PHY2201 Statistical Physics

2007-2008

Aims

The development of statistical mechanics stands as one of the crowning achievements of 19th century science. It was the great contribution of Maxwell, Boltzmann and Gibbs to show that the application of statistical methods could yield accurate predictions for bulk thermal properties of a system from an atomistic description of matter. The theory provides the only tractable means of analysing the almost unimaginable complexity of an N-body system containing 10²³ particles. The classical Second Law of Thermodynamics finds a natural explanation in terms of the evolution of a system from the less probable to the more probable configurations. The module, which builds directly on the Stage 1 core module Thermal Physics (PHY1002), extends the discussion of classical thermodynamics and shows how its laws arise naturally from the statistical properties of an ensemble. Real-world examples of the key ideas are presented and their application in later modules such as Solid State Physics I (PHY3102), Energy and the Environment (PHY3112) and Astrophysics (PHY3132) is stressed. The concepts developed in the module are further extended in the more advanced Statistical Mechanics (PHYM421) module

Intended Learning Outcomes

Students should be able to:

Module Specific Skills

• explain the nature of classical entropy, and its relationship to the second law of thermodynamics;
• determine the maximum efficiency of simple heat-engines and heat pumps;
• calculate the equilibrium energy distribution of a system using the Boltzmann distribution;
• explain the origin of the second law from a statistical viewpoint;
• describe the significance of various thermodynamic potentials and deduce relations between them;
• demonstrate, by calculating certain properties of real gases, an understanding of the limitations of the ideal gas law;
• calculate bulk thermodynamic properties such as heat capacity, entropy and free energy from the partition function;
• predict whether a gas constitutes a classical or a quantal gas, and explain key differences in the behaviour of these;

Discipline Specific Skills

• use calculus to calculate maximum and minimun values of constrained multivariable systems;
• use graphs and diagrams to illustrate arguments and explanations;

Personal and Key Skills

• solve problems;
• apply general concepts to a wide range of specfic systems and situations;
• use on-line (WWW) learning resources to enhance and deepen learning.

Learning and Teaching Methods

Lectures, tutorials, problems classes and provision of comprehensive coverage of the lecture material as e-learning resources.

Assignments

Assignments are set every two weeks by the instructors in a supporting weekly problems class, and also by tutors.

Assessment

One 30-minute test (20%), problems classes (10%) and one 90-minute examination (70%).

Syllabus Plan and Content

1. Entropy and the second law
   1. The Carnot cycle.
   2. Heat engines and heat pumps.
   3. Entropy as a function of state.
   4. The Fundamental Thermodynamic Relationship.
2. Introduction to Statistical Mechanics.
   1. Maxwell-Boltzmann speed distribution.
   2. Boltzmann energy sharing.
   3. Microscopic interpretation of Entropy.
3. Classical thermodynamics.
   1. Thermodynamic potentials and Maxwell relations.
   2. Real gases.
4. Statistical thermodynamics.
   1. The partition function Z.
   2. Macroscopic functions of state in terms of Z.
   3. Equation of state for an ideal monatomic gas.
   4. The equipartition theorem.
   5. Quantum statistical mechanics; the Bose-Einstein and Fermi-Dirac distributions.

Core Text

Supplementary Text(s)

Formative Mechanisms

The problems that students are set on this module are marked and discussed in detail in the problems classes and in tutorials.

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.