H.T. Yau
Logarithmic Sobolev Inequality for Generalized Simple Exclusion Processes
(270K, ps)
ABSTRACT. Let $\bar \mu$ be a
probability measure on the set $\{0, 1, \cdots , R\}$ for some $R\in \NN$ . Let $\L_L$ be
a cube of width $L$ in $\ZZ^d$. Denote by $\mu^{gc}_{\L_L}$ the (grand canonical )
product measure with $\bar \mu$ as
the marginal measure; here the superscript indicates the
grand canonical ensemble.
The canonical ensembles, denoted by $\mu^c_{\L_L, n}$, are defined by
conditioning $\mu^{gc}_{\L_L}$ given that the total number of particle is $n$. Consider the
zero-range exclusion dynamics where each particle performs random walk
with rates depending only on the number of particles
at the same site. The rates are chosen
such that, for every $n$ and $L$ fixed, the measure $\mu^c_{\L_L, n}$ is
reversible. We prove the logarithmic Sobolev inequality in the sense
that $\int f \log f d\mu^c_{\L_L, n} \le \hbox{const.} L^2 D(\sqrt f)$
for any probability density $f$ with respect to $\mu^c_{\L_L, n}$; here the constant
is independent of $n$ or $L$ and $D$ denote the
Dirichlet form of the dynamics. The dependence on $L$ is optimal.