 96205 H.T. Yau
 Logarithmic Sobolev Inequality for Generalized Simple Exclusion Processes
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May 16, 96

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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
zerorange 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.
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