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Journal of the Physical Society of Japan

Collective Coordinate Approach to Statistical Mechanics of Soliton Systems. II. Non-Canonical Method

Mitsuto Fukuma, Satoshi Takada

Published Sep 15, 1986• 1 citation
Publisher

Abstract

Quantum statistical mechanics of soliton systems is developed in the path integral formalizm by making use of the non-canonical collective coordinate method. The free energy is calculated in the ideal-gas approximation up to the order of two-loop. Explicit calculations are given for the sine-Gordon model. The classical limit of the present quantum calculation agrees with the exact result of the low-temperature expansion of the transfer-integral method. Numerical calculation reveals the quantum effect on the specific heat.

DOI
10.1143/jpsj.55.3123

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Quantum statistical mechanics of soliton systems is developed in the path integral formalizm by making use of the non-canonical collective coordinate method. The free energy is calculated in the ideal-gas approximation up to the order of two-loop. Explicit calculations are given for the sine-Gordon model. The classical limit of the present quantum calculation agrees with the exact result of the low-temperature expansion of the transfer-integral method. Numerical calculation reveals the quantum effect on the specific heat.
The collective coordinate method is applied to develop the statistical mechanics of soliton systems. The present perturbation expansion is a direct extention of that done in the quantum field theory and is applicable to the quantum case. A new type of divergence which has never met so far in the many body theory appears in the present treatment but it can be removed by an appropriate treatment giving reasonable results. As the first paper of this series the classical case is investigated and explicit calculation is done in the classical sine-Gordon model. The results agree with the exact low temperature corrections obtained by the transfer-integral method and with those obtained by Miyashita and Maki who applied the collective coordinate approach in a different way.
This paper reviews soliton theories for classical statistical mechanics of the massive-Thirring-sine-Gordon model. These theories attempt not only to reproduce the classical transfer matrix method result but also to provide thermodynamic information about exact nonlinear excitations of the model. Three theories are discussed. Firstly, the soliton–gas phenomenology which is successful in the Toda lattice is shown to be not even approximately correct for the sine-Gordon model. This breakdown is due to the incorrect counting of the number of degrees of freedom. Secondly, the soliton–phonon theory which reproduces the WKB result is also shown to be not exact. This non-exactness is ascribed to the fact that the nonlinear phonons are not exact elementary excitations in the model. Thirdly, a non-gas phenomenological soliton–breather theory is derived as a classical limit of quantum statistical mechanics of the model. The exactness of the last theory is demonstrated through an accurate numerical analysis of its basic equations.
The calculation of the equilibrium free energy of integrable models like the sine-Gordon and attractive nonlinear Schrödinger models is discussed in the context of biological molecules like DNA : the thermalisation process (approach to equilibrium) is also discussed. The sine-Gordon model has a "repulsive" form which is the sinh-Gordon model. The approach to equilibrium of the sinh-Gordon model is described in all completeness in terms of a quantum mechanical master equation at finite temperatures. Although the dynamical evolution of the master equation as written is a solved problem, only the equilibrium solution is examined in this paper. The equilibrium free energy is calculated exactly as an integral equation for certain excitation energies at finite temperatures. Bose-fermi equivalent forms of this integral equation are given. The bose form yields a similar integral equation in classical limit. The iteration of this yields a low temperature asymptotic series for the classical free energy which checks against the result of the transfer integral method (TIM). Results for the zero temperature quantum eigenenergies are found. A further discussion of the dynamics of the approach to thermal equilbrium is made.
We give two fundamental methods for evaluation of classical free energies of all the integrable models admitting soliton solutions; the sine-Gordon equation is one example. Periodic boundary conditions impose integral equations for allowed phonon and soliton momenta. From these, generalized Bethe-Ansatz and functional-integration methods using action-angle variables follow. Results for free energies coincide, and coincide with those that we find by transfer-integral methods. Extension to the quantum case, and quantum Bethe Ansatz, on the lines to be reported elsewhere for the sinh-Gordon equation, is indicated.