Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

Details Tunneling of the Kawasaki dynamics at low temperatures in two dimensions Tunneling of the Kawasaki dynamics at low temperatures in two dimensions

2011-09-13

arxiv.org/abs/1109.2776v1

Mathematics

Probability

Scientific paper

Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature $\beta$ on a two dimensional torus $\Lambda_L=\{0,..., L-1\}^2$ . We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are $n^2\ll L$ particles and that the initial state is the configuration in which all sites of the square $\mb x + \{0,..., n-1\}^2$ are occupied. We show that in the time scale $e^{2\beta}$ the process is close to a Markov process on $\Lambda_L$ which jumps from any site $\mb x$ to any other site $\mb y\not =\mb x$ at a strictly positive rate which can be expressed in terms of the jump rates of simple random walks.

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