Abstract
We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter $L$ may depend on $L$ but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that this logarithmic-Sobolev constant grows as the square of $L$, that is presented in a forthcoming paper.
Highlights
The zero-range process is a system of interacting particles moving in a discrete lattice Λ, that here we will assume to be a subset of Zd
In the zero-range process associated to c(·) particles evolve according to the following rule: for each site x ∈ Λ, containing ηx particles, with probability rate c(ηx) one particle jumps from x to one of its nearest neighbors at random
In some respect zero-range processes may appear simpler than Kawasaki dynamics: the interaction is zero-range rather than finite-range and, as a consequence, invariant measures have a simpler form
Summary
The zero-range process is a system of interacting particles moving in a discrete lattice Λ, that here we will assume to be a subset of Zd. For dynamics with exclusion rule and finite-range interaction (Kawasaki dynamics) relaxation to equilibrium with rate diam(Λ) has been proved (see [7, 2]) in the high temperature regime. In some respect zero-range processes may appear simpler than Kawasaki dynamics: the interaction is zero-range rather than finite-range and, as a consequence, invariant measures have a simpler form. Due to unboundedness of the density of particles various arguments used for exclusion processes fail; in principle the rate of relaxation to equilibrium may depend on the number of particles, as it does in some cases;.
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