Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Collective Coordinate Approach to Statistical Mechanics of Soliton Systems. I. Classical Case

Mitsuto Fukuma, Satoshi Takada

Published Aug 15, 1986• 1 citation
Publisher

Abstract

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.

DOI
10.1143/jpsj.55.2701

Related papers

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

Mitsuto Fukuma, Satoshi Takada

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.

Journal of the Physical Society of Japan•Sep 15, 1986•1 citation

Statistical Mechanics of Soliton Systems

Toshio Tsuzuki, Kazuo Sasaki

The statistical mechanics of the soliton-bearing nonlinear Klein-Gordon systems in one dimension is reviewed. Several different theoretical approaches are discussed: the transfer-integral method, the soliton-gas approach, the Bethe-ansatz method, and a few methods of computing the quantum correction. The recent development in the low-temperature double-series expansion for the free energy is the main subject.

Progress of Theoretical Physics Supplement•Jan 1, 1988•23 citations

SOLITON THEORY FOR CLASSICAL STATISTICAL MECHANICS OF THE MASSIVE-THIRRING-SINE-GORDON MODEL

S. G. Chung

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.

International Journal of Modern Physics B•Aug 15, 1994•7 citations

Quantum-statistical mechanics of extended objects. I. Kinks in the one-dimensional sine-Gordon system

Kazumi Maki, Hajime Takayama

Making use of thermal-Green's-function technique, we study the quantum-statistical mechanics of a sine-Gordon system in 1 + 1 dimensions. In the weak-coupling limit, the temperature dependences of the soliton energy, ${E}_{s}$, the soliton inertial mass, and the soliton density are determined. At high temperatures ($T>m$, where $m$ is the mass of the fundamental field), ${E}_{s}$ decreases monotonically as the temperature increases, and ${E}_{s}$ jumps to zero around $T={T}_{\mathrm{cr}}$ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where ${{E}_{s}}^{0}$ is the soliton energy at $T=0$ K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}>T\ensuremath{\gg}m$, if ${E}_{s}$ in the classical theory is replaced by the temperature-dependent one of the present theory.

Physical review. B, Condensed matter•Oct 15, 1979•128 citations

Classical Limit of Bethe-AnsatzThermodynamics for the Sine-Gordon System

Niu-Niu Chen, Michael Durnan Johnson, Michael Fowler

We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature $t$ and the soliton density ${e}^{\frac{\ensuremath{-}1}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.

Physical Review Letters•Mar 3, 1986•21 citations

SOLITON STATISTICAL MECHANICS AND THE THERMALISATION OF BIOLOGICAL SOLITONS

R. Bullough, D. J. Pilling, YI CHENG +2 more

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.

Le Journal de Physique Colloques•Mar 1, 1989•2 citations
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.
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.
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.
Making use of thermal-Green's-function technique, we study the quantum-statistical mechanics of a sine-Gordon system in 1 + 1 dimensions. In the weak-coupling limit, the temperature dependences of the soliton energy, Es{E}_{s}Es​, the soliton inertial mass, and the soliton density are determined. At high temperatures (T>mT>mT>m, where mmm is the mass of the fundamental field), Es{E}_{s}Es​ decreases monotonically as the temperature increases, and Es{E}_{s}Es​ jumps to zero around T=TcrT={T}_{\mathrm{cr}}T=Tcr​ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where Es0{{E}_{s}}^{0}Es​0 is the soliton energy at T=0T=0T=0 K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}>T\ensuremath{\gg}m$, if Es{E}_{s}Es​ in the classical theory is replaced by the temperature-dependent one of the present theory.
We use the quantum Bethe-Ansatz method to compute the free energy of the classical sine-Gordon system. Previous attempts to do this have failed because the number of coupled integral equations to be solved to find the free-energy diverges in the classical limit. We present a transformation, extending a method of Maki, which reduces this divergent set to only two coupled equations, for the densities of solitons and anharmonic phonons. These equations can be solved iteratively in the temperature ttt and the soliton density ${e}^{\frac{\ensuremath{-}1}{t}}$ to give a double series for the free energy. This series coincides to high order with classical transfer-matrix results.
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.