P212ps3 - Physics 212 Statistical mechanics II Fall 2006 Problem set 3 due Thursday 1 Imagine that you have a large table of fundamental constants

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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Problem set 3 : due Thursday, 10/12/05 1. Imagine that you have a large table of fundamental constants expressed in SI or cgs units and without leading zeros. What fraction of these constants would you expect to have first digit 1? (Hint: if you actually look at such a table, significantly more than 1/9 will have first digit 1.) Formulate a scaling argument to explain this pattern. 2. The point of this problem is to show that the infinite-range Ising model, in a certain limit, corresponds exactly to mean-field theory. Each spin interacts equally strongly with all other spins, and the exponent of the Boltzmann factor is- βE = KS 2 + βHS (1) where S = ∑ N i s i . Write down the partition function in terms of an auxiliary parameter m to eliminate the S 2 term, by applying the Gaussian identity (choose a and φ properly) e φ 2 / 2 a = Z dm √ a √ 2 π e mφ- am 2 / 2 (2) Then carry out the sum over spins, leaving a functional of the auxiliary parameter. Then fixingThen carry out the sum over spins, leaving a functional of the auxiliary parameter....
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This note was uploaded on 08/01/2008 for the course PHYSICS 212 taught by Professor Moore during the Fall '06 term at University of California, Berkeley.

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P212ps3 - Physics 212 Statistical mechanics II Fall 2006 Problem set 3 due Thursday 1 Imagine that you have a large table of fundamental constants

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