Module description
Syllabus
Basic notions of information theory. Entropy as measure of uncertainty. Constrained optimisation with Lagrange multipliers. Maximum entropy inference with constraints. Partition function, free energy as generating function. Collective behaviour in spin systems: from independent voters to the tight-knit model (or Curie-Weiss ferromagnet); phase transitions and spontaneous symmetry breaking. Distributions of functions of random variables using Kronecker delta. Laplace's approximation for integrals. Bolzmann distribution and 1d Ising chain: exact calculation for free energy. Variational approximations and trial (factorized) distributions.
Time permitting: multi-party voters, stochastic dynamics and Markov Chains, models on social networks, traffic flow and epidemic models.
Prerequisites
4CCM141A/B Probability and Statistics I (essential), 4CCM111A Calculus I & 4CCM112A Calculus II, 5CCM221A Real Analysis
Assessment details
Written examination.
Educational aims & objectives
The main aims of the module are:
(i) to introduce the concepts and techniques of the statistical mechanics approach to understanding collective behaviour in large systems of simple interacting units (e.g. atoms, neurons, humans)
(ii) to provide notions of information theory (Shannon entropy and related concepts) and constrained optimisation
(iii) to apply these techniques to simple models of collective behaviour (with emphasis on the concept of phase transitions in voting systems and magnetic materials)
Teaching pattern
Three hours of lectures and one hour of tutorial per week throughout the term
Suggested reading list
Indicative reading list - link to Leganto system where you can search with module code for lists