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**Unformatted text preview: **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 = h ( U-h U i ) 2 i = h ( U ) 2 i . (1) In this notation, h U i = Z- 1 i U ( i ) e- U ( 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 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?...

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