phys212ln20

# phys212ln20 - Physics 212 Statistical mechanics II Fall...

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Unformatted text preview: Physics 212: Statistical mechanics II, Fall 2005 Lecture XX In the last lecture we used the Flory model of the self-avoiding walk (SAW) to obtain an estimate of the exponent ν that governs the typical size of a polymer: R ∼ N ν , ν = 3 d + 2 . (1) However, we still have little understanding of the polymer exponents γ and α , or why these quan-tities might be universal. The main goal of the ﬁrst part of this lecture is to develop a connection between self-avoiding walks and critical points of spin models. Then all the approaches and insights previously developed for spin models can be directly applied to the polymer problems. Note that there is no “energy” or “Hamiltonian” in the original self-avoiding walk problem. Recall the good old Ising model on a lattice, whose partition function is (again the Tr means a sum over all spin conﬁgurations) Z I = Tr e ∑ h ij i Ks i s j = Tr Y h ij i e Ks i s j . (2) Now this can be rewritten using e Ks i s j = cosh K + s i s j sinh K, (3) which holds since either s i s j = 1 or s i s j =-1 on each bond. So we have Z I = (cosh K ) N Tr Y h ij i (1 + s i s j tanh K ) , (4) where N is the number of sites. The prefactor is a smooth function of K for any N , so it cannot aﬀect the critical properties, and will be dropped (it can be restored at the end if desired). The rest takes the simple form in terms of v = tanh K Z I = Tr Y h ij i (1 + vs i s j ) , (5) Now that Z I is in this convenient form, let us carry out an expansion for small K (high tem-perature). All we have to do is write out the term of order v N , then trace over the s i variables to get the result. However, we see immediately that most of these terms vanish. For instance, the linear in v term is Tr X h ij i vs i s j = 0 . (6) Any term which has one of the spin variables appearing an odd number of times will be equal to zero, since the trace over that spin’s values ± 1 will cancel. Hence the nonvanishing terms are those in which every spin variable occurs an even number of times. To make things easier, let us 1 put our Ising model on the hexagonal lattice, in which each spin has three nearest neighbors. The point of doing this is that each spin variable can appear either zero times, one time, two times, or three times. So the only possible nonzero terms have, for each spin s i , either s 2 i or 1. (For the square lattice, we would also have to include s 4 i , as there are four bonds per site: this means that intersections of loops are allowed, and the connection to polymers is more complicated.) intersections of loops are allowed, and the connection to polymers is more complicated....
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phys212ln20 - Physics 212 Statistical mechanics II Fall...

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