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Outline:

This module will suit students who wish to explore applications of the ideas of statistical physics. We discuss analytic and numerical techniques for evaluating the canonical statistics of equilibrium systems, and we investigate the dynamical and philosophical foundations of such an approach. We extend these ideas to nonequilibrium systems, where phase changes and heat and particle flows can take place and entropy is produced. Illustrations will be taken from the field of liquids and gases. The module is intended to be suitable for physics students rather than those with a more formal mathematical background.

Aims:

A particular aim of the module is to convey a modern understanding of entropy and the second law. The module offers exposure to the advanced mathematics used to describe the dynamics of random events such as diffusion, barrier crossing and first order phase transformations.

Intended Learning Outcomes:

• To develop an understanding of statistical physics beyond an introductory module.
• To use equilibrium statistical mechanics to deduce the properties of systems of interacting particles, particularly harmonically bound solids and finite density fluids, with emphasis on criteria for phase coexistence, and to discuss the nucleation of phase transitions.
• To develop master equations and the Langevin and Fokker-Planck equations as mathematical models of stochastic processes, with reference to discrete random walks and drift-diffusion processes in continuous phase space.
• To understand how effective stochastic dynamical models may emerge from underlying deterministic dynamics.
• To learn about the techniques of stochastic calculus, noting the distinct Itô and Stratonovich rules, and to apply them to solve various problems such as Brownian motion, including their use in modelling complex systems outside science such as finance.
• To consider the meaning of entropy and its production in nonequilibrium statistical mechanics, with reference to Loschmidt's reversibility paradox, coarse graining and the demons of Maxwell and Laplace.
• To appreciate the concepts of energy exchange between a system and its environment, the associated entropy production, and fluctuation relations.Ìý

Teaching and Learning Methodology:

This module is delivered via weekly lectures supplemented by a series of workshops and additional discussion.

In addition to timetabled lecture hours, it is expected that students engage in self-study in order to master the material. This can take the form, for example, of practicing example questions and further reading in textbooks and online.

Indicative Topics:

• Systems of interacting particles: Partition functions of harmonic structures; Virial expansions; Thermodynamics of phase coexistence; Potential of mean force; Monte Carlo and Molecular Dynamics simulation techniques; Many-particle phase space dynamics. Liouville’s theorem and the H-theorem
• Stochastic processes: Fluctuations, uncertainty and entropy; Random walks and Brownian motion; Master equations of macrostate probability dynamics; Nucleation and aggregation kinetics; Fokker-Planck equation; Langevin equation, fluctuation-dissipation relation; Stochastic differential equations and Ito’s lemma; Ito-Stratonovich dilemma; Financial modelling
• Irreversibility: Philosophical issues; Entropy production and Loschmidt’s reversibility paradox; Coarse graining and projection of dynamics onto smaller phase spaces; Caldeira-Leggett model; Stochastic thermodynamics and fluctuation relations; Maxwell's Demon