# PX366 Statistical Physics

## Lecturer: Marco Polin

## Weighting: 7.5 CATS

Fluctuations play an essential role in nature. Statistical mechanics is, in essence, a description of the role played by these fluctuations. The module studies the physics of such seemingly diverse problems as diffusion, phase transitions and Fermi-Dirac and Bose-Einstein statistics. It also uses examples from Polymer physics where the behaviour of the polymer chains is driven by configurational entropy, rather than energy.

**Aims:**

To illustrate the important concepts of modern statistical physics using simple examples. It should give an appreciation of the fundamental role played by fluctuations in nature.

**Objectives:**

At the end of the module you should

- Have been reminded about aspects of equilibrium thermodynamics
- Have been introduced to the statistical mechanics of long chain molecules (polymers) and understand why they are examples of systems driven by configurational entropy rather than energy
- Appreciate the importance of phase transitions, primarily the paramagnetic-ferromagnetic transition
- Understand the motivation behind the Ginzburg-Landau theory of continuous symmetry breaking phase transitions
- Understand the physics of diffusive processes. Know the origin of the Langevin and diffusion equations
- Understand the origin of the law of equipartition of energy and see how it may be used to solve real problems where we are interested in the statistics of fluctuations

**Syllabus:**

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

Introduction to phase transitions I: 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). Gas-liquid coexistence region, lever rule, metastable states and spinodal decomposition.

Introduction to phase transitions II: mean-field description. Introduction to the idea of universality, importance of symmetries, concept of order parameter and spontaneous symmetry breaking. Ferromagnetic Ising model: mean field theory, Curie temperature and emergence of spontaneous magnetisation. Introduction to the concept of critical exponent. Ginzburg-Landau description of phase transitions (continuous and first order). Failure of mean field and concept of critical dimension. Polymers. Motivate a treatment of polymers based on statistical physics emphasising an insensitivity to the chemistry. Difference between ideal and non-ideal chains. Critical review of different models for ideal chain -Gaussian chain, lattice chain, freely jointed chain-. Master equation and derivation of diffusion equation. Restricted ideal polymers: entropic elasticity of a single chain (both linear and nonlinear); polymers confined by one wall (polymer brush) and two walls (disjoining pressure).

Brownian motion I: macroscopic description. Introduction to the concept of Brownian motion and its physical origin. Typical orders of magnitude and diffusion timescales. Conservation equation, Fick’s law and its microscopic motivation. Derivation of Fokker-Planck equation; fluctuation-dissipation theorem. Simple solutions to Fokker-Planck equation (free diffusion and link to ideal polymers; one absorbing wall; communicating reservoirs) and discussion of physical meaning of different boundary conditions.

Brownian motion II: microscopic description. Difference between macroscopic and microscopic description, concept of ensemble average. Langevin equation, origin and properties of Brownian noise. Massless diffusion (overdamped case): formal solution; concept of a Markov process; mean square displacement; how to establish the properties of Brownian noise; velocity autocorrelation and diffusivity. Brownian particle in a potential (Ornstein-Uhlenbeck process): autocorrelation and memory of the process; mean square displacement and caging. Free diffusion with mass (underdamped case) and connection with Ornstein-Uhlenbeck process. Issues with Langevin equation at short timescales (hydrodynamic memory).

**Commitment:** 15 Lectures

**Assessment:** 1.5 hour examination

**Recommended Texts:** F. Mandl, *Statistical Physics*, Wiley; David Chandler, *Introduction to Modern Statistical Mechanics*, OUP; P de Gennes *Scaling Concepts in Polymer Physics*, Cornell Univ. Press

This module has its own home page.

**Leads from:** PX265 Thermal Physics II