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

Evgeni Burovski1, Wolfhard Janke2, Maria Guskova1

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Markov chain Monte Carlo simulations reveal that the acceptance rate in spin models linearly correlates with energy, particularly near critical points. This finding offers insights into simulation efficiency for classical spin systems.

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Area of Science:

  • Statistical Mechanics
  • Computational Physics
  • Monte Carlo Methods

Background:

  • Markov chain Monte Carlo (MCMC) simulations are crucial for studying classical spin models.
  • Local update algorithms, like single-spin-flip, are commonly used but their efficiency can vary.
  • Understanding the acceptance rate is key to optimizing MCMC performance.

Purpose of the Study:

  • To analytically derive and numerically investigate the mean acceptance rate of single-spin-flip algorithms.
  • To explore the relationship between acceptance rate and energy in various classical spin models.
  • To analyze the variance of the acceptance rate in relation to specific heat and phase transitions.

Main Methods:

  • Derivation of analytic expressions for the mean acceptance rate.
  • Numerical simulations for the one-dimensional Ising model, and three- and four-state Potts and XY models.
  • Analysis of acceptance rate dependence on energy and variance in relation to specific heat.

Main Results:

  • The average acceptance rate for the Metropolis algorithm is a linear function of energy.
  • For the Potts and XY models, the acceptance rate shows an almost linear dependence on energy in the critical region.
  • The variance of the acceptance rate remains finite even when specific heat exhibits a singularity near phase transitions.

Conclusions:

  • The acceptance rate in local update MCMC simulations of classical spin models exhibits predictable energy dependence.
  • Simulation efficiency can be better understood by analyzing the linear relationship between acceptance rate and energy.
  • The finite variance of the acceptance rate, despite specific heat singularities, suggests robust simulation behavior near phase transitions.