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| [[Image:Ergodicity.gif|thumb|right|250px|In an energy landscape with a potential barrier separating two regions of configuration space (bottom sketch), Monte Carlo sampling may be unable to sample the system over a sufficient range of configurations to accurately calculate thermodynamic data, compared to a favourable energy structure (top plot).]]
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| '''Umbrella sampling''' is a technique in [[computational physics]] and [[chemistry]], used to improve [[Sampling (statistics)|sampling]] of a system (or different systems) where [[ergodicity]] is hindered by the form of the system's [[energy landscape]]. It was first suggested by Torrie and Valleau in 1977 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-4DDR2HH-3V&_user=126524&_coverDate=02%2F28%2F1977&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000010360&_version=1&_urlVersion=0&_userid=126524&md5=950772b2b4de26ad320fd672ec611557]. It is related to [[Importance sampling]] in statistics.
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| Systems in which an energy barrier separates two regions of configuration space may suffer from poor sampling in [[Metropolis Monte Carlo]] runs, as the low probability of overcoming the potential barrier can leave inaccessible configurations poorly sampled – or even entirely unsampled – by the simulation. An easily visualised example occurs with a solid at its melting point: considering the state of the system with an [[order parameter]] ''Q'', both liquid (low ''Q'') and solid (high ''Q'') phases are low in energy, but are separated by a [[Thermodynamic free energy|free energy]] barrier at intermediate values of ''Q''. This prevents the simulation from adequately sampling both phases.
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| Umbrella sampling is a means of "bridging the gap" in this situation. The standard Boltzmann weighting for Monte Carlo sampling is replaced by a potential chosen to cancel the influence of the energy barrier present. The [[Markov chain]] generated has a distribution given by:
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| :<math>\pi(\mathbf{r}^N) = \frac{w(\textbf{r}^N) \exp{(-U(\mathbf{r}^N})/k_B T)}{\int{w(\mathbf{r^\prime}^N) \exp{(-U(\mathbf{r^\prime}^N})/k_B T)} d\mathbf{r^\prime}^N},</math>
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| with ''w''('''r'''<sup>''N''</sup>) a function chosen to promote configurations that would otherwise be inaccessible to a Boltzmann-weighted Monte Carlo run. In the example above, ''w'' may be chosen such that ''w'' = ''w''(''Q''), taking high values at intermediate ''Q'' and low values at low/high ''Q'', facilitating barrier crossing.
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| Values for a thermodynamic property ''A'' deduced from a sampling run performed in this manner can be transformed into canonical-ensemble values by applying the formula:
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| :<math>\langle A \rangle = \frac{\langle A / w \rangle_\pi}{\langle 1 / w \rangle_\pi},</math> | |
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| with the ''<math>\pi</math>'' subscript indicating values from the umbrella-sampled simulation. Series of umbrella sampling simulations can be analyzed using the weighted histogram analysis method (WHAM)<ref>{{cite journal|last=Kumar|first=Shankar|coauthors=Rosenberg, John M., Bouzida, Djamal, Swendsen, Robert H., Kollman, Peter A.|title=THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method|journal=Journal of Computational Chemistry|date=30 September 1992|volume=13|issue=8|pages=1011–1021|doi=10.1002/jcc.540130812}}</ref> or its generalization.<ref>{{cite journal|last=Bartels|first=C|title=Analyzing biased Monte Carlo and molecular dynamics simulations|journal=Chemical Physics Letters|date=7 December 2000|volume=331|issue=5-6|pages=446–454|doi=10.1016/S0009-2614(00)01215-X|bibcode = 2000CPL...331..446B }}</ref>
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| Subtleties exist in deciding the most computationally efficient way to apply the umbrella sampling method, as described in Frenkel & Smit's book ''Understanding Molecular Simulation''.
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| Alternatives to umbrella sampling for computing [[Potential of mean force|potentials of mean force]] or [[reaction rate]]s are [[free energy perturbation]] and [[transition path sampling#Transition interface sampling|transition interface sampling]]. A further alternative which functions in full non-equilibrium is [[transition path sampling#Stochastic process rare event sampling|S-PRES]].
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| == References ==
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| {{Reflist}}
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| [[Category:Monte Carlo methods]]
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| [[Category:Molecular dynamics]]
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| [[Category:Computational chemistry]]
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| [[Category:Theoretical chemistry]]
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