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# mcmc5 - Lecture V Multicanonical Simulations 1...

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Lecture V: Multicanonical Simulations. 1. Multicanonical Ensemble 2. How to get the Weights? 3. Example Runs ( 2 d Ising and Potts models) 4. Re-Weighting to the Canonical Ensemble 5. Energy and Specific Heat Calculation 6. Free Energy and Entropy Calculation 7. Summary 1

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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 re-weighting 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
MC calculation of the interface tension of a first order phase transition provide an example where canonical MC simulation miss the important configurations. Let N = L d be the lattice size. For first order phase transition pseudo-transition temperatures β c ( L ) exist so that the energy distributions P ( E ) = P ( E ; L ) become double peaked and the maxima at E 1 max < E 2 max are of equal height P max = P ( E 1 max ) = P ( E 2 max ) . In-between the maximum values a minimum is located at some energy E min . Configurations at E min are exponentially suppressed like P min = P ( E min ) = c f L p exp( - f s A ) (1) where f s is the interface tension and A is the minimal area between the phases, A = 2 L d - 1 for an L d lattice, c f and p are constants (computations of p have been done in the capillary-wave approximation). The interface tension can be calculated by Binder’s histogram method: f s ( L ) = - 1 A ( L ) ln R ( L ) with R ( L ) = P min ( L ) P max ( L ) (2) 3

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and a finite size scaling (FSS) extrapolation of f s ( L ) for L → ∞ . For large systems a canonical MC simulation will practically never visit configurations at energy E = E min and estimates of the ratio R ( L ) will be very inaccurate. The terminology supercritical slowing down was coined to characterize such an exponential deterioration of simulation results with lattice size. Multicanonical simulations approach this problem by sampling, in an appropriate energy range, with an approximation w mu ( k ) = w mu ( E ( k ) ) = e - b ( E ( k ) ) E ( k ) + a ( E ( k ) ) (3) to the weights w 1 /n ( k ) = w 1 /n ( E ( k ) ) = 1 n ( E ( k ) ) (4) where n ( E ) is the spectral density. The function b ( E ) defines the inverse microcanonical temperature and a ( E ) the dimensionless, microcanonical free 4
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.

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mcmc5 - Lecture V Multicanonical Simulations 1...

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