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Statistical Mechanics 3 (MATH 34300)

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Administrative Information

  1. Unit number and title: MATH 34300 Statistical Mechanics 3
  2. Level: H/6
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 2005/2006
  6. Unit Organiser: Tanniemola Liverpool
  7. Lecturer: Prof T Liverpool
  8. Teaching block: 2
  9. Prerequisites: MATH 11009 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.

Unit aims

Introduction to the mathematical foundations of thermodynamics and statistical mechanics.

General Description of the Unit

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. Both equilibrium and non-equilibrium situations will be covered, ending with a brief discussion of numerical simulation methods.

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 Physics 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 and computational methods.

Teaching Methods

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.

Learning Objectives

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

Assessment Methods

The final assessment mark for Statistical Mechanics 3 is constructed as follows:

  • 20% is based on specified marked homework assignments;
  • 80% is based on the mark in a 2.5 hour examination in May/June, with five questions (calculators are not permitted). The best four answers will be considered for credit.
     

Different homework assignments will be given out for the level 3 and M versions of this unit.

The exam for the level 3 and M versions of the unit will also be different.

Award of Credit Points

Credit points are gained by:

  • either passing the unit;
  • or getting an examination mark of 30 or over and also handing in satisfactory attempts at half of the marked homework assignments.

Transferable Skills

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

Texts

See the unit homepage for advice.

  • Statistical Mechanics, R.K. Pathria, Elsevier 2005, 529 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.Dorfman. Cambridge 1999, 287 pages.

Syllabus

Thermodynamics

  • 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.

Modern Topics

  • Brief introduction to computational techniques.