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1 results

  • 11 - Numerical Simulations

  • Luca Leuzzi, National Research Council of Italy, Enzo Marinari, Sapienza Università di Roma, Giorgio Parisi, Sapienza Università di Roma
  • Book:
    Probability Theory for Quantitative Scientists
    Published online:
    24 July 2025
    Print publication:
    14 August 2025, pp 289-303

  • Summary

    In this chapter we continue to present the theory of Markov chains, specializing in reversible chains, for which the very important relation of detailed balance is shown to hold. This is a basic concept also in continuous processes, and the foundation of the property termed “equilibrium” in thermodynamics and statistical mechanics. The major application we present is the Monte Carlo method for estimating thermodynamic averages, which maps the average on a statistical ensemble into the dynamics through states of a reversible Markov chain. We introduce the Metropolis algorithm and we apply it to the Ising model for ferromagnetism.