The model: In view of our naive physical picture of electrophoresis, we consider ±1 charges

1.1 if B = 0 , but there are other possibilities, too. Therefore it is reasonable to expect v ǫ ≈ f u ǫ in a mean square sense as ǫ → 0 , consequently the limit u of u ǫ 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 ∂ t u ǫ + ∂ x ρ ǫ − u 2 ǫ = 0 , ∂ t ρ ǫ + ∂ x u ǫ − u ǫ ρ ǫ = ǫ −1 Bu ǫ , ρ ǫ F u ǫ − ρ ǫ , 1.2 where x ∈ R , u ∈ [−1, 1] , ρ ∈ [0, 1], Fu := 134−4−3u 2 1 2 and B ≥ 12 , consequently the limiting equation for u reads as ∂ t u + ∂ x F u − u 2 = 0 . The first proof uses h = 12F 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.

1.2. The model: In view of our naive physical picture of electrophoresis, we consider ±1 charges

moving in an electric field on Z 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 ∈ Z , i.e. ω k is interpreted as the charge of the particle at site k ∈ Z , 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 Z , i.e. b ∈ Z ∗ , 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 7→ R 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 L o ϕω := X b ∈Z ∗ c b ω € ϕω b − ϕω Š , 1.3 where c b ω := 12η k + η k+1 + ω k − ω k+1 if b = k, k + 1 . This generator lets ⊕ particles jump to the right, ⊖ particles jump to the left at rate 1 , if allowed, while a collision ⊕⊖ → ⊖⊕ 232 occurs at rate 2 . Both particle number P η k and total charge P 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 λϕ ≡ R 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 L ∗ := L o + βG ∗ , where β 0 and G ∗ ϕω := X b ∈Z ∗ c + b ω € ϕω b+ − ϕω Š + X b ∈Z ∗ c × b ωϕω b × − ϕω , 1.4 where c + b ω := 1[η k = 0, η k+1 = 0] = 1 − η k 1 − η k+1 , c × b ω := 1[ω k = 1, ω k+1 = −1] = 14η k + ω k η k+1 − ω k+1 if b = k, k+1 , and 1[A] denotes the indicator function of the event A ⊂ Ω . For any bond k, k+1 = b ∈ Z ∗ the elementary action ω ↔ ω b ∗ is defined by ω b ∗ := ω b+ if ω k , ω k+1 = 0, 0 , ω b ∗ = ω b × if ω k , ω k+1 = +1, −1 , while ω b ∗ = ω otherwise. Since c + b + c × b = 1 if ω b ∗ 6= ω , we can rewrite G ∗ as G ∗ ϕω = X b ∈Z ∗ € ϕω b ∗ − ϕω Š . Only total charge P 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: λ ρ,u [ω k = 1, ω k+1 = −1] = λ ρ,u [ω k = ω k+1 = 0] . This means that Cρ, u = 0 , where C ρ, u : = 1 − ρ 2 − 14ρ 2 − u 2 = 14 € 3 ρ 2 − 8ρ + u 2 + 4 Š = 34 ρ − Fu ρ − F ∗ u , 1.5 F u := 1 3  4 − p 4 − 3u 2 ‹ , F ∗ u := 1 3  4 + p 4 − 3u 2 ‹ . 1.6 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 L ∗ 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 G ∗ , 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 com- pactness, 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 S ϕω := X b ∈Z ∗ € ϕω b − ϕω Š . 1.7 This process is reversible with respect to any λ ρ,u , and both P η k and P ω k are preserved. HDL of the process generated by L σ := L o + σS 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 L := L o + βG ∗ + σS , see 1.3, 1.4 and 1.7 for definitions, where β and σ are positive 233 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 L κ := L o + αG κ + σS was investigated. The spin - flip generator G κ reads as G κ ϕω := X k ∈Z η k − κω k € ϕω k − ϕω Š , 1.8 where κ ∈ −1, 1 is a constant, and ω ↔ ω k means that ω k k = −ω k , while ω k j = ω j other- wise. Since G κ violates conservation of total charge, and 1 − κλ ρ,u [ω k = 1] = 1 + κλ ρ,u [ω k = −1] in equilibrium, i.e. 1 − κρ + u2 = 1 + κρ − u2 , we have u = κρ . 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. 1.3. 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 L ? . By direct computations we get L o ω k = j ωo k −1 − j ωo k , where j ωo k ω := 1 2 η k + η k+1 − 2ω k ω k+1 + ω k η k+1 − η k ω k+1 + ω k − ω k+1 , 1.9 whence J uo ρ, u := λ ρ,u j ωo k = ρ − u 2 and λ ∗ u j ωo k = f u := F u − u 2 . Similarly, L o η k = j ηo k −1 − j ηo k , where j ηo k ω := 1 2 ω k + ω k+1 − ω k η k+1 − ω k+1 η k+1 + η k − η k+1 , 1.10 whence J ρo ρ, u := λ ρ,u j ηo k = u − uρ and λ κ ρ j ηo k = κ ρ − ρ 2 . The case of S is trivial: S ω k = ∆ 1 ω k and S η k = ∆ 1 η k , where ∆ 1 ξ k := ξ k+1 + ξ k −1 − 2ξ k for any sequence, ξ indexed by Z , 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 G κ η k = 0 . Moreover, the identity G κ ω k = G κ ω k − κη k = 2κη 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 G ∗ ω k = j ω∗ k −1 − j ω∗ k , where j ω∗ k := c × b ω − c + b ω , b = k, k + 1 , 1.11 thus J u ∗ ρ, u := λ ρ,u j ω∗ k = −Cρ, u , cf. 1.5. Since λ ∗ u is reversible with respect to G ∗ , the effect of j ω∗ vanishes in the hydrodynamic limit; λ ∗ u j ω∗ k = 0 . On the other hand, G ∗ η k = c + b ω − c × b ω + c + b − ω − c × b − ω , 1.12 where b = k, k + 1 and b − := k − 1, k , whence λ ρ,u G ∗ η k = 2Cρ, u = 32ρ − Fuρ − F ∗ u . Observe now that the second factor is negative because F ∗ u ≥ 53 , thus we have a good 234 reason to suspect that the terms G ∗ η k give rise to relaxation: ρ ≈ Fu in the scaling limit. This problem, however, is more difficult than the previous one because G ∗ 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.

1.4. Macroscopic equations: Under hyperbolic scaling of space and time, at a level ǫ ∈ 0, 1] of