Nonlinear Lattice Statistical Mechanics: The Liouville-Horii Theorem

Statistical mechanics involves two universes at linear and nonlinear regimes. One is the Liouville universe at the linear range. The other is the Toda universe at the nonlinear range. For the universes, the above basic concept is well satisfied. Randon states in the Liouville universe, where degrees of freedom in kinetics are extremely high, become, if the exponential interactive potential energy between two nearest neighbor particles are introduced, a deterministic system in the Toda universe, whcih is subordinated to the nonlinear wave theory. The degrees of freedom in the motions are reduced by the constraints due to the potential energy.