Energy Sharing and Equilibrium for Nonlinear Systems

Abstract

A study is made of a one‐dimensional system of identical particles in which the forces between neighbors are linear. The system is nonlinear because it is assumed that collisions occur between adjacent particles, which each have an effective diameter d. The energies E_i in the linear normal modes are computed numerically to show that energy is freely exchanged between all the modes in the system, as predicted by the theory. Furthermore, the time averages 〈E_i〉 of these energies show a strong tendency towards equipartition of energy among the modes. This is in distinct contrast to the computations of Ulam, Fermi, and Pasta, which showed that some nonlinear systems appear to be nonergodic. An equation of state and an expression for the total energy of the system as a function of thermodynamic coordinates are derived via statistical mechanics. Expected values for the pressure and temperature of the assembly may then be computed. A comparison of these with the numerical values of those variables arising from the computations shows that the nonlinear system approaches equilibrium.