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Hydrodynamical limit for asymmetric attractive particle systems on $$\mathbb{Z}{}^ d$$. (English) Zbl 0751.60097
The author is interested in the conservation of local equilibrium for the zero range process on $$Z^ 2$$ involving undistinguishable particles moving on $$Z^ 2$$. Let $$g: N\to R$$ be nondecreasing and such that $$g(0) < g(1)$$, and let $$P(x,y)$$ be a transition probability on $$Z^ 2$$ such that $$P(x,y)=p(y-x)$$, $$x,y\in Z^ 2$$, and, for some $$A\in N$$, $$p(x)=0$$ if $$\| x\|\geq A$$. If $$k$$ particles are at site $$x\in Z^ 2$$, they wait a mean $$(g(k))^{-1}$$ exponential time at the end of which one of them jumps to $$y$$ with probability $$P(x,y)$$. Suppose that the corresponding Markov process on $$N^{Z^ 2}$$ with semigroup $$(S_ t)$$ has an infinite family of extremal invariant measures $$\nu_ p$$ depending on a parameter $$p$$ in some open set $$P\subset R^ n$$. It is said that, for a sequence of probability measures $$\mu_ \varepsilon$$ $$(\varepsilon > 0)$$ on $$N^{Z^ 2}$$ there is conservation of local equilibrium if there are a time renormalization $$T(\varepsilon)$$ and a regular function $$f: R_ +\times R^ 2\to P$$ such that, in the weak* sense, $$\lim_{\varepsilon\to 0}\mu_ \varepsilon S_{T(\varepsilon)t}\tau_{[x\varepsilon^{-1}]}=\nu_{f(t,x)}$$ for every continuity point $$(t,x)$$ of $$f$$ (here, $$\tau_ y$$ denotes translation by $$y$$ on $$N^{Z^ 2}$$, and $$[r]$$ denotes the integer part of $$r\in R$$). Throughout the paper, the Euler rescaling $$T(\varepsilon)=\varepsilon^{-1}$$ is considered. The author obtains two theorems specifying conditions which are sufficient for the conservation of local equilibrium.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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