Nonlinearity and Disorder in the Statistical Mechanics of Integrable Systems

Abstract

Attention is drawn to a theory of the statistical mechanics (SM) of the integrable models in 1+1 dimension — a theory of ‘soliton statistical mechanics’ classical and quantum [1–17]. This SM provides a generic example of integrable nonlinearity interacting with disorder. In the generic classical examples, such as the classical SM of the sine-Gordon model, phonons provide disorder in which sit coherent structures — the kink-like solitons. But these solitons are dressed by the disorder, in equilibrium, while the breather-like solitons break up to form the disordered structures which are the phonons in thermal equilibrium. On the other hand quantum solitons, dressed by both the vacuum and finite temperature effects, are nevertheless coherent ordered structures. In particular, the quantum breathers have a degree of freedom, and a consequent internal structure,which is localised in k-space, and through which the quantum breathers exhibit some features of quantum chaos. The disorder of the phonons is eliminated as such in these quantum systems. But the translational motions of these quantum particles continue to be disordered by the Brownian motion thereby contributing to the finite entropies of these particles.KeywordsBrownian MotionCoherent StructureQuantum ChaosFermi OscillatorClassical SolitonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.