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Unformatted text preview: Lecturenotes MCMC IV – Contents 1. Multicanonical Ensemble 2. How to get the Weights? 3. Example Runs ( 2 d Ising and Potts models) 4. ReWeighting to the Canonical Ensemble 5. Energy and Specific Heat Calculation 6. Free Energy and Entropy Calculation 7. Summary 1 Multicanonical Ensemble One of the questions which ought to be addressed before performing a large scale computer simulation is “What are suitable weight factors for the problem at hand?” So far we used the Boltzmann weights as this appears natural for simulating the Gibbs ensemble. However, a broader view of the issue is appropriate. Conventional canonical simulations can by reweighting techniques only be extrapolated to a vicinity of this temperature. For multicanonical simulations this is different: A single simulation allows to obtain equilibrium properties of the Gibbs ensemble over a range of temperatures. Of particular interest are two situations for which canonical simulations do not provide the appropriate implementation of importance sampling: 1. The physically important configurations are rare in the canonical ensemble. 2. A rugged free energy landscape makes the physically important configurations difficult to reach. 2 Multicanonical simulations approach these problem by trying to sample, in an appropriate energy range, with an approximation b w mu ( k ) = w mu ( E ( k ) ) = e b ( E ( k ) ) E ( k ) + a ( E ( k ) ) (1) to the weights b w 1 /n ( k ) = w 1 /n ( E ( k ) ) = 1 n ( E ( k ) ) (2) where n ( E ) is the spectral density. The function b ( E ) defines the inverse microcanonical temperature and a ( E ) the dimensionless, microcanonical free energy . The function b ( E ) has a relatively smooth dependence on its arguments, which makes it a useful quantity when dealing with the weight factors. Instead of the canonical energy distribution P ( E ) , one samples a new multicanonical distribution, which does not longer depend strongly on the energy, P mu ( E ) = c mu n ( E ) w mu ( E ) ≈ c mu . (3) 3 The desired canonical probability density is obtained by reweighting P ( E ) = c β c mu P mu ( E ) w mu ( E ) e βE . (4) This relation is rigorous, as the weights w mu ( E ) used in the simulation are exactly known. Statistical errors are those of P mu ( E ) times an exactly known factor. The multicanonical method requires two steps: 1. Obtain a working estimate b w mu ( k ) of the weights b w 1 /n ( k ) . Working estimate means that the approximation has to be good enough to ensure movement in the desired energy range, while deviations from a flat P mu ( E ) are tolerable. 2. Perform a Markov chain MC simulation with the fixed weights b w mu ( k ) . The thus generated configurations constitute the multicanonical ensemble. Canonical expectation values are found by reweighting to the Gibbs ensemble and jackknife methods allow reliable error estimates....
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 Fall '08
 Berg
 Energy, Statistical Mechanics, Entropy, Heat, Potts, 10state Potts, 10state Potts model

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