Abstract Set-Up and Metastability in the Zero-Temperature Limit

Abstract

Chapter 16 describes the metastable behaviour of lattice systems in small volumes at low temperatures subject to a Metropolis dynamics. These theorems are derived under two hypotheses on the energy landscape, i.e., on the interaction Hamiltonian. The theorems themselves are model-independent, and therefore amplify the universal nature of metastability (in the setting considered here). However, they involve a number of quantities that are model-dependent. In Sect. 16.1 we define Metropolis dynamics on a general configuration space with respect to a general Hamiltonian and for a general set of allowed moves, we state the theorems subject to the hypotheses, and we place the results in their proper context. Section 16.2 explains how this abstract set-up fits into the potential-theoretic framework. The proofs of the theorems are given in Sect. 16.3. In Sect. 16.4 we take a brief look at two other dynamics, namely, heat-bath dynamics and probabilistic cellular automata, and we indicate how these can be included into the same abstract set-up with only minor modifications.

Talking is a wonderful smoother-over of difficulties. When I come upon anything—in Logic or in any other hard subject—that entirely puzzles me, I find it a capital plan to talk it over, aloud, even when I am all alone. One can explain things so clearly to one’s self! And then, you know, one is so patient with one’s self: one never gets irritated at one’s own stupidity!  (Lewis Carroll, A Selection from Symbolic Logic)