Guo, M. Z.; Papanicolaou, G. C.; Varadhan, S. R. S. Nonlinear diffusion limit for a system with nearest neighbor interactions. (English) Zbl 0652.60107 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 Cited in 21 ReviewsCited in 139 Documents 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 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Kipnis, C., Varadhan, S. R. S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys.104, 1-19 (1986) · Zbl 0588.60058 · doi:10.1007/BF01210789 [2] De Masi, A., Ianiro, N., Pellegrinotti, A., Presutti, E.: A survey of the hydrodynamical behavior of many-particle systems. In: Nonequilibrium phenomena II. From stochastics to hydrodynamics. Lebowitz, J. L., Montroll, E. N. (eds.) pp. 123-294. Amsterdam: North-Holland 1984 · Zbl 0567.76006 [3] Dobrushin, R., Seigmund-Schultze, R.: The hydrodynamic limit for systems of particles with independent evolution, Akademie der Wissenschaften der DDR, Berlin, 1981 · Zbl 0474.60084 [4] Spohn, H.: Equilibrium fluctuations for interacting Brownian particles. Commun. Math. Phys.103, 1-33 (1986) · Zbl 0605.60092 · doi:10.1007/BF01464280 [5] Fritz, J.: On the hydrodynamic limit of a scalar Ginzburg-Landau lattice model, IMA Vol.9. Berlin, Heidelberg, New York: Springer 1987 · Zbl 0681.76089 [6] Funaki, T.: preprint 1987 [7] Donsker, M., Varadhan, S. R. S.: preprint 1987 [8] Varadhan, S. R. S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.19, 261-286 (1966) · Zbl 0147.15503 · doi:10.1002/cpa.3160190303 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.