The second relaxation inequality: Using S

where G + a = ν0, a if a ≥ 0 , and G − a = νa, 0 if a 0 . To demonstrate relaxation to 1.16, we need ∂ ρ S ν ρ, uCρ, u ≤ −b ρ − Fu 2 5.1 with some universal constant b 0 , which implies ∂ ρ S ν ρ, u = 0 if ρ = F u . In terms of the Riemann invariants this means 0 = C ρ, u = 14z 2 + w 2 + 3z 2 w 2 − 1 , whence z 2 = 1 − w 2 1 + 3w 2 . Therefore setting G + z = z 2 , ν {0} = −1 and G − w = 1 − 1 − w 2 1 + 3w 2 , we obtain ∂ ρ S ν ρ, u = 1 − z 2 − w 2 − 3z 2 w 2 1 + 3w 2 = − 4C ρ, u 1 + 3w 2 5.2 by a direct computation. However, w 2 ≤ 1 and |Cρ, u| ≥ 12|ρ − Fu| , consequently the above choice of ν yields 5.1 with b = 14 .

5.6. The second relaxation inequality: Using S

ν instead of the trivial Liapunov function H = 12F u − ρ 2 , we can improve Lemma 3.6 as follows. Lemma 5.1. There exist two universal constants l ∈ N and C 5 +∞ such that ǫ 2 X |k|rǫ Z τǫ Z € ¯ η l,k − F ¯ ω l,k Š 2 d µ ǫ,t d t ≤ C 5 ‚ rτǫ β + r τǫl 2 σ Œ whenever r, τ ≥ 1 and ǫl 3 ≥ σ ≥ l ≥ l . Proof. Let us consider X ν ǫ ψ := − Z ∞ Z ∞ −∞ € ψ ′ t t, xS ν ˆ v ǫ + ψ ′ x t, xΦ ν ˆ v ǫ Š d x d t 5.3 for 0 ≤ ψ ∈ C 1 c R 2 , where ˆ v ǫ t, x = ˆ ρ ǫ t, x, ˆ u ǫ t, x := € ˆ η l,k tǫ, ˆ ω l,k tǫ Š if |x − ǫk| ǫ2 ; the process is generated by L = L o + βǫG ∗ + σǫS . The evaluation of X ν ǫ 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 process ˆ v ǫ consists of two components, cf. [FT04], the crucial issue is to control the effect of G ∗ . Due to formal conservation of entropy, the contribution of L o is compensated by the mesoscopic flux Φ ν ˆ v ǫ , the vanishing ǫl 2 σ is the order of this difference. Indeed, apart from some quadratic remainders we have L S ν ˆ v ǫ ≈ ∂ ρ S ν ˆ v ǫ L ˆ ρ ǫ + ∂ u S ν ˆ v ǫ L ˆ u ǫ , 258 where L ˆ ρ ǫ = ∇ ∗ l ¯j ηo l,k , L ˆ u ǫ = ∇ ∗ l ¯j ωo l,k , and O ǫ −1 l −3 → 0 is the total contribution of the remainders. The microscopic currents are replaced by their canonical expectations via Lemma 3.3 at a total cost of O lσ , that is ¯j ηo l, j ≈ ¯ ω l,k − ¯ ω l,k ¯ η l,k and ¯j ωo l, j ≈ ¯ η l,k − ¯ ω l,k 2 . In a similar way, by means of Lemma 3.4 we get Φ ν ˆ η l,k −1 , ˆ ω l,k −1 − Φ ν ˆ ω l,k , ˆ η l,k ≈ ∂ ρ S ν − ¯ η l,k ∂ ρ S ν − 2 ¯ ω l,k ∂ u S ν ∇ ∗ l ¯ ω l,k + ∂ u S ν − ¯ ω l,k ∂ ρ S ν ∇ ∗ l ¯ η l,k ; again O lσ is the total order of our error terms. By Lemma 4.1 the numerical error coming from the substitution of ψ ′ x by ǫ −1 ψt, x + ǫ − ψt, ǫ also vanishes, thus 1.18 implies the desired cancelation. Finally, again by Lemma 3.4 the contribution of σS S ν is bounded, see also Lemma 4.2, while the martingale component in the evolution equation of S ν has zero expectation, consequently O 1 is the total order of those terms of X ν ǫ which have been considered so far. Therefore, just as in case of Lemma 3.6, we have to concentrate on the effect of G ∗ . Apart from the usual quadratic remainders G ∗ S ν ˆ η l,k , ˆ ω l,k ≈ ∂ ρ S ν ˆ η l,k , ˆ ω l,k G ∗ ˆ η l,k + ∂ u S ν ˆ η l,k , ˆ ω l,k G ∗ ˆ ω l,k , and G ∗ ˆ ω l,k is a difference of mesoscopic block averages, therefore Γ ν ǫ ψ := ǫ X k ∈Z Z ∞ ψ k t∂ ρ S ν ˆ η l,k , ˆ ω l,k ˆ C ν∗ l,k ω d t is the critical term, where ˆ C ν∗ l,k ω := 2 l 2 − l l −1 X m=1 X b ⊂[k−m,k+m] € c × b ω − c + b ω Š ; remember that ˆ C ν∗ l,k 2 is a mean value of arithmetic averages with weights 2m − 1l 2 − l . Since |G ∗ ˆ η l,k − ˆ C ν∗ l,k ω| = O 1l , moreover | ˆ ω b ∗ l,k − ˆ η l,k | = O l −2 while | ˆ η b ∗ l,k − ˆ η l,k | = O 1l , we obtain − EΓ ν ǫ ψ ≤ K ν ψ 1 β + 1 ǫl 5.4 by a direct computation, see the derivation of 3.7. Note that σβǫl 2 ≈ ǫ −12 β is much bigger then the corresponding 1 β above. Now we are in a position to continue the argument of Section 3.4. Having in mind 5.2, let us introduce W ν ǫ ψ : = 2ǫ X k ∈Z Z ∞ ψ k t∂ ρ S ν ˆ η l,k , ˆ ω l,k C ˆ η l,k , ˆ ω l,k d t ≤ − ǫ 2 X k ∈Z Z ∞ ψ k tF ˆ ω l,k − ˆ η l,k 2 d t and ˆ D k t := ˆ C ν∗ l,k ω − 2C ˆ η l,k , ˆ ω l,k , whence 2Γ ν ǫ − W ν ǫ = 2Γ ν ǫ − 2W ν ǫ + W ν ǫ ≤ ǫ 4 X k ∈Z Z ∞ ψ k t ˆ D 2 k t d t . 259 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 type ˆ ξ l,k ; note that we have to control squared differences. Lemma 3.4 allows us to replace C ˆ η l,k , ˆ 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 ≤ m l . On the other hand, due to Cauchy D k ≤ 4 l −1 X m=1 2m − 1 l 2 − l ˜ C m − C m 2 , where ˜ C m := 1 2m − 1 X b ⊂[k−m,k+m] € c × b ω − c + b ω Š ≈ C m is a consequence of Lemma 3.3 if m ≥ l , therefore 2Γ ν ǫ − W ν ǫ = O l 2 σ ; the constant does depend on l , too. Since −W ν ǫ = 2Γ ν ǫ − W ν ǫ − 2Γ ν ǫ , 5.4 implies a preliminary version of Lemma 5.1: instead of F ¯ ω l,k − ¯ η l,k 2 we have F ˆ ω l,k − ˆ 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 = βǫ . Theorem 5.2. The conclusion of Theorem 1.1 holds true even if its condition ǫσ 2 ǫβ 2 ǫ → +∞ is replaced with βǫσǫ → +∞ as ǫ → 0 . Proof. Applying the above bound instead of Lemma 3.6, and following the proof of Lemma 4.4, we see that lim ǫ→0 ǫ β = lim ǫ→0 ǫβ 2 l 2 σ = lim ǫ→0 1 βσ = lim ǫ→0 β 2 l 2 σ 2 = 0 are the requirements of an effective relaxation, which completes the proof as l 2 ≈ σ p ǫ . Of course, Theorem 5.1 also holds true under the above condition on β = βǫ of Theorem 5.2.

5.7. Entropy for the spin - flip model: In view of Section 5.5, in case of the spin - flip dynamics