Hydrodynamic Limit of One-Dimensional Exclusion Processes with Speed Change

Hydrodynamic limit for a facilitated exclusion process

“…In particular, when the density is close to 1 2 , these measures exhibit spatial correlations, and adapting the entropy method for these non-product grand canonical measures requires significant extra technical work. We note that both papers [6,22] also cover cases with Gibbs measures which are not necessarily product, although for non-degenerate dynamics. Besides exclusion processes, microscopic models with non-product Gibbs measures for which the hydrodynamic limit is rigorously established are very few.…”

“…As discussed in the next section, the model is reversible and gradient. Moreover, it can be formally understood as the specific case of Examples 1 or 2 in [6,Section 5], specifically the case with parameters b(1) = −∞, b(k) = 0 for k 2 in Example 1, or α = ∞ and β = 0 in Example 2 under proper normalization. In particular, the exclusion processes in the class described in Example 1 of [6] can be mapped to zero-range processes by a simple but non-linear transformation of configurations, which we discuss precisely in Section 3.…”

Hydrodynamics of a particle model in contact with stochastic reservoirs

“…In the case of the translation invariant space (the torus T N instead of the interval Λ N ) a common way to prove the one-block estimate (see e.g. [6]) is to show an ergodic representation for a limiting measure µ ∞ of {μ N t } in terms of the extreme invariant measures of the process, which, in our case, are the Gibbs measures. To justify application of the ergodic representation theorem, one needs to prove existence of a probability measure w on the class of Gibbs measures, which specifies which convex combination of the Gibbs measures equals to the limiting measure µ ∞ :…”

“…As a result, in contrast with [14,15], we can not neglect the evolution inside the reservoirs. Our derivation of the parabolic equation (1.2) is based on the papers [4,5,6]. Although, for the convenience of the reader we attempted to repeat the relevant material from the papers in order to make our exposition self-contained, some details were omitted, as it required us to repeat the articles almost completely.…”

“…Proof. The proof follows from the one of[6, Proposition 4.2], where instead of applying the Birkhoff theorem (which the authors call the individual ergodic theorem), we use the one-block estimate (Lemma 7.1). Two-block estimate).For Λ N ∈ {[−N, −1], [0, N ], [N + 1, 2N ]}, and any F : [0, 1] → R + -bounded continuous, as N → ∞, l → ∞, ε → 0 + , max {x 0 , z 0 : |x 0 −z 0 |≤2εN B l (x 0 )∪B l (z 0 )⊂Λ N } X |F (M B l (x 0 ) (η)) − F (M B l (z 0 ) (η))|μ N t (dη) → 0.…”