Title: Hydrodynamical limit and relaxation rate for lattice gases
Speaker: Professor Horng-Tzer YAU
Speaker Info: Courant
Brief Description:
Special Note:
Abstract:
Denote the configuration of particles by $X = (x_1, \cdots, x_j, \cdots), x_j \in {\bf R}^d$. A Gibbs state is formally given by $\mu \sim \exp \big [ - \beta \sum_{i ot = j} V(x_i-x_j) \big ] $ with $V$ a two-body interaction. A reasonable class of dynamics describing the relaxation to this measure is obtained by the infinite dimensional differential operator $L$ formally given by $$ \int f (-L) f d \mu = \sum_{j=1}^\infty \int a_j (X) ( abla_{x_j} f)^2 d \mu $$ Our goal is to prove a relaxation estimate $$ \| e^{t L} f \|_{L^2}^2 \le C_f t^{-d/2} \eqno(1) $$ and an off-diagonal estimate $$ e^{t L} (X, Y) \sim t^{-d/2} \exp \big [ -{ (X-Y)^2 \over t D } \big ] \eqno(2) $$ which holds for the usual heat kernel in finite dimension. The semigroup $e^{t L}$ have no pointwise kernel and thus (2) has to be understood weakly. We can consider the lattice version of the dynamics, called lattice gas dynamics, by replacing the Brownian motions with random walks on the standard lattice. For these dynamics, we proved the decay estimate (1) (with C. Landim, J. Quastel) and the hydrodynamic limit (with S. Varadhan) provided that the Gibbs states are away from phase transition regions. The hydrodynamic limit proves (2) in average sense in certain region of space-time and it determines the diffusion coefficient $D$. The technical inputs of this work are the spectral gap, the logarithmic Sobolev inequality, nongradient system methods and a new idea to prove $L^2$ decay estimates of the heat kernel in infinite dimension which is independent of the Nash-Moser methods or the maximum principle.Date: Friday, October 23, 1998