Home > MAM4001W > Applied Mathematics Modules > Stochastic processes and non-equilibrium systems

Stochastic processes and non-equilibrium systems

Dr Christian Rohwer
1st semester 2020
20 credits / 30 lectures

Complex systems (with many constituents, randomness, correlations) often require statistical descriptions. This course covers methods for modelling stochastic processes involving noise and fluctuations, in and out of equilibrium. Important applications include statistical physics of active matter, fluctuation-induced forces (classical and quantum) and financial models. Outline:

  1. Fundamentals
  • Statistical mechanics: ensembles, partition functions
  • What is equilibrium? Even better: what isn't?
  • Making sense of noise: averages, correlations, fluctuations
  1. Effective dynamical models
  • Langevin equations: noise makes you go places
  • Diffusion: a friendly primer on dynamics and conservation laws
  • Fokker-Planck and Smoluchowski: the evolution of probability
  • Stochastic calculus: Ito vs Stratonovich – when does it matter?
  • Markov, nonlinear dynamics, memory: when to forget quickly, and when not
  1. Linear response and the Fluctuation-Dissipation Theorem: wiggle and wait
  1. Path integrals
  • Statistical field theory: functionals, actions, variations
  • Martin-Siggia-Rose and Jarzynski (time permitting)
  1. Applications (time permitting; can also flow into Honours projects)
  • Active matter: self-propulsion, phoresis
  • Critical phenomena: when wiggling here causes wiggles far away
  • Casimir forces in classical and quantum systems: two plates – bring snacks
  • Financial models, e.g., Black-Scholes: show me the (time evolution of the) money!
  • Langevin simulations

The course is mostly self-contained. Concepts covered in MAM2046W and MAM3040W, and a basic knowledge of differential equations, probability theory, and Hamiltonian mechanics would be helpful.