Nonequilibrium Aspects of Transfer Matrices
Per Arne Rikvold · Progress of Theoretical Physics Supplement · 1989
A set of functions associated with the eigenstates of the transfer matrix, referred to as 'generalized probability densities', are defined.It is shown that, under a set of reasonable regularity conditions, these functions lead to constrained nonequilibrium generalizations of the entropy, internal energy, magnetization, and free energy for a discrete Ising or lattice-gas system.The behavior of these 'generalized state functions' is studied numerically for a simple model system with variable interaction range, N, which is known to approach the mean-field limit as N -HX>.At a temperature below the mean-field critical temperature clear indications of stable, metastable, and unstable states are observed.The approach appears promising, and topics for future research are suggested.§ 1. Introduction 95The transfer-matrix method for obtaining the partition function and correlation functions of a discrete Ising or lattice-gas system 1 > has received considerable attention in recent years through its applications to the finite-size scaling (phenomenological renormalization) technique for studying equilibrium phase transitions.2 H> Many workers have contributed to developing the combination of these two techniques into a powerful computational tool, and a large number of papers have been published in this process.A comprehensive review of the method and its applications is given in Ref. 9).Although much work has gone into understanding the equilibrium aspects of the transfer-matrix formalism, contained in its few largest eigenvalues and their eigenvectors, comparatively little effort has been directed towards unraveling the nonequilibrium information presumably provided by the smaller eigenvalues and their eigenvectors.The idea of a "restricted equilibrium ensemble" was introduced by Penrose and Lebowitz,I 0 >,n> and combined with the transfer-matrix formalism by Privman and Schulman.12 >' 13 > The latter authors constructed renormalized, free energy-like functions from combinations of pairs of nearly degenerate eigenvalues.Within this conceptual framework Novotny, Klein and the present author have studied the emergence of a classical spinodal in a long-range Ising model that approaches the classical mean-field model as the interaction range increases.14 > In spite of encouraging preliminary results, a number of unanswered questions concerning the interpretation of the eigenvectors and eigenvalues of the transfer matrix in terms of nonequilibrium 'constrained states' remain unsolved.We are pursuing these