mcmc1 - Contents of Lecturenotes II 1 Statistical Physics...

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Contents of Lecturenotes II 1. Statistical Physics and Potts Models 2. Sampling and Re-weighting 3. Importance Sampling and Markov Chain Monte Carlo 4. The Metropolis Algorithm 1
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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, O = O ( β ) = O = Z - 1 K k =1 O ( k ) e - β E ( k ) (1) where Z = Z ( β ) = K 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
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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 ) 0 + H M ( k ) (3) where E ( k ) 0 = - 2 ij J ij ( q ( k ) i , q ( k ) j ) δ ( q ( k ) i , q ( k ) j ) + 2 d N q (4) with δ ( q i , q j ) = 1 for q i = q j 0 for q i = q j and M ( k ) = 2 N i =1 δ (1 , q ( k ) i ) . 3
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The sum ij 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 q -state 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 = 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 > 0 (conventionally J = 1) (5) the original model is recovered and q = 2 becomes equivalent to the Ising ferromagnet. The Ising case of Edwards-Anderson spin glasses and quadrupolar Potts glasses are obtained when the exchange constants are quenched random variables. Other choices of the J ij include anti-ferromagnets and the fully frustrated Ising model. For the energy per spin the notation is: e s = E/N . 4
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The normalization is chosen so that e s agrees for q = 2 with the conventional Ising model definition, β = β Ising = β Potts / 2 . For the 2 d Potts models a number of exact results are known in the infinite volume limit, mainly due to work by Baxter. The phase transition temperatures are 1 2 β Potts c = β c = 1 T c = 1 2 ln(1 + q ) , q = 2 , 3 , . . . . (6) At β c the average energy per state is e c 0 s = E c 0 /N = 4 q - 2 - 2 / q . (7) The phase transition is second order for q 4 and first order for q 5 for which the exact infinite volume latent heats e 0 s and
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