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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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