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.