The aims of the module are for students to gain an understanding of the foundations of statistical mechanics and thermodynamics, based on statistical ensembles and the laws of thermodynamics, as well as to gain an understanding of key applications including diffusion (analysed using numerical computer programming), ideal gases, thermodynamic cycles, and phase transitions.

(LO1) Use the central limit theorem to analyse macroscopic diffusion emerging from stochastic microscopic dynamics.

(LO2) Numerically analyse macroscopic behaviour emerging from an underlying stochastic process.

(LO3) Apply the micro-canonical, canonical, and grand-canonical ensembles to analyse statistical systems subject to the corresponding constraints.

(LO4) Use the concepts of work, heat, and the laws of thermodynamics to analyse thermodynamic processes and thermodynamic cycles, determining the efficiency of the latter.

(LO5) Derive the equation of state for classical and quantum ideal gases.

(LO6) Carry out routine calculations for interacting statistical systems, including the use of order parameters to distinguish phases separated by a phase transition.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

(S4) Communication skills

(S5) IT skills

(S6) Organisational skills

(S7) Teamwork

Probability foundations: Central limit theorem; Random walks; Diffusion; Numerical computer programming

Micro-canonical ensemble: Statistical ensembles; Thermodynamic equilibrium; Entropy; Laws of thermodynamics; Temperature

Canonical ensemble: Partition function; Boltzmann distribution; Helmholtz free energy; Physics of information

Ideal gases: Continuous variables; Mixing entropy and irreversibility; Equations of state

Thermodynamic cycles: Work and heat; Isotherms and adiabats; Carnot cycle; Efficiency

Grand-canonical ensemble: Chemical potential; Landau free energy; Particle number

Quantum statistics: Energy levels and occupation numbers; Bose--Einstein; Fermi--Dirac; Relation to classical Maxwell--Boltzmann limit

Quantum gases: Photons and Planck spectrum; Phonons and Debye model; Fermion gases and degeneracy pressure

Interacting systems: Ising model; Phases; Phase transitions and critical p henomena; Mean-field approximation; Exact results

Broader applications: Monte Carlo importance sampling; Universality (van der Waals, liquid--gas, Ising); Biophysics; Sociophysics; Lattice quantum field theory