Statistical Mechanics

Overview

Fundamentals of classical thermodynamics: systems, phases, thermodynamic variables, equilibrium, equations of state, distinction between intensive and extensive thermodynamic variables, work and heat, Carnot cycle, Gibbs phase rule, first and second laws of thermodynamics, definitions of thermodynamic potentials, derivation of Maxwell relations, response functions, thermodynamic stability and Ehrenfest classification of phase transitions.
Equilibrium statistical mechanics: microstates and phase space, role of information and connection with entropy, method of Lagrange multipliers, generalised partition function, microcanoncial, canonical, isothermal-isobaric and grand-canonical partition functions, ensemble averages and connection between fluctuations and response functions.
Computer simulation: importance sampling, Monte Carlo algorithm, molecular dynamics.

Learning Objectives

On completion of the module students should be able to:

Explain how the terms system, state, equilibrium and phase are used within thermodynamics and state the 1st and 2nd laws. They should then be able to use these definitions, together with the theory of partial differential equations, to derive Maxwell relations and constraints on the values of the response functions.

Explain the meanings of the terms microstate and phase space and give an expression for the probability of being in a particular microstate for the microcanonical, canonical, isothermal-isobaric and grand-canonical ensembles.

Write a Monte Carlo program in Python that calculates ensemble averages and fluctuations for a simple model Hamiltonian. Students should be able to use this program to calculate response functions such as the heat capacity by finite differences and from the fluctuations.

Evaluate publications on recent developments in atomistic simulation critically by making comments on issues of model accuracy and sampling methodologies.

Skills

By the end of this module students will have developed the following skills:

An ability to write computer programs to calculate integrals by means of Monte Carlo simulation.

Improved presentation skills

Critical reading skills

Assessment

None