Statistical Mechanics - Physics at Oregon State University

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204 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS. or χ = Tr S0βMe −β(H−hM) Tr e −β(H−hM) This can be transformed to − Tr S0e −β(H−hM) TrβM e −β(H−hM) Tr e −β(H−hM) 2 (9.60) χ = β {〈S0Si〉 − 〈S0〉〈Si〉} (9.61) i χ = β 〈S0 − m〉〈Si − m〉 (9.62) i which clearly shows that χ is directly related to the fluctuations in the spin variables. Here we used 〈S0〉 = 〈Si〉 = m. If fluctuations are uncorrelated we have < AB >=< A > which would seem to imply that for uncorrelated fluctuations χ = 0. That is not correct, however. Fluctuations on the same site are always correlated, and in that case we would get χ =< (S0 − m) 2 >, which is always positive indeed. It is customary to define a spin-correlation function Γi by Γi(T ) = 〈S0Si〉T − 〈S0〉T 〈Si〉T (9.63) If the spins at site 0 and site i are uncorrelated, Γi = 0. This spin-correlation function can also be expressed in terms of real coordinates r, as long as we understand that the value Γ(r, T ) is actually an average over a number of atomic sites in a volume ∆V which is small compared to the total volume of the crystal. In that case we find χ = β d 3 rΓ(r, T ) (9.64) If for large values of r the spin-correlation function is proportional to r −p the integral is well-defined only if p > 3. Therefore, at Tc where the spin-correlation function diverges we need to have p 3. At temperatures away from the critical temperature the spin correlation function does not diverge. If the correlation function would always be a power law, Γ ∝ r −p(T ) , this would imply p = 3 at the critical temperature, since p(T ) > 3 away from the critical point. That is not correct, however, because we also have an exponential factor, which disappears at the critical temperature. In general we write Γ ∝ r −p e −α(T )r and we use α(T ) > 0 away from the critical point and hence α(Tc) = 0. Since these are the only two possibilities and since we know from experiment that the first one is wrong, we find that correlations have to die out exponentially. The length scale corresponding to this exponential decay is called the correlation length, and at the critical point the correlation length diverges and becomes infinite. If we have no correlations we find χ = β(< S 2 0 > −m 2 ) = β(1 − m 2 ). Here we know < S 2 0 >= 0 because the value of the spin variable is ±1 and hence
9.5. EXACT SOLUTION FOR THE ISING CHAIN. 205 the value of the square is always one. But the susceptibility diverges even in mean-field theory! This means that the spin-correlation in mean field theory for large distances is non-zero. There seems to be a contradiction here, but that is not true. The only assumption we make in mean field theory is that Γi = 0 for nearest-neighbors! For further neighbors we do not make any assumptions. We are not able to calculate the correlation function at larger distances. Similar, in the Bethe approximation we can only calculate the spin correlation function inside the cluster. In this case the condition involving h ′ is not readily expressed in terms of the spin-correlation function. 9.5 Exact solution for the Ising chain. The Ising model in one dimension can be solved exactly. The same statement is true in two dimensions, but the solution is much more difficult. We will not do that here. First consider a chain of N atoms without an external field. The energy of a configuration {σ1, · · · , σN} is given by The partition function is H(σ1, · · · , σN) = −J Z(T, N) = σ1,···,σN N−1 i=1 which can be calculated by starting at the end via σiσi+1 (9.65) e βJ i σiσi+1 (9.66) Z(T, N) = e βJσ1σ2 · · · e βJσN−1σN (9.67) σ1 σ2 Since the last factor is the only place where σN plays a role, the sum over σN can be performed giving σN e βJσN−1σN = 2 cosh(βJ) (9.68) σN Note that σN−1 drops out since it only takes the values ±1 and since the hyperbolic cosine is symmetric! Therefore the partition function is Z(T, N) = 2 cosh(βJ) e βJσ1σ2 · · · e βJσN−1σN (9.69) This process is now repeated and leads to σ1 σ2 Z(T, N) = 2 N−1 cosh N−1 (βJ) and the free energy at h = 0 is σ1 σN−1 = 2 N cosh N−1 (βJ) (9.70)
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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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140 CHAPTER 7. CLASSICAL STATISTICA
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