Statistical mechanics of the Toda lattice based on soliton dynamics

Abstract

A classical theory of statistical mechanics of the Toda lattice is presented on the basis of soliton dynamics. Following the inverse spectral theory, the partition function of the Toda lattice is reconstructed from one-particle partition functions of soliton and ripple modes. Discussions are made on the contribution of these modes to the thermodynamic properties of the Toda lattice. At low temperatures, it is shown that the average number of excited solitons has the temperature dependence $T^{\frac{1}{3}}$. With the comparison of our results with those from the exact theory, several problems to be worked out are pointed out in our soliton-ripple gas-mixture model.