Wang-Landau method: Difference between revisions
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whereas for the Wang-Landau procedure we can write: | whereas for the Wang-Landau procedure we can write: | ||
<math> P(X) \propto \exp \left[ | <math> P(X) \propto \exp \left[ f(E(x)) \right] </math> ; | ||
where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes | where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes | ||
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a uniform distribution); this is done by modifying the values of <math> f(E) </math> | a uniform distribution); this is done by modifying the values of <math> f(E) </math> | ||
to reduce the probability of the energies that have been already ''visited''. | to reduce the probability of the energies that have been already ''visited''. | ||
Such a simple scheme is continued until the shape of the energy distribution | |||
approaches the prefixed one. | |||
Revision as of 12:14, 8 July 2008
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The Wang-Landau method was proposed by F. Wang and D. P. Landau (Ref. 1) to compute the density of states, , of Potts models; where is the number of microstates of the system having energy .
The Wang-Landau method in its original version is a simulation technique designed to reach an uniform sampling of the energies of the system in a given range. In a standard Metropolis Monte Carlo in the canonical ensemble the probability of a given microstate, is given by:
;
![{\displaystyle P(X)\propto \exp \left[-E(X)/k_{B}T\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/958003b4d2a8e698ba9b8134763dfa7d0325696c)
whereas for the Wang-Landau procedure we can write:
;
![{\displaystyle P(X)\propto \exp \left[f(E(x))\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2af2791cb09b32c7c624dbc8a2784622e80b7aa6)
where is a function of the energy. changes during the simulation in order get a prefixed distribution of energies (usually a uniform distribution); this is done by modifying the values of to reduce the probability of the energies that have been already visited. Such a simple scheme is continued until the shape of the energy distribution approaches the prefixed one.
References
- Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E 64 056101 (2001)
- D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics 72 pp. 1294-1302 (2004)
- Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics 127 154504 (2007)
- R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics 127 184105 (2007)
- R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E 75 046701 (2007)