Diffusive limit of lattice gas with mixing conditions. (English) Zbl 0947.60089

Varadhan, S. R. S.; Yau, Horng-Tzer

Asian J. Math. 1, No. 4, 623-678 (1997).

Summary: We prove, under certain mixing conditions, that the hydrodynamical limit of a stochastic lattice gas on the cubic lattice \(\mathbf Z^d\) is governed by a nonlinear diffusion equation. Following S. R. S. Varadhan [in: Asymptotic problems in probability theory: Stochastic models and diffusions on fractals. Pitman Res. Notes Math. Ser. 283, 75-128 (1993; Zbl 0793.60105)], we characterize the diffusion coefficient by a variational formula, which is equivalent to the Green-Kubo formula. The fluctuation-dissipation equation is established rigorously as an important step of the proof. Our mixing conditions are implied by the Dobrushin-Shlosman mixing conditions which are always valid at high temperatures.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82C22	Interacting particle systems in time-dependent statistical mechanics

Keywords:

hydrodynamical limit; nonlinear diffusion equation; Green-Kubo formula; fluctuation-dissipation equation; mixing conditions

Citations:

Zbl 0793.60105