We study properties of Markov chain Monte Carlo simulations of classical spin models with local updates. We derive analytic expressions for the mean value of the acceptance rate of single-spin-flip algorithms for the one-dimensional Ising model. We find that for the Metropolis algorithm the average acceptance rate is a linear function of energy. We further provide numerical results for the energy dependence of the average acceptance rate for the 3- and 4-state Potts model, and the XY model in one and two spatial dimensions. In all cases, the acceptance rate is an almost linear function of the energy in the critical region. The variance of the acceptance rate is studied as a function of the specific heat. While the specific heat develops a singularity in the vicinity of a phase transition, the variance of the acceptance rate stays finite.
Our result for the acceptance rate of the heat-bath algorithm for the one-dimensional Ising model can be viewed as an addition to the Glauber paper on the dynamics of the one-dimensional Ising model. The acceptance rate in the Glauber dynamics is the frequency of the spin flips.

Lev Shchur is a computational physicist working in many areas of physics. His scientific life is connected with Landau Institute for Theoretical Physics since the master student time. He got his PhD in 1983 and Habilitation degree in 1997, both from Landau Institute. He is vice-chairman of Science Center in Chernogolovka, and Professor at Moscow Institute for Electronics and Mathematics at the National Research University Higher School of Economics. He elected as APS fellow in 2017 for his innovative use of computer simulations and the development of superior random number generators for their use in statistical physics.