Prerequisites

There are no formal prerequisites, as all students are expected to have attended the Part II Thermal & Statistical Physics course (or an equivalent course, for the MASt cohort). It would be helpful to have attended the Part II Soft Condensed Matter option; for those who have not, the lectures will cover the essential material.

Learning Outcomes and Assessment

This is a theoretical course that focuses on the microscopic processes that drive fluctuations, examining the fundamentals and modern techniques of statistical mechanics, with applications in non-equilibrium thermal physics, genetics and evolution, and stock markets. However, the in-depth topics are all physical, covering the concepts of mean first-passage time, non-Markov noise and anomalous diffusion, and the crossover between quantum and statistical uncertainty.

Synopsis

1. Introduction: Equilibrium statistical mechanics revised via partition functions / free energy, canonical / microcanonical states, and rates / detailed balance.
2. Stochastic processes: Definitions, Central Limit theorem, propagators, detailed balance; Markov process: Poisson and Wiener process; Stochastic differential equations; Fluctuation-dissipation theorem; Geometric Brownian motion. Example 1 – Black-Scholes model for the price of options.
3. Applications of Langevin dynamics: Diffusion in external potentials, Ornstein-Uhlenbeck process. Fokker Planck and Smoluchowski equations; Diffusion in shear flow. Example 2 – viscosity of a simple fluid. Kramers escape theory and mean first-passage time; Periodic and random potentials. Example 3 – “hitting a small target”.
4. Generalised Langevin dynamics: Multi-variable stochastic processes, Fokker-Planck and Smoluchowski equations; Non-Markov stochastic process, viscoelasticity and memory functions; Generalised (viscoelastic) Langevin equation and anomalous diffusion; Liouville operator (Heisenberg and Schrodinger formalisms); Mori equation.
5. Quantum crossover: Quantum and stochastic probabilities; statistics of the density matrix; Dissipation as coupling to an ensemble of oscillators. Example 4 – Marcus equation for electron transfer. Example 5 – rate of escape over and tunnelling under the potential barrier

KEY BOOKS

Brownian Motion: Fluctuations, Dynamics, and Applications, Mazo R. (OUP 2002)

Nonequilibrium Statistical Mechanics, Zwanzig R. (OUP 2001)

Handbook of Stochastic Methods for Natural Sciences, Gardiner , C. W.  (Springer, 2004)

Statistical Mechanics: Entropy, Order Parameters and Complexity, Sethna J. P. (OUP 2007)