phys212ln19 - Physics 212 Statistical mechanics II Fall...

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Physics 212: Statistical mechanics II, Fall 2006 Lecture XIX Now we begin Part III of the course, which will examine a few applications beyond phase transitions of the ideas and methods already developed. Lattice models of polymer physics This first lecture will focus on some standard models of homopolymers (polymers in which every unit is the same), and on the classic “Flory theory” which is essentially a mean-field theory. The second lecture will explain how all of this is connected with magnetic phase transitions. Most of the polymer material covered here can be found in Cardy, but a specialist reference is C. Vanderzande, “Lattice models of polymers”, Cambridge UP. Polymers are the first example we’ll see of nontrivial scaling that is purely a consequence of a geometrical constraint like self-avoidance, rather than tuning of parameters in a free energy. Real polymers usually live in continuous space, but we will deal with lattice versions such as lattice random walks and SAW’s. Experimentally this seems to be adequate for some “universal” properties determined by long length scales, just as the Ising model and binary fluid seem to have the same universal properties. For example, consider random walks on a square lattice. Let the walk has steps of length 1 and be at position x,y after step N . Then its mean squared displacement grows at the next step as h R 2 i N +1 - h R 2 i N = 1 4 ± ( x + 1) 2 + ( x - 1) 2 - 2 x 2 + ( y + 1) 2 + ( y - 1) 2 - 2 y 2 ² = 1 4 (4) = 1 . (1) So h
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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 Berkeley.

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phys212ln19 - Physics 212 Statistical mechanics II Fall...

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