# Statistical Mechanics 34

## Unit aims

Introduction to the mathematical foundations of thermodynamics and statistical mechanics.

## Unit description

The unit begins with a discussion of thermodynamics, the macroscopic (large scale) laws of heat. In contrast to mechanical systems, thermodynamics is fundamentally irreversible, so for example processes like thermal equilibration, combustion, and mixing can occur spontaneously, but the reverse processes never occur without external input. This leads to fixed constraints on the capabilities of (for example) engines, fridges and living organisms.

The remainder of the unit ("statistical mechanics") deals with the microscopic basis for thermodynamics, that is, explaining large scale properties from properties of individual molecules. Although the dynamical equations can be solved exactly in only a very few cases, the very large number of particles means that statistical assumptions are often justified, making a strongly predictive and irreversible theory from reversible mechanics. Mainly equilibrium situations and some non-equilibrium situations will be covered.

## Relation to other units

Statistical mechanics is a branch of mathematical physics, along with mechanics, quantum mechanics and relativity. Its molecular treatment of fluids is complementary to the continuum approaches in the fluids units. There are also connections with information theory and chaotic dynamics. Connections with probability and statistics exist, but are not strong. Some parts of the unit are similar to Thermal Physics and Condensed Matter and Statistical Mechanics offered in physics; the approach here is more mathematical, and more directed towards research interests of the department, including fluids, dynamical systems, biological physics, nonequilibrium systems.

This is a double-badged version of the Level 6 Mathematics unit MATH34300 Statistical Mechanics 3, sharing the lectures but with differentiated problems and exam.

## Learning objectives

By the end of the unit the students should be familiar with the main concepts of equilibrium thermodynamics and statistical mechanics, understand thermodynamic limitations of systems, and be able to derive thermodynamic properties of systems of weakly interacting particles.

## Transferable skills

Clear, logical thinking and an ability to comprehend and solve problems of mathematical physics.

## Syllabus

- State variables, laws, potentials.
- Applications in phase and chemical equilibria, heat engines, fridges, the atmosphere.

Equilibrium statistical mechanics

- Classical ensembles (microcanonical, canonical, grand canonical), entropy of mixing, quantum statistics, derivation of thermodynamic quantities.
- Computations for the ideal gas (classical, Fermi and Bose) and applications.

Dynamical foundations

- Review of Hamiltonian mechanics, Liouville equation, ergodicity, mixing, Poincare recurrence.

Nonequilibrium statistical mechanics

- Boltzmann equation and H-theorem.

## Reading and References

See the unit homepage for advice.

- Statistical Mechanics, R.K. Pathria, Elsevier 2005, 529 pages.
- Equilibrium Thermodynamics, C.J. Adkins, Cambridge 1983, 285 pages.
- Introduction to Modern Statistical Mechanics, D. Chandler, Oxford 1987, 274 pages.
- Equilibrium and non-equilibrium statistical thermodynamics, M. LeBellac, F. Mortessagne and George Batronni, Cambridge 2004, 616 pages.
- An introduction to chaos in non-equilibrium statistical mechanics , J.R.Doffman. Cambridge 1999, 287 pages.
- Statistical Physics of Particles, M. Kardar, Cambridge 2007, 330 pages.

**Unit code:** MATHM4500

**Level of study:** M/7

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturers: **Professor Tannie Liverpool and Dr Karoline Wiesner

## Pre-requisites

MATH11009 Mechanics 1. However some of the concepts introduced in the course will be more familiar to those who have taken MATH21900 Mechanics 2 and MATH35500 Quantum Mechanics.

## Co-requisites

None

## Methods of teaching

A standard chalk-and-talk lecture unit of about 30 lectures, with occasional problems classes or informal discussion to meet the needs of individual students.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 100% from a 2 hour 30 minute exam in May/June

NOTE: Calculators are not allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.