Introductory description

The collective behaviour of large numbers of interacting particles, or 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 material from Thermal Physics II, as statistical mechanics 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.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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

Introduce ideas in non-equilibrium statistical physics leading to the flucutation-dissipation theorem and a description of Brownian motion based on Langevin's equation. Focus on the meaning and importance of ensemble averages.

Motivate a treatment of polymers based on statistical physics emphasising an insensitivity to the chemistry. Calculate the statistics of ideal and self-avoiding chains. Give the ideas behind the reptation and tube models for the dynamics of overlapping chains.

Explain the definition and role of an order parameter using examples including ferromagnets and liquid crystals. Introduce the notion of broken symmetry and the Ginzburg-Landau theory of phase transitions explaining carefully its structure. Obtain solutions for the simple phase transitions and discuss the results.

Calculate the kinetic energy of a gas molecule. Extend the result to obtain the law of equipartition of energy.

Look at the role of fluctuations and how they relate to the law of equipartition of energy. Examples including fluctuating interfaces. Qualitative discussion of the importance of fluctuations in phase transitions.

Learning outcomes

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

• Work with equilibrium thermodynamics
• Describe the statistical mechanics of long chain molecules (polymers)
• Characterise phase transitions
• Explain and work with the Ginzburg-Landau theory of continuous symmetry breaking
• Use the law of equipartition of energy to solve problems where the statistics of fluctuations is important

Indicative reading list

F. Mandl, Statistical Physics, Wiley;
D. Chandler, Introduction to Modern Statistical Mechanics, OUP;
J.M. Yeomans, Statistical Mechanics of Phase Transitions, Oxford Science Publications;
P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press;
R.A.L. Jones, Soft Condensed Matter, OUP

View reading list on Talis Aspire

Subject specific skills

Knowledge of mathematics and physics. Skills in modelling, reasoning, thinking

Transferable skills

Analytical, communication, problem-solving, self-study