Advanced Statistical Mechanics

Examenvragen

1 Sept 2017

Theory:

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.

Exercices:

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 $p(x,0)=\delta (x)$ 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 $\delta (x)=\frac{1}{2a}+\frac{1}{a}\sum _{n=1}^{\mathrm{\infty }}\cos \left(\frac{n\pi x}{a}\right)$

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).