Nonlinear diffusion limit for a system with nearest neighbor interactions. (English) Zbl 0652.60107

Guo, M. Z.; Papanicolaou, G. C.; Varadhan, S. R. S.

Commun. Math. Phys. 118, No. 1, 31-59 (1988).

The problem of passage to the hydrodynamic limit is investigated for a system of interacting diffusions. The variables are located at the sites of a periodic one-dimensional lattice and are regarded as charges of indeterminate sign. It is supposed that the exchange of charges takes place between adjacent sites according to the diffusion law and that the algebraic sum of the charges is preserved.
The main result states that under certain conditions on the initial density of the distributions of charges \(f\) \(0_ N\) defined in terms of so-called asymptotic macroscopic charge density, \(m_ 0(\theta)\), there exists for every \(t\geq 0\) an asymptotic deterministic charge density m(t,\(\theta)\) which describes limit properties of \(f\) \(t_ N\) (N\(\to \infty)\). Moreover m(t,\(\theta)\) is the unique weak solution of a nonlinear diffusion equation with the initial condition \(m_ 0(\theta)\).

Reviewer: S.Pogosian

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B05	Classical equilibrium statistical mechanics (general)
60J60	Diffusion processes

Keywords:

entropy estimates; local Gibbs states; hydrodynamic limit; interacting diffusions; weak solution of a nonlinear diffusion equation

References:

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