PHYM421 Statistical Mechanics

2011-2012

Code: PHYM421
Level: M
Title: Statistical Mechanics
Instructors: Dr E. Mariani
CATS Credit Value: 10
ECTS Credit Value: 5
Pre-requisites: N/A
Co-requisites: N/A
Duration: T2:01-11
Availability: unrestricted
Background Assumed: Thermal Physics (PHY1002) and Statistical Physics (PHY2201)
Directed Study Time: 22 lectures
Private Study Time: 78 hours
Assessment Tasks Time: -

Aims

The module builds upon the Thermal Physics (PHY1002) and Statistical Physics (PHY2201) modules already taken by students, and examines how the time-symmetric laws of quantum mechanics obeyed by all systems can be linked, through a chain of statistical and thermodynamic reasoning, to the (apparently time-asymmetric) natural processes occurring in macroscopic systems.

Four aspects of statistical physics are emphasised, and illustrated by applying them to a number of physical systems in equilibrium. Firstly, it is shown that a knowledge of the thermodynamic state depends upon an enumeration of the accessible quantum states of a physical system; secondly, that statistical quantities such as the partition function can be found directly from these states; thirdly, that thermodynamic observables can be related to the partition function, and fourthly, that the theoretical results relate to experimental observations.

This module furnishes the theoretical background in statistical mechanics for a number of other modules e.g. Solid State Physics (PHY3102), Quantum and Classical Fluids (PHYM423), and Semiconductors and Heterostructures (PHYM424).

Intended Learning Outcomes

After completing this module, the student should be able to:

• describe the role of statistical concepts in understanding macroscopic systems;
• deduce the Boltzmann distribution for the probability of finding a system in a particular quantum state;
• apply statistical theory to determine the magnetisation of a paramagnetic solid as a function of temperature;
• deduce the Einstein and Debye expressions for the heat capacity of an insulating solid and compare the theory with accepted experimental results;
• deduce the equation of state and entropy for an ideal gas;
• extend the theory to deal with open systems where particle numbers are not constant.
• deduce the Fermi-Dirac and Bose-Einstein distributions;
• describe superfluidity in liquid helium, Bose-Einstein condensation and black body radiation.
• deduce the heat capacity of a electron gas.

Transferable Skills

Information retrieval from the WWW and problem-solving. Communication skills via discussion of statistical mechanics in lectures. Students are required to meet deadlines for completion of work to be discussed in class and must therefore develop appropriate time-management strategies. Knowledge of the laws and applications of thermodynamics and statistical mechanics.

Learning / Teaching Methods

Lectures, problem sessions, e-learning resources.

Assignments

Solutions to problem sheets available on WWW will be handed in at regular intervals.

Assessment

One 90-minute examination (100%).

Syllabus Plan and Content

1. Introduction
   aims and methods of thermodynamics and statistical mechanics; differences between thermodynamics and mechanics
2. Thermodynamic equilibrium
   internal energy; hydrostatic and chemical work; heat; the first law of thermodynamics
3. Reversible, irreversible and quasistatic processes
   entropy; the Clausius and Kelvin statements of the second law
4. Criteria for equilibrium
   enthalpy; the Helmholtz and Gibbs free energies; the grand potential
5. Statistical mechanics
   microstates and macrostates; assumption of equal a priori probabilities
6. The canonical ensemble and the Boltzmann distribution
   partition functions; derivation of thermodynamic quantities
7. Systems in contact with a heat bath
   vacancies in solids; paramagnetism
8. Reversible quasistatic processes
   statistical meaning of heat and work; Maxwell's relations; the generalised Clausius inequality; Joule-Thomson effect; the thirdlaw of thermodynamics
9. Heat capacity of solids
   the Einstein and Debye models
10. Partition function for ideal gas
    validity of classical statistical mechanics; Maxwell velocity distribution; kinetic theory; approach to equilibrium
11. Diffusion of particles between systems
    the grand canonical ensemble; the grand partition function; application to the ideal gas; chemical reactions
12. Quantum gases
    Bose-Einstein, Fermi-Dirac and Boltzmann statistics; Black-body radiation; Bose-Einstein condensation; The degenerate electron gas
13. A selection of more-advanced topics:
    phase equilibria; Monte Carlo methods; mean-field theory of second-order phase transitions; the kinetics of growth

Core Text

Mandl F. (1971), Statistical Physics, John Wiley, ISBN 0-471-56658-6 (UL: 530.132 MAN)

Supplementary Text(s)

Bowley R. and Sanchez M. (1996), Introductory Statistical Mechanics, Oxford Science Publications, ISBN 0-19-851794-7 (UL: 530.13 BOW)
Goodstein D.L. (2002), States of Matter, Dover, ISBN 978-0486495064 (UL: 530.4 GOO)

Formative Mechanisms

Discussion of performance in set problem papers.

Evaluation Mechanisms

The module will be evaluated using information gathered via the student representation mechanisms, the staff peer appraisal scheme, and measures of student attainment based on summative assessment.