## EVENT DETAILS AND ABSTRACT

**Twentieth Midwest Probability Colloquium**
**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

**Time:** 4:30 PM

**Where:** Anneberg G15

**Contact Person:** Professor Mark Pinsky

**Contact email:** pinsky@math.nwu.edu

**Contact Phone:**

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