Advanced Statistical Mechanics
1 Sept 2017
1) Consider a discrete Markov process in continuous time. Write down the master equation. Determine sufficient conditions for the existence of a stationary probability distribution. Derive the form of the stationary distribution function under these conditions. Is a uniform PDF possible?
2) Consider the Landau-Ginzburg Hamiltionian in dimension d and for an n-component order parameter m. Discuss the critical behavior (t<0 and t>0) of the (ferromagnetic) order parameter and of the specific heat in mean-field theory. Consider subsequently small fluctuations of the order parameter components and treat them in a Gaussian approximation (take n=2). Calculate and discuss the fluctuation corrections to the specific heat and derive the Gaussian approximation for the critical exponent of the specific heat. Interpret your result.
1) Consider diffusion in one dimension in a finite region (-a<x<a) with impenetrable and fully reflecting endpoints x=+/-a. Use seperation of variables to find the PDF solutions p(x,t) of the diffusion equation for the PDF on (-a,a), respecting these boundary conditions, as well as the initial condition and normalized to unity on (-a,a).
Hints: -The boundary conditions imply that the current on the endpoints vanish at all times. -On the interval (-a,a) the Dirac delta can be represented as
2) Real-space RG. Consider the Migdal-Kadanoff (bond-moving) transformation of the square Ising lattice (with nearest-neighbour coupling K) with rescaling factor b (b=integrer\geq 2).