the space of continuously differentiable ψ : 2 → with compact support in the interior of 2 + . The notion of Lax entropy plays a fundamental role in the study of weak solutions for all convex entropy -flux pairs (h, J) and compactly supported 0 ≤ ψ ∈ C 1 co ( 2 + ) . The scaled density field of charge ω is defined as The initial conditions are specified in terms of a family µ ,0 of probability measures, we are assuming that in probability for all ϕ ∈ C c ( ) , where −1 ≤ u 0 ≤ 1 is a given measurable function. We are considering the process generated by = o + β( ) * + σ( ) , in its simplest form our main result can be stated as follows.
in probability for all ψ ∈ C c ( 2 ) ; this u(t, x) is the uniquely specified weak entropy solution to (1.16) with initial value u 0 .

Some remarks:
The empirical process u shall be redefined in terms of block averages, in that case we get convergence in the strong local topology of L 1 ( ) , see Theorem 5.1.
It is quite natural to fix the value of β > 0 because it is a parameter of the basic model, our conditions mean that β( ) can not be too small or too large. We can not improve the upper bound (growth condition) of β , it is needed to control βj ω * k , and β * η k in particular. However, the lower asymptotic bound σ 2 β 2 → ∞ of β can be relaxed to σ( )β( ) → +∞ by using a Lax entropy of (1.13) instead of the trivial Liapunov function H = (1/2)(ρ − F (u)) 2 , see Theorem 5.2 in the last section.
The relaxation of the spin -flip dynamics to the Burgers equation is discussed in Sections 5.4. In this case H κ = (1/2)(u − κρ) 2 is a nice Liapunov function, but the result is weaker that that of [FN06]. We are sorry to tell that entropy is not really helpful here because, unless κ = 0 , we can not construct the required Lax entropy -flux pair of the LeRoux system.

An Outline of the Proof
As far as possible we follow the argument of [FN06], which is based on the stochastic theory of compensated compactness, while the necessary a priori bounds follow from the logarithmic Sobolev inequalities we do have for and κ . In fact, in [FN06] the second LSI is used to replace block averages of ω k with those of κη k ; in the present case this second step should consist in a replacement of block averagesη l,k of η with F (ω l,k ) , whereω l,k denotes the corresponding block average of ω . However, we do not have any effective LSI involving * ; the required replacement will be carried out by exploiting relaxation of the microscopic system.

Block averages:
As it is more or less obligatory in the microscopic theory of hydrodynamics, first we rewrite the empirical process in terms of block averages. For l ∈ and for any sequence ξ indexed by we define two sequences of moving averages, namelyξ l = (ξ l,k : k ∈ ) and The usual arithmetic meanξ l,k is preferred in computing canonical expectations, the reason for using the "more smooth" averagesξ l,k is rather technical. 3 For convenience the size l = l( ) of these blocks is chosen as the integer part of (σ 2 / ) 1/4 , then our conditions on σ = σ( ) imply while l 2 ( ) → +∞ as → 0 . Because of technical reasons we have to assume that l( ) exceeds a certain threshold l 0 ∈ . For convenience we may, and do assume that and β( )l 2 ( ) → +∞ as → 0 ; the first and last relations are responsible for the conditions of Theorem 1.1. From now on the block size l = l( ) is specified as above, and these relations will frequently be used in the next coming computations, sometimes without any reference.
Our first statements on HDL will be formulated in terms of a modified empirical processû . It is defined asû (t, x) :=ω l,k (t/ ) if |x − k| < /2 , andP denotes the distribution ofû ; several topologies can be introduced to study limit distributions ofP as → 0 . The usual L p norm of a measurable ψ : 2 + is denoted by ψ p , and 〈ϕ, ψ〉 is the scalar product in L 2 ( 2 + ) . In general û 1 = +∞ , therefore we have to localize convergence by multiplying with a test function φ ∈ C 1 co ( 2 + ) , say. The local strong convergence of ψ n to ψ in L p means that φψ n − φψ p → 0 for all φ , while the local weak convergence ψ n ψ is defined by 〈φ, ψ n 〉 → 〈φ, ψ〉 whenever φ ∈ C 1 co ( 2 + ) . The allowed class of test functions φ can be enlarged by means of the Banach -Steinhaus theorem. Since |û (t, x)| ≤ 1 for all (t, x) ∈ 2 + , the set of all realizations of the empirical process is relative compact in the local weak topology of L 2 ( 2 + ) , therefore is a separable metric space, and the family {P : > 0} is tight in this sense.

Measure -valued solutions:
The notion of Young measure is a most convenient tool for the description of all limit distributions of our empirical processû , cf. [Tar79], [Daf05] or [Ser00]. Let Θ denote the set of measurable families, θ of probability measures θ = {θ t,x (du) : (t, x) ∈ 2 + } such that θ t,x is a probability measure on [−1, 1] for each (t, x) ∈ 2 + , and θ t,x (h) is a measurable function of (t, x) whenever h : [−1, 1] → is measurable and bounded; θ t,x (h) denotes expectation of h with respect to θ t,x . We say that θ ∈ Θ is a measure -valued solution to the macroscopic equation , 1] is represented by a family θ ∈ Θ of Dirac measures such that θ t,x is concentrated on the actual value u(t, x) of u ; this θ is called the Young representation of u . Therefore any weak solution is a measure -solution. On the other hand, any θ ∈ Θ can be identified as a locally finite measure m θ by d m θ := d t d x θ t,x (du) on := 2 + × [−1, 1] ; let M θ ( ) denote the set of such measures m θ equipped with the associated local weak topology. In view of the Young representation, our empirical processû can be considered as a random elementm θ , of M θ ( ) ; letP ,θ denote its distribution. The family {P ,θ : > 0} is obviously tight because the configuration space, Ω is compact; here and also later onˆ θ denotes the set of weak limitsP θ := lim n→∞P (n),θ obtainable as (n) → 0 . In Section 4 we prove the following preliminary result.
Proposition 2.1. Any limit distributionP θ ∈ˆ θ of the Young representation ofû is concentrated on a set of measure -valued solutions.
This is easy, but uniqueness of measure -valued solutions is rather problematic, see e.g. [Rez91] with further references. By means of the stochastic theory of compensated compactness, first we prove the Dirac property of the limiting Young measure, which means that limit distributionsP θ ∈ θ ofP ,θ are sitting on a set of measurable functions. Therefore we have convergence to a set of weak solutions, and uniqueness of weak solutions to a single conservation law is a well settled issue.

Entropy production:
The microscopic version of entropy production X = ∂ t h + ∂ x J is defined as a distribution: for ψ ∈ C 1 c ( 2 ) and entropy -flux pairs (h, J) of (1.16) we introduce This follows by a formal integration by parts if ψ ∈ C 1 c ( 2 + ) and ψ(0, x) = 0 ∀ x ∈ . Calculating the stochastic differential of where L , J and M are defined as follows.
while N (ψ, h) is a numerical error due to the lattice approximation of the space derivative, see (4.1).
Calculation of X is quite easy when h(u) ≡ u because ω k is a difference of currents along adjacent bonds, thus rearranging sums by performing discrete integration by parts, the test function nicely absorbs the factor −1 of . This is the way of proving convergence to the set of measure -valued solutions, see the proof of Proposition 2.1 in Section 4.5. Compensated compactness is applied then to show the Dirac property of these measure -valued solutions, it is based on a delicate evaluation of entropy production for some couples of entropy -flux pairs.

Compensated compactness:
In view of the stochastic version of the Tartar -Murat theory of compensated compactness, cf. [Fri01], [Fri04] or [FT04], we have to find a decomposition and B (φ) are random variables depending only on φ and h . We say that a random functionalX (ψ, h) is correctly decomposed ifX =Ŷ +Ẑ as summarized above, i.e.Ŷ satisfies (2.8), whileẐ satisfies (2.9). The stochastic version of the celebrated Div-Curl Lemma reads as follows.
Proposition 2.2. Let (h 1 , J 1 ) and (h 2 , J 2 ) denote a couple of continuously differentiable entropyflux pairs of (1.16), and suppose that both X (ψ, h 1 ) and X (ψ, h 2 ) are correctly decomposed. With probability one with respect to any limit distribution,P θ ∈ˆ θ of the Young representation ofû we have for almost every (t, x) ∈ 2 + .
Our main task now is the verification of conditions (2.8) and (2.9) above. Most terms on the right hand side of (2.6) will be split into further ones, and we shall show in Section 4 that each of them satisfies either (2.8) or (2.9). The martingale component and the numerical error both vanish, but in a regime of shocks σ( )L s 0 as → 0 . The crucial part of the proof is to show that L o and J cancel each other, while L * disappears when → 0 . The logarithmic Sobolev inequality for , and the relaxation mechanism induced by creation and annihilation are applied at these steps. The slightly sophisticated construction of the empirical process,û is also relevant.
A slightly simplified version of the proof yields lim sup →0 X (ψ, h) ≤ 0 in probability if h is convex and ψ ≥ 0 , whence by tightness of the family {P ,θ : > 0} we get the Lax inequality in terms of the Young measure. With probability one with respect to any limit distributionP θ ∈ˆ θ ofP θ , we have whenever (h, J) is a convex entropy -flux pair and 0 ≤ ψ ∈ C 1 co ( 2 + ) . The Dirac property of the Young measure follows from (2.10) in several situations. Since we are considering a single conservation law, to prove convergence of the empirical processû to a set of weak solutions, it is sufficient to apply (2.10) to two entropy -flux pairs only. Let h 1 (u) :≡ u , J 1 (u) := F (u) − u 2 , and h 2 (u) := F (u) − u 2 with the associated flux J 2 ; it is defined by J 2 (0) = 0 and J 2 (u) = J 2 1 (u) . In view of Theorem 16.4.1 and Theorem 16.4.2 in [Daf05], the following statement holds true. Proposition 2.3. Suppose that the above couple of entropy -flux pairs satisfies (2.10), then any limit distributionP θ ∈ˆ θ of the Young measure is Dirac, i.e. it is concentrated on a set of measurable functions.
As a direct consequence we obtain (1.22) from Proposition 2.1, while (2.11) turns into the weak entropy condition (1.23), which imply uniqueness of the limit by Main Theorem of [CR00].
The verification of the conditions of Proposition 2.2 is mainly based on inequalities involving relative entropy and the associated Dirichlet form; the basic ideas go back to [GPV88], see also [Var94] and [Yau97] with further references. These computations are supplemented with an extension of the method of relaxation schemes to microscopic systems, see Lemma 3.6 and Sections 5.4 and 5.6. Since we are going to treat hydrodynamic limit in infinite volume, we have to control entropy flux by means of its rate of production, cf. [Fri90], [Fri01] and [FN06] for some earlier results in this direction. This a priori bound allows us to apply the robust LSI and the relaxation scheme of the microscopic process, see Lemma 3.3, Lemma 3.4 and Lemma 3.6. Most technical details of this estimation procedure have been elaborated in [Fri01], [Fri04] and [FT04]; [FN06] is our basic reference. First we substitute the microscopic time derivative o h(û ) with the spatial gradient of a mesoscopic flux depending on the block averagesη l,k andω l,k . The replacement ofη l,k with its empirical estimator F (ω l,k ) is based on the microscopic relaxation scheme, see Lemma 3.6 below, and also Section 5.6.

Entropy, Dirichlet Form, LSI and Relaxation
In this section we derive some fundamental estimates based on entropy and the associated Dirichlet forms. The parameters β, σ and l are almost arbitrary here, their dependence on the scaling parameter > 0 is not important. We only need β > 0, σ ≥ 1 and l ∈ . We follow calculations of [FN06] with slight modifications. (3.1) ϕ = log f is the condition of equality. Note that As a reference measure we can choose any of the equilibrium product measures λ = λ * u with −1 < u < 1 fixed, say u = 0 . At a level > 0 of scaling, µ ,t denotes the evolved measure, µ ,t,n is the distribution of the variables {ω k : |k| ≤ n} , and f n = f ,t,n := dµ ,t,n /dλ . Entropy in the box Λ n := [−n, n] ∩ is defined as

Entropy and its temporal derivative
respectively. By convexity, S n and any of D q n , q ∈ {o, * , s} are nondecreasing sequences. The Kolmogorov equation yields Taking into account inequality (3.2), it is easy to recover the local Dirichlet forms from the segments −n ≤ k < n of the corresponding sums, the rest gives the so called boundary terms.

Entropy flux:
Our basic a priori bound on local entropy and Dirichlet forms is the content of Lemma 3.1. If σ ≥ 1 and β ≤ σ then we have a universal constant C 0 such that for any initial distribution µ ,0 , n ∈ and t > 0 .
Proof. We have to estimate the boundary terms of ∂ t S n by means of S n and its associated Dirichlet forms D q n . Following the lines of the proof of Lemma 3.1 in [FN06], we arrive at a system (3.4) of differential inequalities with some universal constant K 1 . The cases of boundary term induced by o and are the same as in [FN06], the asymmetric o yields the flux S n+1 − S n on the right hand side. When we estimate boundary terms induced by , we are exploiting its following properties. The jump rates are constant, and the elementary actions, ω → ω b all preserve λ in a reversible way. Since * possesses all of these features, replacing ω b with ω b * in the computations concerning the boundary terms induced by * , we get (3.4). This system can explicitly be solved by means of the following lemma, which is a simple generalization of Lemma 3 in [Fri90]. Proof. Just as in [Fri90], a clever cutoff function g n (r) can be defined by

Lemma 3.2. Suppose that s n+1
Using βu n and σv n to estimate the corresponding square roots on the right hand side, we get ∂ s(t, r) provided that the decay of r(t) is fast enough. More precisely, let r solve r d r/d t + M 1 r + M 1 σ = 0 with terminal condition r(t) = n . Since r(0) = (t + (n 2 + σt) 1/2 ) in this case, and g n (r) ≥ e −n/r if r ≥ 1 , this completes the proof by a direct calculation.
Now we are in a position to complete the proof of Lemma 3.1. Since our reference measure λ * 0 is the uniform distribution on {0, 1, −1} 2n+1 , S n (0) ≤ (2n + 1) log 3 . Therefore choosing s n = S n , u n = D * n and v n = D n we see that (3.4) really implies Lemma 3.1.
This lemma is the fundamental a priori bound we need to materialize hydrodynamic limit in infinite volume. Since σ → 0 , t ≈ τ/ and n ≈ r/ in its following consequences, (r + τ)/ is the order of the bound, and r + τ ≤ rτ if r, τ ≥ 1 . To simplify formulae, from now on we are assuming that σ ≤ 1 and l 3 ≥ σ ≥ l ≥ 1 .

One block and two blocks estimates:
The first replacement lemma for microscopic currents is based on the logarithmic Sobolev inequality for stirring . Givenω l,k = u andη l,k = ρ , letλ l,k ρ,u denote the conditional distributions of ω k , ω k+1 , ..., ω k+l−1 with respect to λ . In view of Proposition 4 of [FT04], we have a universal constant, ℵ such that wheneverμ λ l,k ρ,u is a probability measure and f := dμ/dλ l,k ρ,u . It is very important that ℵ does not depend on ρ, u, l and f . This LSI allows us to estimate canonical expectations via the basic entropy inequality. The moment generating part is also a conditional expectation, consequently its bound should be independent of the conditions. Let us remark that Lemmas 3.3, 3.4 and 3.6 are consequences of the local entropy bound Lemma 3.1, therefore methods of [FN06] work also in case of the creation -annihilation process, and the results are valid for the spin -flip dynamics, too. Lemma 3.3. Let j(ω 0 , ω 1 ) denote a given function, J(ρ, u) := λ ρ,u (j) and j k (ω) := j(ω k , ω k+1 ) . We have a threshold l 0 ∈ and a universal constant C 1 depending only on C 0 and j such that Proof. In view of Lemma 3.1 the statement is more or less a direct consequence of the first inequality of Proposition 1 in [FT04], where notation is slightly different from ours. Lemma 3.2 of [FN06] treats the case of j k = j ηo k , our problem is essentially the same.
This is a sharp form of the so called One Block Lemma of [GPV88], the explicit rate due to LSI is needed for the evaluation of microscopic currents in the expression of the Lax entropy production X . The following comparison of block averages of typeξ andξ follows also via LSI in much the same way as the previous lemma did.
The statement on block averages of η is a direct consequence of Lemma 3.3 in [FN06], the bound for ξ l,k =ω l,k −ω l,k can be proven in the same way. The argument works even if ξ l,k =ω l,k+m −ω l,k , or ξ l,k =η l,k+m −η l,k , but an integration by parts trick yields better results in these cases when m is large. By means of the Cauchy inequality the deviation of blocks of different size can also be estimated.
Lemma 3.5. We have universal constants l 0 ∈ and C 3 < +∞ such that if l 0 ≤ l ≤ m but l 3 ≥ σ ≥ l and r, τ ≥ 1 , then For block averages of η the statement is proven in [FN06], see Lemma 3.4 and Section 6 there, the case of ω is the same. To estimate the deviation of consecutive block averages, set m = 2l . We see that m = δ σ/ , δ → 0 is the maximal size of the bigger block, but σ → 0 , thus large microscopic block averages can not be replaced with small macroscopic ones: a strong compactness argument in not available.

The first relaxation inequality:
To complete the evaluation of entropy production, we have to replaceη l,k with its empirical estimator F (ω l,k ) .
Lemma 3.6. For all r, τ ≥ 1 we have universal constants l 0 ∈ and C 4 < +∞ such that Proof. Let us consider the evolution of (3.6) Here, and also later on, most calculations are done at the microscopic level. Sincev is a step function, the integral mean, ψ k (t) := 1 of ψ appears in such expressions. We write V k (t) := V (η l,k (t),ω l,k (t)) whenever V is a function of the empirical process; for instance H u,k (t) = H u (v ( t, k)) . In the forthcoming calculations notation and facts from Sections 1.2 and 1.3 may be used without any reference.
The first and second terms, EH(0, ψ) and EĪ(∞, ψ t ) on the right hand side of (3.6) are bounded; to evaluate the third one, let us consider its decomposition: where B k is a quadratic form with bounded coefficients, we getL s = R s + Q s , and Q s (ψ) is a quadratic form of the differencesη b l,k −η l,k andω b l,k −ω l,k . Remember now that * H(η l,k ,ω l,k ) = Having in mind also Lemma 3.3, we now decomposeL * asL is the critical term here, it looks much larger than the others: its order seems to be 1/ .
However, due to some cancelations we have where K(ψ) < +∞ is a constant depending only on ψ . To prove this, observe first that . The derivation of (3.7) reduces to these deterministic bounds by simple computations.
The full power of our tools has not been exploited in the proof of (3.7), it is possible to improve Lemma 3.6, thus also Theorem 1.1 a bit. Namely, the lower bound σ 2 ( )β 2 ( ) → +∞ can be replaced with σ 2 ( )β 3 ( ) → +∞ as → 0 . We do not go into details because by means of a clever Lax entropy -flux pair a much better condition, σ( )β( ) → +∞ can be proven, see Sections 5.5 and 5.6.

Estimation of Entropy Production
The main part of this section is devoted to the verification of the conditions of Proposition 2.2, first of all the components L , M , J and N of entropy production X have to be evaluated. We are assuming that h, J ∈ C 2 ( ) with bounded first and second derivatives; is the relation of h and J . Our calculations are based on the a priori bounds of Section 3, the argument follows [FN06] with some modifications. Although h and J here are now functions of ω , considerable differences are coming only from the replacement of α κ with β * . Spin -flips are simple because κ η k = 0 , and the linear expression, κ ω k = −2(κη k − ω k ) is also easily controlled; * η k and * ω k are more complicated, cf. Section 1.3. Nevertheless, the main lines of our computations are essentially the same as in [FN06], just Lemma 3.6 of this paper is used instead of Lemma 3.5 of [FN06] at the final evaluation of entropy production. Among others, we have to show that due to reversibility of * , the microscopic current j ω * vanishes in the limit.

The numerical error:
This is the easiest case, by a direct calculation Since J is bounded, without any modification of the argument of the proof of Lemma 4.1 of [FN06], we obtain that the numerical error, N satisfies (2.8) with a vanishing bound.

The martingale:
We estimate the H −1 norm of M (∞, ψ, h) as follows. The stochastic differential dh(û ) = −1 h(û ) + d m defines a martingale m = m (t, x) for each x ∈ such that m (t, x) = m (t, k) if |x − k| < /2 and The martingale m is identified by the intensity q of its quadratic variation: Proof. Letṁ (t, x) denote the time derivative of m in the H −1 sense, we have to show that consequently we have to estimate However, |ω b l,k −ω l,k | ≤ 2/l 2 and |ω b * l,k − h(ω l,k | ≤ 1/l 2 , thus independently of the configuration we have q (t, x) = ((σ + β)/ l 3 ( )) , which completes the proof.

The microscopic current:
The starting point of the estimation of L is an identity, it is the contribution of σ . We have L ωs = Y ωs + Z ωs , where Since the entropic Dirichlet form of also has a factor σ( ) in our fundamental a priori bound, Lemma 3.1, we have Proof. It is exactly the same as that of the second part of Lemma 4.2 in [FN06]. First we separate the factors by means of the Cauchy -Schwarz inequality to let Lemma 3.5 work. For example, suppose that φ is supported in the rectangle It is plain that Ψ 1 ≤ 2 φ 2 ψ 2 +1 , while Q 1 = ( /σ) follows by Lemma 3.5 because ∇ 1ωl,k = ∇ lωl,k . This trick will be used several times in the next coming computations.
By means of the one and two blocks estimates the contributions of the microscopic currents j ωo and j ω * can be reduced as follows. Let (4.5) (4.6) and introduce their mesoscopic counterparts: where J uo (ρ, u) = λ ρ,u (j ωo k ) and J u * (ρ, u) = λ ρ,u (j ω * k ) , see Section 1.3. Now we split the corresponding differences by doing discrete integration by parts such that L ωo − V ωo = Y ωo + Z ωo and The following bounds are more or less direct consequences of Lemma 3.3 and Lemma 3.4 or 3.5; β can not be too large here.
Proof. It follows the argument of the first part of Lemma 4.2 in [FN06]. First we separatej ωo l,k − J uo (η l,k ,ω l,k ) andj ω * l,k − J u * (η l,k ,ω l,k ) by means of the Cauchy inequality from their factors, then we can use Lemma 3.3 and Lemma 3.4 or Lemma 3.5 in both cases. The procedure is terminated by the elementary computation of ∇ l ψ k . We get So far we have replaced the dominant parts of the microscopic currents ofω k,l with their canonical expectations, whenη k,l andω l,k are given. The crucial step of the whole proof follows right now, it is the replacement ofη l,k with F (ω l,k ) .

Relaxation in action:
As we have indicated above, the last step of the evaluation of entropy production consists in a comparison of J and V ωo , see (2.7) and (4.7). Indeed, as total charge is preserved by the creation -annihilation mechanism, we expect that V ω * vanishes in the limit, while J and V ωo cancel each other. By a direct calculation we get for the definition of F . We are going to replace J k−1 − J k with h k ∇ * l f (ω l,k ) . We have with some intermediate valuesω l,k andω l,k such that the quadratic remainders on the right hand side can be neglected. More precisely, the contribution of these remainders to J satisfies (2.9) with a vanishing bound; this follows by Lemma 3.4 and Lemma 3.5 in the usual way. Therefore (4.10) is the essential component of J , we split J jo − V ωo into two parts: , see Section 1.3, we are now in a position to apply Lemma 3.6.
Lemma 4.4. Y jo and Y j * satisfy (2.8), while Z jo and Z j * satisfy (2.9); all bounds do vanish.
Proof. In much the same way as in the proof of Lemma 4.3, by means of Lemma 3.6 we get Finally, as in the case of Z ω , Lemma 3.4 and Lemma 3.6 imply Therefore we need which complete the proof as l 2 ≈ σ/ , see also (2.2) and (2.3).
The results of this section can be summarized as follows. We have decomposed entropy production X (ψ, h) in a correct way, therefore Proposition 2.2 applies. Apart from Z s , all terms of the decomposition vanish, while Z s (ψ, h) ≤ 0 if h is convex and ψ ≥ 0 , thus lim sup X (ψ, h) ≤ 0 in probability as → 0 holds true in this case.
Proof of Theorem 1.1: First we prove Proposition 2.1. In fact we have to evaluate X (ψ, h) when h(u) = u , which is easy. Non -gradient analysis is not needed at all, and Z s is missing from the decomposition of X . For ψ ∈ C 1 c ( 2 ) by Kolmogorov where M is the terminal value of a martingale, andj is a block average,j (t, x) :=j ωo l,k (t/ ) + β( )j ω * l,k (t/ ) + σ( )j ωs l,k (t/ ) if |x − k| < /2 , see Section 1.3. In view of Lemma 4.2, EM 2 → 0 as → 0 , and the replacement ofj with f (ū ) is a consequence of Lemma 3.3 and Lemma 3.6; remember that f (u) = F (u) − u 2 is just the flux of (1.16). Finally, Lemma 3.4 implies thatū −û also vanish in the limit, thus we have E|X (ψ, u)| → 0 as → 0 , which completes the proof because the distributionsP ,θ of the Young measures form a tight family. Now we are in a position to finish the proof of Theorem 1.1. Proposition 2.1 and Proposition 2.3 imply that any limit distribution,P θ of the Young measures is concentrated on a set of weak solutions. On the other hand, lim sup X (ψ, h) ≤ 0 in probability if ψ ≥ 0 and h is convex, thus first we get (2.11), whence Proposition 2.3 yields (1.23) almost surely with respect to anyP θ . Therefore the uniqueness of the limiting solution follows by the Main Theorem of [CR00] on uniqueness of entropy solutions. Since the limit is deterministic, for ψ ∈ C c ( 2 ) we have in probability. The space integral on the left hand side is actually a sum, thus the block average can be transposed on ψ , which completes the proof of Theorem 1.1 because ψ is uniformly continuous.

Concluding Remarks
Here we summarize some improvements and explanations of our main result including further remarks on the method of relaxation schemes.

Strong convergence:
The last step of the argument yields a stronger form of Theorem 1.1, cf. [FT04]. Letū denote the empirical process of Section 3.4,ū (t, x) =ω l,k (t/ ) if |x − k| < /2 , l = l( ) as in (2.2). This version is certainly more natural thanû , which has been introduced because of technical reasons: l∇ * 1ĵ ωo is a difference of block averages, andj ωo is well controlled by where u denotes the unique entropy solution to (1.16) with initial value u 0 .

Microscopic block averages:
By means of Lemma 3.5 we can fill in the gap between large microscopic and small macroscopic block averages of the evolved configuration, see [FN06]. Let u ,l (t, x) :=ω l,k (t/ ) if |x − k| < /2 , where l ∈ does not depend on , then for all τ, r > 0 we have

Measure -valued solutions:
Convergence of the empirical processū ,l to a set of measurevalued solutions holds true under fairly general conditions. Let us consider CRANNI, that is the process generated by * = o + β * with a fixed value of β > 0 . Let ψ ∈ C 1 c ( 2 ) , then following the proof of Proposition 2.1 we write whereM is the terminal value of a martingale, andj ,l is a block average,j ,l (t, x) :=j ωo l,k (t/ ) + β( )j ω * l,k (t/ ) if |x − k| < /2 ; remember that l ∈ does not depend on > 0 . It is easy to show that EM 2 → 0 as → 0 , and the replacement ofj ,l with f (ū ,l ) follows by the standard One -Block Lemma of [GPV88]. For τ, r > 0 we have because the usual entropy argument shows that every translation invariant stationary measure is a superposition of product measures of type λ * u . Therefore all limit distributions of the Young representation ofū ,l are concentrated on a set of measure -valued solutions, cf. (1.22); these limit distributions are obtained by sending → 0 first, and l → +∞ next. The weak Lax inequality (2.11) can not be proven in this easy way, we are facing with a non -gradient problem there.

Relaxation of the spin flip model:
, and consider the evolution ofH where the empirical processv is the same as in Section 3.4, but it is now generated by κ = o + α( ) κ + σ( ) . We follow the lines of the derivation of (3.7), the only difference consists in the evaluation of the effect of κ . Due to κ η k = 0 , these computations are considerably simpler than those of Section 3.4. Denoteω j l,k the block average of the sequence ω j , then is an identity, whence by a direct calculation we obtain the final a priori bound: Let us remark that the second LSI in [FN06] yields a considerably better order, (1/l + /α) , see This means α( )σ 2 ( ) → +∞ as → 0 ; the condition α( )σ( ) → +∞ of Theorem 1.1 in [FN06] has not been improved in this way.
Proof. Let us consider for 0 ≤ ψ ∈ C 1 c ( 2 ) , wherê v (t, x) = ρ (t, x),û (t, x) := η l,k (t/ ),ω l,k (t/ ) if |x − k| < /2 ; the process is generated by = o + β( ) * + σ( ) . The evaluation of X ν follows the lines of the proof of Lemma 3.6 in Section 3.4; considerably better bounds are obtained in the first part of the argument. Concerning the decomposition of entropy production and estimation techniques we refer to Section 2.3 and Section 4. However, just as in case of Lemma 3.6, it suffices to control expectations only, which simplifies the argument. The main steps are outlined below with a concrete emphasis on nontrivial differences. It is not really problematic that our empirical processv consists of two components, cf. [FT04], the crucial issue is to control the effect of * .
Due to formal conservation of entropy, the contribution of o is compensated by the mesoscopic flux Φ ν (v ) , the vanishing l 2 /σ is the order of this difference. Indeed, apart from some quadratic remainders we have where 0ρ = ∇ * In order to estimate the right hand side via Lemma 3.3, first of all we have to get rid of the block averages of typeξ l,k ; note that we have to control squared differences. Lemma 3.4 allows us to replace C(η l,k ,ω l,k ) by C k := C(η 2l,k−l+1 ,ω 2l,k−l+1 ) , while C m ≈ C l follows by Lemma 3.5 if l 0 ≤ m < l . On the other hand, due to Cauchy is a consequence of Lemma 3.3 if m ≥ l 0 , therefore 2Γ ν − W ν = (l 2 /σ) ; the constant does depend on l 0 , too.
Since −W ν = 2Γ ν − W ν − 2Γ ν , (5.4) implies a preliminary version of Lemma 5.1: instead of (F (ω l,k ) −η l,k ) 2 we have (F (ω l,k ) −η l,k ) 2 in the sum on the left hand side of the second relaxation inequality. The required substitution is immediate, Lemma 3.4 completes the proof of this lemma.
The second relaxation inequality allows us to improve the previous lower bound on β = β( ) .
Proof. Applying the above bound instead of Lemma 3.6, and following the proof of Lemma 4.4, we see that (1 −η l,k ) 2 (ω l,k − κη l,k ) 2 dµ ,t d t ≤C 4 rτ + rτ α would follow, see Lemma 4.3 in [FN06] and Section 5.4 here, provided that ν and S ν are well defined. The bound above is effective if α( )σ( ) → +∞ as → 0 ; this condition on α is the very same as that of [FN06]. Unfortunately, unless κ = 0 , no version of our construction is correct: the underlying measure ν can not be defined because the map z = (κ − w)/(1 − κw) does not interchange the ranges of z and w if κ = 0 . The case of κ = 0 is trivial at the level of hyperbolic scaling, nevertheless it is interesting to note that relaxation via entropy may be just as strong as LSI.