Indian Journal of Economics and Development

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Metropolis and Wang-Landau Algorithms


Affiliations
1 Presidency College, Kamarajar Salai, Triplicane, Chennai 600 005, Tamilnadu, India
2 Chennai Mathematical Institute (CMI), H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai 603 103, Tamilnadu, India
 

Objectives: We review two algorithms developed for simulating macroscopic systems. The first is the Metropolis and the second is the Wang-Landau algorithm.

Methods: Metropolis algorithm has been extensively employed for simulating a canonical ensemble and estimating macroscopic properties of a closed system at any desired temperature. A mechanical property, like energy can be calculated by averaging over a large number of micro states of the stationary Markov chain generated by the Metropolis algorithm. However thermal properties like entropy, and free energies are not easily accessible. A method called umbrella sampling was proposed some forty years ago for this purpose. Ever since, umbrella sampling has undergone several metamorphoses and we have now multi canonical Monte Carlo, entropic sampling, flat histogram methods, Wang-Landau algorithm etc.

Findings: In this paper we review Metropolis algorithm for estimating mechanical properties and Wang-Landau algorithm for estimating both mechanical and thermal properties of an equilibrium system.

Applications: We shall make the review as pedagogical and as self-contained as possible. These algorithms can be applied to a variety of problems in physics, astrophysics, chemistry, biology, soft matter, computer science, etc.


Keywords

Monte Carlo Simulation, Metropolis Algorithm, Entropic Sampling, Flathistogram Methods, Detailed Balance, Markov Chain.
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Notifications

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  • Metropolis and Wang-Landau Algorithms

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Authors

T. Pramananda Perumal
Presidency College, Kamarajar Salai, Triplicane, Chennai 600 005, Tamilnadu, India
K. P. N. Murthy
Chennai Mathematical Institute (CMI), H1, SIPCOT IT Park, Siruseri, Kelambakkam, Chennai 603 103, Tamilnadu, India

Abstract


Objectives: We review two algorithms developed for simulating macroscopic systems. The first is the Metropolis and the second is the Wang-Landau algorithm.

Methods: Metropolis algorithm has been extensively employed for simulating a canonical ensemble and estimating macroscopic properties of a closed system at any desired temperature. A mechanical property, like energy can be calculated by averaging over a large number of micro states of the stationary Markov chain generated by the Metropolis algorithm. However thermal properties like entropy, and free energies are not easily accessible. A method called umbrella sampling was proposed some forty years ago for this purpose. Ever since, umbrella sampling has undergone several metamorphoses and we have now multi canonical Monte Carlo, entropic sampling, flat histogram methods, Wang-Landau algorithm etc.

Findings: In this paper we review Metropolis algorithm for estimating mechanical properties and Wang-Landau algorithm for estimating both mechanical and thermal properties of an equilibrium system.

Applications: We shall make the review as pedagogical and as self-contained as possible. These algorithms can be applied to a variety of problems in physics, astrophysics, chemistry, biology, soft matter, computer science, etc.


Keywords


Monte Carlo Simulation, Metropolis Algorithm, Entropic Sampling, Flathistogram Methods, Detailed Balance, Markov Chain.

References