Relative free energy and its application to speed change model

Abstract

The purpose of this paper is to study a continuous spin system by introducing the relative free energy* which plays a role of the free energy in discrete spin systems. Although the usual free energy is a very important functional,it can be available to only identically distributed 2-spin lattice systems. About such systems, refer to Holley [1], [2]. On the other hand, our relative free energy enables us to deal with spin systems with state space not necessarily discrete nor identically distributed. In Section 3, we consider a 1-dimensional lattice spin system with continuous state space to show that the relative free energy does not increase with time and especiallyit strictly decreases if the initial state is not Gibbsian; this is one of the main results of this paper. From this fact, moreover we conclude that the time evolution of any shift invariant non-Gibbsian state converges to an equilibrium state. The precise definitions of the relative free eners-v et al are erivenin Section 2.