Convergence of the iterated Boltzmann map
Consider the Boltzmann equation of degree \(d\geq 2\) for a discrete sample space \(\Omega\) and discrete time; in one 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 of \(\Omega\) are equally spaced and scattering conserves energy, the iterated map \(\{\tau^ m\mu \}_{m=1,2,...}\) converges in \(\ell_ 2\) 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.
microcanonical or grand canonical ensemble
scientific article; zbMATH DE number 3885086