An Introduction to Monte Carlo Simulations in Statistical by K. P. N. Murthy

By K. P. N. Murthy

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It consists of sampling microstates randomly, independently and with equal probabilities. Assemble an ensemble of N equi-probable microstates; carry out (Boltzmann) weighted average of the macroscopic quantity, say magnetization M, over the ensemble as given below. M = NLim. →∞ M N = N i=1 M(Ci ) exp[−βE(Ci )] . e. V shall be of the order of 1023 . Let us consider a modest model system with some hundred spins on a two dimensional 10 × 10 square lattice. The number of spin configurations is 2100 ≈ 1030 .

Consider a two dimensional square lattice of sites. The first site in a row is considered as the right nearest neighbour of the last site in the same row and the last site in a row is considered as the left nearest neighbour of the first site in the same row. The same holds for the top and bottom sites in each column. Periodic boundaries are known to give least finite size effects; we shall see about finite size effects sometime later. g. , that are implemented depending on the nature of the problem.

Assemble an ensemble of N equi-probable microstates; carry out (Boltzmann) weighted average of the macroscopic quantity, say magnetization M, over the ensemble as given below. M = NLim. →∞ M N = N i=1 M(Ci ) exp[−βE(Ci )] . e. V shall be of the order of 1023 . Let us consider a modest model system with some hundred spins on a two dimensional 10 × 10 square lattice. The number of spin configurations is 2100 ≈ 1030 . Even if we assume optimistically that it takes a nano second to generate a spin configuration, the total time required to sample all the spin configurations is nearly of the order of thirty thousand billion years!

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