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Unformatted text preview: Contents of Lecturenotes II 1. Statistical Physics and Potts Models 2. Sampling and Reweighting 3. Importance Sampling and Markov Chain Monte Carlo 4. The Metropolis Algorithm 1 Statistical Physics and Potts Model MC simulations of systems described by the Gibbs canonical ensemble aim at calculating estimators of physical observables at a temperature T . In the following we choose units so that β = 1 /T and consider the calculation of the expectation value of an observable O . Mathematically all systems on a computer are discrete, because a finite word length has to be used. Hence, b O = b O ( β ) = hOi = Z 1 K X k =1 O ( k ) e β E ( k ) (1) where Z = Z ( β ) = K X k =1 e β E ( k ) (2) is the partition function . The index k = 1 ,...,K labels all configurations (or microstates) of the system, and E ( k ) is the (internal) energy of configuration k . To distinguish the configuration index from other indices, it is put in parenthesis. 2 We introduce generalized Potts models in an external magnetic field on d dimensional hypercubic lattices with periodic boundary conditions. Without being overly complicated, these models are general enough to illustrate the essential features we are interested in. In addition, various subcases of these models are by themselves of physical interest. Generalizations of the algorithmic concepts to other models are straightforward, although technical complications may arise. We define the energy function of the system by β E ( k ) = β E ( k ) + H M ( k ) (3) where E ( k ) = 2 X h ij i J ij ( q ( k ) i ,q ( k ) j ) δ ( q ( k ) i ,q ( k ) j ) + 2 dN q (4) with δ ( q i ,q j ) = 1 for q i = q j 0 for q i 6 = q j and M ( k ) = 2 N X i =1 δ (1 ,q ( k ) i ) . 3 The sum h ij i is over the nearest neighbor lattice sites and q ( k ) i is called the Potts spin or Potts state of configuration k at site i . For the qstate Potts model q ( k ) i takes on the values 1 ,...,q . The external magnetic field is chosen to interact with the state q i = 1 at each site i , but not with the other states q i 6 = 1 . The J ij ( q i ,q j ) , ( q i = 1 ,...,q ; q j = 1 ,...,q ) functions define the exchange coupling constants between the states at site i and site j . The energy function describes a number of physically interesting situations. With J ij ( q i ,q j ) ≡ J > (conventionally J = 1) (5) the original model is recovered and q = 2 becomes equivalent to the Ising ferromagnet. The Ising case of EdwardsAnderson spin glasses and quadrupolar Potts glasses are obtained when the exchange constants are quenched random variables. Other choices of the J ij include antiferromagnets and the fully frustrated Ising model. For the energy per spin the notation is: e s = E/N ....
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This note was uploaded on 11/10/2011 for the course PHY 5157 taught by Professor Berg during the Fall '08 term at University of Florida.
 Fall '08
 Berg
 Physics

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