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# p212ps5 - Physics 212 Statistical mechanics II Fall 2006...

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Physics 212: Statistical mechanics II, Fall 2006 Problem set 5 : due 11/9/06 1. Suppose that one wants to calculate the specific heat of a system by Monte Carlo. One approach would be to calculate the free energy F or internal energy U at different temperatures, then differentiate numerically. A better way is to use the following relation at fixed temperature to relate the specific heat to fluctuations in U : k B T 2 C V = ( U - U ) 2 = U ) 2 . (1) In this notation, U = Z - 1 i U ( i ) e - βU ( i ) , where the sum is over the states of the system and U ( i ) is the energy of microstate i . Prove the above relation. How must the typical fluctuation Δ U scale with system size? Assume that C V is proportional to the volume of the system, as is appropriate except possibly right at a critical point. What does this say about the relative size of fluctuations if we are away from a critical point? 2. Wick’s theorem for the Gaussian model states that expectation values of products of n fields factorize into products of 2-body correlations. As an example, for

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p212ps5 - Physics 212 Statistical mechanics II Fall 2006...

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