Convergence of the iterated Boltzmann map

by R. F. Streater; Publ. Res. Inst. Math. Sci., Kyoto, 20, 913-927, 1984.

ABSTRACT

Consider the Boltzmann equation of degree d\geq 2 for a discrete sample space Omega and discrete time; in one time-step it defines a non-linear map tau on the space of measures on Omega. We show that the entropy increases under tau. When the energy levels are equally spaced and scattering conserves energy, the iterated map {tau^m mu}_{m=1,2\ldots} converges in l_1 to a canonical state mu_\beta for any initial measure mu, if the scattering matrix mixes each energy shell. If Omega is finite, beta depends only on the energy of mu. Under other mixing conditions, tau^m mu converges to the microcanonical or grand canonical ensemble.

REMARKS

This method for constructing useful dynamical systems was generalised to the quantum case, and in other ways, in subsequent work. See my book Statistical Dynamics for a systematic treatment of the discrete case.