Statistical Mechanics - Physics at Oregon State University

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176 CHAPTER 8. MEAN FIELD THEORY: CRITICAL TEMPERATURE. The results of this section are identical to the mean-field results. In this section we derived the mean-field results via the density matrix and approximated the density matrix by ignoring all correlations. This is another way of understanding how mean-field theory ignores correlations. The form of the function G is very similar to forms we used in our treatment of Landau theory in thermodynamics. We can show this in more detail by expanding the function in powers of m for values of m near zero. Using ∞ (−1) (1+m) log(1+m) = (1+m) k+1 k mk ∞ (−1) = k+1 k mk ∞ (−1) + k+1 k k=1 k=1 k=1 mk+1 (8.94) and the fact that we have to add the same expression with negative m shows that even exponents are the only ones to survive. Hence (1+m) log(1+m)+(1−m) log(1−m) = 2 or or ∞ k=2,even (1 + m) log(1 + m) + (1 − m) log(1 − m) = 2 (−1) k+1 k mk +2 ∞ k=2,even This is an even function of m as needed, and we have ∞ k=2,even 1 k(k − 1) mk (−1) k k − 1 mk G(h, T, N; m) = − 1 2 JNqm2 − hNm + NkBT − log(2) + 1 2 m2 + 1 12 m4 (8.95) (8.96) (8.97) 1 N G(h, T, N; m) = [−hm] + [−NkBT log(2)] + 1 2 [kBT − Jq]m 2 + 1 4 [kBT 3 ]m4 (8.98) This is exactly the model for a second order phase transition that we discussed in Thermodynamics. Hence the Ising model in the mean field approximation shows a second order phase transition at a temperature Tc given by kBTc = qJ. The critical exponent β is therefore 1 (Note that β is in statistical mechanics used for 1 kBT 2 , but also denotes a critical exponent. Do not confuse these two!). The relation between the macroscopic Landau theory and statistical mechanics is very important, it shows how the parameters in a thermodynamical model can be derived from a statistical mechanical theory on a microscopic level.
8.5. CRITICAL TEMPERATURE IN DIFFERENT DIMENSIONS. 177 In the previous sections we have derived the same result in two different ways, and therefore have shown that these two approximations are equivalent. In Mean Field Theory we replace the quadratic coupling terms in the Hamiltonian by the product of an average and a variable. This is equivalent to coupling each spin to the average field due to the neighbors. In the Bragg-Williams approximation we replace the density matrix by a form that does not contain coupling between neighboring sites. As a result, in the Hamiltonian again the average field of the neighbors plays a role only. 8.5 Critical temperature in different dimensions. In the mean-field solution of the Ising model the critical temperature does not explicitly depend on the dimensionality of the problem. Indirectly it does, of course, because the possible values of q depend on the number of dimensions. For a mono-atomic, periodic lattice in one dimension q has to be two, in two dimensions q can be two, four, or six, and in three dimensions q can be two, four, six, eight, or twelve. Hence the higher the number of dimensions, the larger the critical temperature can be. For closed packed systems the number of neighbors is maximal, and the critical temperature would increase according to the pattern 2:6:12 going from one to three dimensions. This is also true in general. When the number of neighbors increases, the total strength of the interactions driving an ordered state increases and the ordered state can persist up to a higher temperature. The mean-field solution of the Ising model is only an approximation. In general, the errors in the mean-field solutions become larger when the number of dimensions decreases. In one dimension, for example, there are no phase transitions in the Ising model! What causes the error in mean-field theory? In essence, we did something dangerous by interchanging two limits. We first assumed that N → ∞ in order to deduce that 〈Si〉 is the same for all positions i. Or that ρi is independent of i. Then we used this common value to solve the equations for finite values of N. The results made of this procedure made sense in the thermodynamic limit, since m did not depend on N. The error is actually in the first step. We have to solve the whole problem at finite N and calculate mi(N, T ). In this expression we should take the limit N → ∞. The results will be different in that case. We cannot interchange these limits! There is an old argument, due to Landau (who else?), that the Ising chain does not show a phase transition. Consider a finite chain of N sites, numbered one through N sequentially. Assume that h = 0. There are two completely ordered states, either all spins are up or down. Next we consider all states which include a so-called Bloch wall. The first m spins have one direction, while the remaining N − m spins have the opposite direction. The number of these states is 2(N −1). The energy of a completely ordered chain is E0 = −J(N −1). The energy of a Bloch wall state is different only due to the contribution of the two spin variables at opposite sites of the Bloch wall. Therefore the energy of a Bloch wall state is −J(N − 2) + J = E0 + 2J. At a fixed temperature and at
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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90 CHAPTER 5. FERMIONS AND BOSONS
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120 CHAPTER 6. DENSITY MATRIX FORMA
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226 CHAPTER 9. GENERAL METHODS: CRI
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230 APPENDIX A. SOLUTIONS TO SELECT
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