Poincaré Cycles, Ergodicity, and Irreversibility in Assemblies of Coupled Harmonic Oscillators

The transport coefficients (diffusion constant, electrical conductivity, etc.) associated with irreversible processes in an assembly of particles can be expressed as integrals over certain time relaxed correlation functions between small numbers of variables of the assembly. The scattering of slow neutrons is also a measure of time relaxed correlation functions.Irreversibility is a consequence of the vanishing of the correlation coefficients as the relaxation time becomes infinite. On the other hand these coefficients have Poincare cycles so that any value which they take on is repeated an infinite number of times. It is shown that, in the case of fluctuations of 0(N−½) from zero (N being the number of degrees of freedom), the period of Poincare cycles is of the order of the mean period of normal mode vibrations while for fluctuations of a magnitude independent of N the period is of the order of CN where C is a constant which is greater than 1.The time relaxed correlation coefficients of a pair of particl...