By Johannes Berg, Gerold Busch
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Extra info for Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12)
1) 2 = 2 cosh J(1 + s1 s3 x )2 cosh J(1 + s3 s5 x )... 3) for the sum over s2 stems only from even powers of s2 here. 1: RG-flow of the Ising model in 1d (left) and 2d (right). You can easily see that the 1d model has only one stable fixed point. The 2d model has to stable and one unstable fixed point. This implies the existence of a phase transition. If we start at exactly x = 1, we will stay there, otherwise, we will go to x = 0. This RG-flow of the coupling constant implies that at increasing length scales, the system looks increasingly disordered.
E. a set of transition rates from configuration s to s , Ws s . Our goal is to choose these rates such that the equilibrium distribution peq (s) is the Boltzmann distribution. An outline of a typical MCMC algorithm is: 1. pick an initial configuration s (in any way) 2. choose a new configuration s with probability Ws s 3. repeat ”many times” until equilibrium has been reached. 26 The rates Ws s specify an artificial dynamics of the system with no relation to the actual dynamics of the system we model.
4) the Gaussian model of course lacks a key term, the quartic m4 (x). 1: The table sums up the values of the critical exponents for different methods. corresponds to the expansion parameter of -expansion, a method of pertubative renormalization group. in detail, see for instance Kardar. 25) For d > 4, the coefficient of the m4 -term decreases under renormalization, for d < 4 it increases. In the latter case, the treatment of the m4 -term as a pertubation is inherently unstable. One speaks of the m4 -term as a relevant (irrelevant) term in the Hamiltonian for d < 4 (d > 4).