PX423 Kinetic Theory

Weighting: 7.5 CATS

'Kinetic Theory' is the theory of how distributions change and is therefore essentially about non-equilibrium phenomena. The description of such phenomena is statistical and is based on Boltzmann's equation (the same Boltzmann who sorted out the equilibrium statistical mechanics you met in Thermal Physics II) and on the related Fokker-Planck equation. 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 of the system, which means specifying particles' position and momenta.) The module establishes relations between conductivity, diffusion constants and viscosity in gases. It looks at molecular simulation and 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.

Aims:

To introduce Maths/Physics students to non-equilibrium statistical physics and to give them an appreciation of mathematical physics as separate sub-discipline

Objectives:
At the end of the module you should

• Be able to derive and solve the Boltzmann equation as it applies to simple problems
• Understand and be able to set up and solve simple boundary value problems involving diffusion
• Understand the significance of the Fokker-Plank approximation

Syllabus:

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.

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.

Commitment: 15 Lectures

Assessment: 1.5 hour examination