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Extra resources for Advanced Statistical Physics: Lecture Notes (Wintersemester 2011/12)
The Boltzmann factor can be written as e−H[s] = i eJsi si+1 = eJs1 s2 eJs2 s3 ... To construct e−H [s ] , take s1 = s1 , s2 = s3 , ... and sum over s2 , s4 , .... Writing eJs1 s2 = cosh J(1 + s1 s2 tanh J) for s1 s2 = ±1 with x = tanh J results in e−H[s] = cosh J(1 + s1 s2 x) cosh J(1 + s2 s3 x) cosh J(1 + s3 s4 x)... 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.
Consider a(z, t) = a = const. Divide a time interval T in many small intervals ∆t. The probability not to jump away in a time interval ∆t is (1 − a∆t), the probability not to jump away in the intervall T is then (1 − a∆t)T /∆t → e−aT . This means, the probability to make a jump at some time is = 1. e. the probability for the final position x to be finitely different from z goes to zero faster than ∆t goes to zero. 1) |x−z|< which is called a ”drift“. 2) |x−z|< which is called ”diffusion“. g. 1 ∆t dx(x − z)3 p(x, t + ∆t|z, t) ≤ |x−z|< < 1 ∆t ∆t ∆t→0 → dx|x − z|(x − z)2 p(x, t + ∆t|z, t) |x−z|< dx(x − z)2 p(x, t + ∆t|z, t) |x−z|< B(z, t) = O( ) Hence, knowing drift and diffusion are sufficient to characterize the system.
The existence of power laws and a scaling function strongly reminds us of RG! 3: Growth of w with time for the BD-model. We can distinguish two regimes: For t we have a power law behavior, for t tx we have saturation. 2 An even simpler model: Random deposition Modify the BD rule so that a particle moves down until it touches a particle (or surface) below it. Then, there are no correlations of hi (t) accross the columns. hi grows by 1 when a particle is dropped above column i. The probability P (h, N ) that a given column has height h after N particles have been dropped follows a binomial distribution P (h, N ) = N h ph (1 − p)N −h with p = 1/L, the probability that a given particle is released above a certain column.