Introductory description

Kinetic Theory is the theory of how distributions change. It is based on Boltzmann's equation (the same Boltzmann who sorted out the equilibrium statistical mechanics you met in Thermal Physics II) and other PDEs. These study the evolution in time of a distribution function, which gives the density of particles in the system's phase space. (Phase space is the space of states which specify particles' position and momenta.) The module establishes relations between conductivity, diffusion constants and viscosity in gases. It looks at applications to financial modelling (many of the concepts are also the basis for models of the 'motion' of stock and option prices in financial markets).

An additional motivation of this module is to illustrate how some of the mathematics you learnt in second year applied mathematics modules is used in theoretical physics.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Time dependent distribution functions: Notation, phase space coordinates and volumes. Liouville’s theorem. Boltzmann equation and the assumption of molecular chaos.

Collisions: Connection between microscopic particle mechanics and the distribution function. Detailed balance, scattering cross section for a rigid ball and classical binary collision integral. Mean free path.

Equilibrium: Derivation of equilibrium statistical mechanics via phase space distributions. Hamilton’s equations as a symplectic transformation.

Transport in gases: Linear transport theory and the Einstein relation between conductivity and diffusion coefficient. Thermal transport, viscosity in gases.

Diffusion: Derivation of the diffusion equation from the Boltzmann equation. Validity of the diffusion approximation. Example boundary problems for the diffusion equation. Mention of Fokker-Plank equation and connection to Langevin dynamics. Applications in molecular simulation and financial modelling.

Some of:

Derivation of Euler equations from kinetics and discussion of viscosity in Navier-Stokes, Fokker-Planck equation

Non Hamiltonian dynamics: Outline treatment of connection to phase space distributions. Extended ‘thermostat’ systems obeying the canonical distribution within the particle subspace. Pathogenic failure in the absence of molecular chaos.

Learning outcomes

By the end of the module, students should be able to:

• Derive and solve the Boltzmann equation
• Set up and solve boundary value problems describing the time-development of distributions in phase space
• Apply the theory to model systems

Indicative reading list

Physical Kinetics, EM Lifshitz and LP Pitaevskii : Chapters 1-3
Statistical Mechanics: Theory and molecular simulation, ME Tuckerman : Chapters 4, 13 and 15.

View reading list on Talis Aspire

Subject specific skills

Knowledge of mathematics and physics. Skills in modelling, reasoning, thinking.

Transferable skills

Analytical, communication, problem-solving, self-study