# 2022/23 Taught Postgraduate Module Catalogue

## MATH5424M Advanced Entropy in the Physical World

### 20 creditsClass Size: 54

Module manager: Dr Mike Evans; Dr Steve Fitzgerald
Email: R.M.L.Evans@leeds.ac.uk;S.P.Fitzgerald@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

[MATH1005 or (MATH1010 and MATH1012)] or [MATH1050 and MATH1400 and (MATH1060 or MATH1331)].

Students should have some prior experience of using the concepts of energy, and conservation of energy, in modelling mechanical or physical systems. Such experience could, for example, have been gained in MATH1012, or in A-level mechanics modules. Knowledge of statistics is assumed, although a statistics primer document will be provided, including probability distributions and density functions, expectation values, variance and standard deviation.

### This module is mutually exclusive with

 MATH3424 Introduction to Entropy in the Physical World

This module is approved as an Elective

### Module summary

The material world is composed of countless microscopic particles. When three or more particles interact, their dynamics is chaotic, and impossible to predict in detail. So, why is it that the materials around us behave in predictable and regular ways? The answer lies in the fact that disordered behaviour on the microscopic scale gives rise to collective behaviour that can be predicted with practical certainty, guided by the principle that the total disorder (or entropy) of the universe never decreases. This module studies calculations involving entropy, as applied to the matter that makes up our world.

### Objectives

Upon completion of the module, students will:
- have gained an understanding of how the macroscopic properties of matter emerge from the microscopic processes within it;
- appreciate the need for a statistical approach to the dynamics of interacting systems involving large numbers of degrees of freedom;
- understand what entropy is and how it is related to disorder;
- be able to apply the methods of statistical mechanics in calculations for microscopic models of classical physical systems;
- understand how entropy determines the "arrow of time" via the direction of physical processes.

Learning outcomes
By the end of the module, students should be able to:
- define and evaluate entropies;
- know and use the statistical definition of temperature;
- calculate expectation values in microscopic and continuum models using Boltzmann's law;
- find the partition function for a number of idealized models;
- apply mathematical descriptions of diffusion, including use of lto calculus.

### Syllabus

Motivation.
Introduction to statistical solution of the many-body problem.
Qualitative introduction to phase transitions and ergodicity.
Classical definitions of entropy.
The arrow of time.
Microcanonical and canonical ensembles; Stirling's formula, Boltzmann's law and the partition function.
Calculation of expectation values.
Specific microscopic models of physical systems, to include:
- 2-state isolated system;
- Classical ideal gas;
- Lattice models.
Introduction to mean-field theory.
Mathematical description of diffusion and Brownian motion, and relationship with concepts of entropy (i) relationship between diffusion and random walks; (ii) the diffusion equation; (iii) Wiener processes and stochastic calculus; (iv) the Langevin equation and Einstein relation; (v) diffusion in a potential.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of formative written exercises (problem sheets).

### Opportunities for Formative Feedback

Feedback on formative written exercises (problem sheets).

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Open Book exam 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated