# Acceptance rate is a thermodynamic function in local Monte Carlo algorithms

## TL;DR

This paper explores the acceptance rate in local Monte Carlo algorithms, deriving analytic expressions for the 1D Ising model and numerically analyzing other models, revealing a near-linear relationship with energy.

## Contribution

It provides the first analytic expression for the acceptance rate in the 1D Ising model and extensive numerical analysis across various models and dimensions.

## Key findings

- Acceptance rate is a linear function of energy in the 1D Ising model.
- Acceptance rate remains nearly linear with energy in critical regions of other models.
- Variance of acceptance rate stays finite near phase transitions despite singularities in specific heat.

## Abstract

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.