an:03852183 Zbl 0536.60097 Andjel, Enrique D.; Kipnis, Claude Derivation of the hydrodynamical equation for the zero-range interaction process EN Ann. Probab. 12, 325-334 (1984). 00149050 1984
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60K35 60J70 hydrodynamic equation; local equilibrium; zero range process In continuation of the reviewer's paper in Z. Wahrscheinlichkeitstheor. Verw. Geb. 58, 41-53 (1981; Zbl 0451.60097) the property of local equilibrium for the asymmetric zero range process with particular initial states is established and the evolution of the parameter characterizing the local equilibria is derived. The model is given by the Markov process on $$N^ Z$$ whose generator is $$Lf(\eta)=\sum_{i}1_{\{\eta_ i>0\}}.(f(\eta^ i)-f(\eta))$$, where $$\eta^ i$$ is obtained from $$\eta$$ by moving a particle from i to $$i+1$$. The equilibria are the product measures $$\nu_{\rho}$$, $$0<\rho<\infty$$, with $$\nu_{\rho}(\eta_ i=k)=(\rho /1+\rho)^ k\cdot(1+\rho)^{-1}$$, $$k\geq 0.$$ The result may roughly be stated as follows: Rescale space and time by the same factor $$\epsilon$$ ; then, as $$\epsilon$$ tends to zero, the macroscopic density profile $$\rho$$ (x,t) evolves according to the equation $\frac{\partial \rho}{\partial t}+\frac{\partial \rho}{\partial x}\cdot(1+\rho)^{-2}=0,$ if the initial density $$\rho$$ (0,.) satisfies one of the following two conditions: (a) it is an increasing step function taking on only two values (Th.2.4); (b) it is decreasing and differentiable (Th.3.3). H.Rost Zbl 0451.60097