Lecturer: Gareth Alexander

Weighting: 15 CATS

The collective behaviour of large numbers of interacting components in a system can lead to the emergence of novel structures and patterns. Phase transitions, the configurations taken up by polymers, and stock market trends are examples. This module looks at how we classify this behaviour, how the different classes of behaviour come about, and how we model it quantitatively.

We will revise the statistical mechanics from year 2, as this is the natural starting point for describing how patterns are nucleated and grow from initial fluctuations. We will then discuss how collective behaviour can be related to order parameters and how these can change across phase transitions.

Aims:
The module should illustrate the important concepts of statistical physics using simple examples. It should give an appreciation of the fundamental role played by fluctuations in nature.

Objectives:

By the end of the module, students should be able to:

• Work with equilibrium thermodynamics
• Describe the statistical mechanics of long chain molecules (polymers)
• Work with the Ginzburg-Landau theory of continuous symmetry breaking phase transitions and scaling theory
• Appreciate a range of emergent phenomena including some of quantum phase order and/or the Fermi liquid state, non-equilibrium phenomena such as turbulence, growth patterns, forest fires, crowd & congestion models

Syllabus:

Review of the fundamental principles underlying conventional statistical mechanics and thermodynamics.

Phase Transitions: thermodynamic description. PVT system: coexistence lines, triple point, critical point, Gibbs phase rule. First order transitions (latent heat) and continuous phase transitions (no latent heat, divergence of susceptibilities). Mean-field description. Universality, importance of symmetries, concept of order parameter and spontaneous symmetry breaking. Mean field theory, Curie temperature and emergence of spontaneous magnetisation. Critical exponents. Ginzburg- Landau description of phase transitions (continuous and first order). Failure of mean field and concept of critical dimension.

Further topics in Collective Phenomena selected from:

1. Polymers: Motivate a treatment of polymers based on statistical physics emphasising an insensitivity to the chemistry. Ideal and non-ideal chains. Different models for ideal chains -Gaussian chain, lattice chain, freely jointed chain. Master equation and derivation of diffusion equation.

2. Extended quantum systems: Fermi liquids, quantitative modelling through Density Functional Theory. Emergent quantum phase ordering: Superconductivity and/or Superfluidity.

3. Non-equilibrium systems: Turbulence, growth patterns, forest fires, crowd & congestion models

Commitment: 30 Lectures

Assessment: 2 hour examination

Recommended Texts: F. Mandl, Statistical Physics, Wiley David Chandler, Introduction to Modern Statistical Mechanics, OUP P-G de Gennes Scaling Concepts in Polymer Physics, Cornell Univ. Press G Rowlands, Non-Linear Phenomena in Science and Engineering, Ellis Horwood James P. Sethna Statistical mechanics: entropy, order parameters, and complexity OUP 2007