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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2006 Lecture XI The next step is to understand correlation functions from meanfield theory, just on the disor dered side of the transition. First we rewrite the correlation function in a simpler form, where Tr denotes the sum over spin configurations: h i i = Tr i e K h ij i i j Tr e K h ij i i j = Tr i e K h ij i i j Tr e K h ij i i j , (1) where the second equality is restricted to spin configurations with = 1. Here the step used to show the second equality is inserting 1 = , 1 + , 1 , and using updown symmetry. This exact relation means that the correlation function can be viewed as the response of other spins to fixing one spin up. Now we use the meanfield approximation. The meanfield equation is now more complicated because we must allow m to depend on i : summing over nearest neighbors, m i = tanh( X j K (  i j  ) m j ) (2) where before we had just m = tanh zKm . We are generalizing slightly to the possibility of inter actions beyond nearest neighbor, in order to make the generalization to the continuum limit more simple. For spins j far away from the original spin at site 0, we assume m j is nearly the equilibrium value (0, when we are on the disordered side of the transition) and linearize the above, in order to get a tractable problem. This calculation of correlation functions is a good example of a case for which it is more useful to work in the continuum (below) than on the lattice. Then m i = X j K (  i j  ) m j + (corrections when  j  small) . (3) This is a linear equation, and has a simple Fourier transform: m ( k ) = K ( k ) m ( k ) + C, (4) where in writing the constant C we have assumed that the corrections localized around the origin are nearly a deltafunction on large scales. (Otherwise, with no source, the resulting solution would just be m = 0.) Finally, we need an assumption about K ( k ) = J ( k ). This is nearly localized, and hence nearly a function. Assuming it is shortranged, we can expand J ( k ) J (1 R 2 k 2 ) where R 2 = r r 2 J ( r ) / (2 r J ( r )) is a measure of the range of the interaction. (See Appendix below for some more information on this method.) Finally m ( k ) C 1 J (1 R 2 k 2 ) . (5) The last step is to Fouriertransform this back to real space. First rewrite it as G ( k ) C KR 2 k 2 + 1 K = CR 2 k 2 + tR 2 = CR 2 k 2 +  2 (6) 1 with = Rt 1 / 2 , and t = ( T T c...
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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.
 Fall '06
 MOORE
 mechanics, Statistical Mechanics

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