Statistical mechanics of one-dimensional solitary-wave-bearing scalar fields: Exact results and ideal-gas phenomenology

The statistical mechanics of one-dimensional scalar fields governed by nonlinear wave equations having solitary-wave (soliton) solutions are discussed in detail. Previously neglected "phase-shift" interactions between phonons and solitary waves (kinks) are taken into account and it is shown that these interactions provide the mechanism for sharing of energy and degrees of freedom among the "elementary" excitations of the nonlinear system. In particular, the ideal-gas phenomenology proposed by Krumhansl and Schrieffer for the $``{\phi }^4\text{'}\text{'}$ model is corrected and extended to the entire class of nonlinear Klein-Gordon models having solitary-wave solutions (e.g., ${\phi }^4$, sine-Gordon, double-quadratic, etc.). By a comparison of the results of the phenomenological approach to those obtained via the exact transfer-operator method, it is found that the ideal-gas phenomenology gives exact results for the various low-temperature thermodynamic functions and correlation lengths.