Statistical Mechanics - Physics at Oregon State University

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226 CHAPTER 9. GENERAL METHODS: CRITICAL EXPONENTS. constant. The deviation δ ˜ J ′ corresponds to a different temperature T ′ according to δ ˜ J = α(T ′ − Tc) and we have (T ′ − Tc) = λ(T − Tc) (9.182) On the other hand the correlation length depends on the temperature via ξ(T ) = c(T − Tc) −ν . By construction the primed system corresponds to a rescaled system with correlation length ξ f . The scaling factor f is equal to two in our example above. Hence we also have combining these two results gives (T ′ − Tc) −ν −ν 1 = (T − Tc) f (T − Tc) −ν λ −ν −ν 1 = (T − Tc) f (9.183) (9.184) or λ ν = f. Hence if we calculate the matrix M at the fixed point the eigenvalues are related to critical exponents. We could also vary h to get similar results. The one dimensional Ising model is again too simple. The fixed points are ˜ J ∗ = 0 or ∞ with a value of λ = 0 or 1. If λ approaches zero from above the value of ν approaches zero. This means that the correlation length is independent of the temperature. In other words, the correlation length is not important and can be taken to be zero for all practical purposes, as expected.On the other hand, for the other fixed point λ approaches one from above, and this means that ν goes to infinity. The correlation length increases faster than a power law, which is what we already found from the exact results. 9.8 Problems for chapter 9 Problem 1. Consider the one-dimensional Ising model in a cluster approximation. A cluster contains a central atom 0 and the neighboring sites j = ±1. A. Calculate < σ0σj > in this approach. B. Assume that there is an additional potential in this model of the form V = λ j=±1 σ0σj. Show that we indeed have ∂G ∂λ =< V >λ. Problem 2. Suppose the earth magnetic field is due to a linear chain of atoms with spin one-half polarized along the chain. Take J = 1 eV and N = 10 16 . Use the results for the one-dimensional Ising model to estimate the time between spontaneous reversals of the earth magnetic field.
9.8. PROBLEMS FOR CHAPTER 9 227 Problem 3. Evaluate 〈σ0σi〉 in the Bethe cluster approximation for h = 0. Use this expression to calculate the free energy for T > Tc. Derive the value of the critical exponent α. Problem 4. Consider a one-dimensional chain of spin one particles. Use the transfer matrix method to obtain the free energy of this system.
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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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140 CHAPTER 7. CLASSICAL STATISTICA
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Statistical Mechanics - Physics at Oregon State University

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