MPhys Physics with Astrophysics

Advanced Statistical Physics (M)

To understand the nature and scope of the dynamical description of the macroscopic world based on statistical principles.

On completion successful students will:

1. Be able to explain what a Markov process is and to use analytical methods to study the dynamics of Markovian systems.
2. Understand the origin of the irreversibility seen at the macroscale including examples which illustrate the essential ideas behind the fluctuation-dissipation theorem; be familiar with modern concepts relating equilibrium and non-equilibrium statistical physics. Bbe able to show how different kinds of description of stochastic processes are related, especially the idea of a microscopic model and its relation to a macroscopic model.
3. Be able to perform straightforward calculations for systems which are described by stochastic dynamics, determining stationary probability distributions from master or Fokker-Planck equations and correlation functions from Langevin equations.
4. Be familiar with the basic numerical methods used to simulate stochastic dynamical systems.

1. Stochastic variables and stochastic processes

Revision of the basic ideas of probability theory; probability distribution functions; moments and cumulants; characteristic functions; the central limit theorem and the law of large numbers.

2. Markov processes

The Chapman-Kolmogorov equation; Markov chains; Applications: (random walk, birth-death process); the master equation; methods of solution of the master equation; efficient simulation methods for Markov processes with discrete states.

3. Drift and diffusion

The Fokker-Planck equation: derivation and methods of solution; relation to Schrödinger’s equation; applications to barrier crossing, activation and mean-first-passage times.

4. Stochastic differential equations

The Langevin equation and its generalisations; analytical and numerical methods of solution; applications to Brownian motion.

5. Modern topics in statistical physics

Fluctuation theorems; statistical physics of small systems; applications to complex systems modelling.

Feedback will be available on any students’ request.

Gardiner, C. Stochastic Methods, A Handbook for the Natural and Social Sciences (Springer)
Jacobs, K. Stochastic Processes for Physicists, Understanding Noisy Systems (Cambridge University Press)
Reichl, L.E. A Modern Course in Statistical Physics, 2nd ed, (Wiley)