Module Identifier PHM1510
Module Title STATISTICAL PHYSICS
Co-ordinator Professor Keith Birkinshaw
Semester Semester 1
Other staff Dr L Grischuck (Cardiff)
Pre-Requisite Successful Completion of Year 3 of the MPhys Scheme
Course delivery Lecture   20 lectures
Seminars / Tutorials   3 seminars / tutorials
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours  79%
Semester Assessment Course Work: Examples Class Deadline: Week 4 of the Semester7%
Semester Assessment Course Work: Examples Class Deadline: Week 8 of the Semester7%
Semester Assessment Course Work: Examples Class Deadline: Week 11 of the Semester7%

#### Learning outcomes

After taking this module students should be able to:
PHASES TRANSITIONS
By completion of the course the student will have:
1. learned to apply familiar formalism in a diverse range of physical situations
2. gained some appreciation of the more fundamental issues underlying the subject.
The student will have been made particularly aware of the increased complexity of statistical treatments of interacting systems, the value of scaling and order-of-magnitude arguments, and the relation between thermodynamics and the more abstract principles underlying Information Theory.

INFORMATION THEORY

• Justify the use of log(p) as a definition of information.
• Calculate the entropy/uncertainty associated with a probability distribution.
• Apply the concept of entropy in an analysis of communication systems with and without noise.
• Analyse the performance of a 1D array of detectors and the information content of a mass spectrum peak in terms of information theory.
• Show relationship between ''''Information Theory entropy'''' and ''''Thermodynamic entropy''''.

ASTROPHYSICAL APPLICATIONS
Students should be able to derive and apply the hydrostatic equilibrium equation for spherically-symmetric stars. They should be able to distinguish between normal stars governed by the Maxwell-Boltzmann law and degenerate stars governed by the laws of the Fermi-Dirac statistics, and be capable of formulating the condition of degeneracy of the stellar gas in terms of the participating physical parameters. Using the Fermi-Dirac distribution function, you will be able to derive the equation of state for degenerate non-relativistic and relativistic electron gases. You wil be able to perform a qualitative derivation of the Chandrasekhar limit for masses of white dwarfs and neutron stars in terms of fundamental constants. Students should be able to use the statistical mechanics of solid bodies for evaluation of heat capacity and cooling times of white dwarfs. You will be capable of naming, describing and explaining in details various phenomena in laboratory and cosmic physics which are governed by the universal laws of quantum statistical mechanics.

#### Brief description

This module will be taught jointly with the Department of Physics at Cardiff and Swansea, using the University of Wales video network. It consists of three blocks of lectures covering different applications of statistical physics:
(a) phase transitional and critical phenomena (Swansea)
(b) Information Theory (Aberystwyth)
(c) astrophysical applications (Cardiff)

#### Content

Introduction:
One lecture to review required statistical mechanics formalism.

Phase Transition and Critical Phenomena:
Phenomenology of phase transitions, eg. liquid-vapour, ferromagnetic
Classical thermodynamics conditions for phase equilibrium
The Clausius-Clapeyron equation
First and Second Order Phase transitions, the order parameter
Statistical mechanics approach: the Ising Model, observables, correlation functions
Mean Field Approximation
Critical Exponents and Universality
Introduction to the Renormalisation Group

Information Theory:
Information - the relation to probability
The message, the bit, message transmission - source, channel, destination, channel capacity, noise
Entropy and information rate
Mutual information
The binary symmetric channel (BSC)
Application in Communications, Spectroscopy and Statistical Mechanics

Astrophysical Applications:
Equilibrium and stability of stars. Gravitational forces and pressure gradients. Normal and degenerate stars.
Breakdown of Maxwell-Boltzmann gas law. Fermi-Dirac/Bose-Einstein statistics. Equation of state for ideal Fermi gas.
White dwarfs. Simple equations of state. Nonrelativistic and ultrarelativistic electrons.
Masses and radii of white dwarfs. The Chandrasekhar limit. Qualitative derivation of the Chandrasekhar limit.
Statistical mechanics of solids and cooling of white dwarfs. Comparison with observations.
Neutron stars. Masses and radii of neutron stars. Pulsars. Observations.

#### Transferable skills

Example classes, Tutorials.