homework7 - Physics 262 Statistical Physics Fall 2008 Problem Set 7 due Wed Nov 12 Reading Pathria Sections 11.1-11.5 12.1-12.3 1 Consider an Ising

homework7 - Physics 262 Statistical Physics Fall 2008...

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Physics 262 : Statistical Physics Fall 2008 Problem Set 7 due: Wed Nov 12 Reading: Pathria, Sections 11.1-11.5, 12.1-12.3 1. Consider an Ising model on a system of N sites. Let N be the number of up spins. Calculate, Γ( N , N ), the total number of ways these N spins could have been placed among the N sites. Obtain the entropy S = k B ln Γ as a function of the magnetization m = ( N - N ) /N , where N = N - N . 2. Ising antiferromagnet: We consider the Ising antiferromagnet on a square lattice. The Hamiltonian is H I = J X h ij i σ i σ j (1) where i, j extend over the sites of a square lattice, h ij i refers to nearest neighbors, and σ i = ± 1. Note that there is no minus sign in front of J , as in class. We take J > 0, so the ferromagnetic state, with all σ i parallel, is the highest energy state. The ground states have the pattern of a chess board: σ i = 1 on one sublattice (A) and σ i = - 1 on the other sublattice (B), and vice versa. Use mean field theory to describe the phase diagram of this model. Argue that the mean field Hamiltonian should have two fields,

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