  Non-Gaussian distributions in cosmology are commonly evaluated with Monte
Carlo Markov-chain methods, as the Fisher-matrix formalism is restricted to the
Gaussian case. The Metropolis-Hastings algorithm will provide samples from the
posterior distribution after a burn-in period, and the corresponding
convergence is usually quantified with the Gelman-Rubin criterion. In this
paper, we investigate the convergence of the Metropolis-Hastings algorithm by
drawing analogies to statistical Hamiltonian systems in thermal equilibrium for
which a canonical partition sum exists. Specifically, we quantify
virialisation, equipartition and thermalisation of Hamiltonian Monte Carlo
Markov-chains for a toy-model and for the likelihood evaluation for a simple
dark energy model constructed from supernova data. We follow the convergence of
these criteria to the values expected in thermal equilibrium, in comparison to
the Gelman-Rubin criterion. We find that there is a much larger class of
physically motivated convergence criteria with clearly defined target values
indicating convergence. As a numerical tool, we employ physics-informed neural
networks for speeding up the sampling process.
