# ENERGY, ENTROPY AND NUMERICAL PHYSICS - 2024/5

Module code: PHY2063

## Module Overview

This module considers develops both the thermodynamic and statistical descriptions of energy and entropy. In addition it builds on the introductory Level FHEQ 4 computing modules to develop the skills needed for computational physics. The module will explore various meanings and definitions of entropy. Knowledge of thermodynamics will then be applied to problem solving. The module will build upon the knowledge obtained of the laws of thermodynamics introduced in Properties of Matter at Level FHEQ 4. It will introduce additional thermodynamic theory and show how statistical physics allows us to calculate thermodynamic functions such as the entropy. The computational physics component will develop the student’s skills in solving both ordinary and partial differential equations, in the context of both quantum and thermal physics.

### Module provider

Mathematics & Physics

### Module Leader

ERKAL Denis (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 5

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 40

Lecture Hours: 33

Tutorial Hours: 11

Laboratory Hours: 22

Guided Learning: 11

Captured Content: 33

## Module Availability

Semester 1

## Prerequisites / Co-requisites

None

## Module content

Indicative content includes:

The module will build on the Level HE1 module “Properties of Matter” by developing our understanding of entropy within thermodynamics, and by showing how statistical physics allows us to calculate the properties of matter, such as the entropy, by averaging huge numbers of states of the matter’s constituent atoms.

The statistical nature of the 2nd Law of Thermodynamics will be shown.

The concept of a free energy and the Helmholtz and Gibb’s free energy functions will be covered. The statistical physics part of the course will introduce Shannon’s expression for the entropy, the partition function at constant temperature, the Boltzmann weight of a state at constant temperature, and also the weight of a state at constant chemical potential/Fermi level.

Fluctuations will be studied and the Central Limit Theorem will be introduced. The relationship between fluctuations and thermodynamic quantities such as heat capacities will be shown.

An application to a simple system: a two-level system at fixed temperature, will be described in detail.

Classical statistical mechanics will be introduced, with the simple example of the partition function of a simple classical particle, as well as the equipartition theorem.

Microscopic models of Phase Transitions, and phenomenological models (Landau theory) will be introduced at the end of the course.

The computational part of the module will include:

Euler’s method for the solution of ordinary differential equations.

Applications of this method to: a single first-order differential equation; two coupled first-order differential equations; and a single second-order equation, expressed as a pair of coupled first-order equations.

The treatment of boundary conditions.

Elementary discussion of finite difference methods for the solution of partial differential equations: application to the solution of partial differential equations in two spatial dimensions, and in one spatial dimension plus time.

Introduction to the Monte Carlo numerical calculation technique.

## Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

Coursework | Energy and Entropy Bi-weekly Questions | 10 |

Coursework | Numerical Physics Questions | 16 |

Coursework | Numerical Physics Coursework | 14 |

Examination | End of Semester Examination - 2 hours | 60 |

## Alternative Assessment

None.

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate

recall of subject knowledge

ability to apply subject knowledge to unseen problems in mathematics and physics

ability to solve mathematical problems by writing computer programs

Thus, the

__summative assessment__for this module consists of:

- Bi-weekly questions on Surrey Learn on the material covered in the lectures on Energy & Entropy

Four questions assessing students' ability to write correct functioning code in computational physics

One computing coursework assignment in which ability to write correct functioning code, and to plot and discuss results.

A final 2-hour exam with section A consisting of compulsory questions, worth a total of 20 marks, and section B consisting of a choice of 2/3 questions for a total of 40 marks.

__Formative assessment and feedback:__

Students receive verbal feedback on their problem solving in tutorials and during the supervised computation sessions. Model solutions are provided for the questions on the problem sheets to provide students with feedback on their problem solving ability.

Written feedback is given on the computational assignments, with feedback on each assignment being given before the next is due.”

## Module aims

- Introduce thermodynamic and statistical descriptions of entropy in a coherent way
- Introduce the basic statistical physics ideas and tools needed to understand and to calculate the properties of matter
- Develop computational and problem solving skills.

## Learning outcomes

Attributes Developed | ||

001 | Understand the connection between the statistical definitions of entropy (Boltzmann and Gibbs) and thermodynamic definitions of entropy. Assess how entropy is related to uncertainty as to the state of the system, the direction of time, and heat flow. | KC |

002 | Derive both the Boltzmann weight of a state at constant temperature, and also the weight of a state at constant chemical potential/Fermi level. Recall both the partition function for a simple classical particle and the equipartition theorem, and determine which one is required for a given system. | KC |

003 | Explain the role of fluctuations, and estimate their size in a range of contexts. | C |

004 | Calculate the properties of the two-level system | KC |

005 | Analyse phase transitions such as the ferromagnetic phase transition using statistical physics methods | KC |

006 | Solve ordinary differential equations numerically using simple finite difference algorithms. Solve simple partial differential equations by discretising space and solving the differential equation on a grid. In both cases, the student will be able to assess the accuracy of the solutions, judge what accuracy is required, and be able to plan simple computational approaches to relevant problems in physics. Use the solution to show an understanding of a simple physical system. | KCT |

007 | Solve a simple problem using the Monte Carlo algorithm, and perform a basic analysis on the results. Apply Bayes' theorem to analyse a simple data set. Use the solution to show understanding of a simple physical system, or or to analyse a simple data set. | KCT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to:

equip students with subject knowledge

develop skills in applying subject knowledge to physical situations and to solve mathematical problems

develop self-reflection skills by teaching students how to check results for consistency

develop skills in writing computer programs to solve problems in mathematics and physics

The

__learning and teaching__methods include:

- Lectures and tutorials
- Supervised computational labs
- Lectures are recorded using audiovisual equipment in the lecture room, providing captured content.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **PHY2063**

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Employability, Digital Capabilities, Global and Cultural Capabilities, Sustainability, and Resourcefulness and Resilience. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

**Digital Capabilities: **The Numerical Physics component of this module teaches students about advanced programming techniques in Python to solve complex physical problems. perform data analysis, and provides an introduction to machine learning. There is also an emphasis on good coding practices during the weekly programming sessions to help students write efficient code.

**Resourcefulness and Resilience: **The module is focused on problem-solving in lectures, during tutorials, and on the assessed coursework. Students are shown how to break down complex problems into smaller chunks, how to be self-critical and check if a result makes sense, and how to interpret their results both qualitatively and quantitatively.

**Employability: **The problem-solving techniques and critical self-reflection practiced in this module provide the students with critical skills for their employability. In addition, the Python coding and machine learning skills taught in this module are in high demand in the fields of data science and machine learning.

## Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Physics with Astronomy BSc (Hons) | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Physics BSc (Hons) | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Physics with Nuclear Astrophysics BSc (Hons) | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Physics with Nuclear Astrophysics MPhys | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Physics with Astronomy MPhys | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Physics MPhys | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Physics with Quantum Computing BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Physics with Quantum Computing MPhys | 1 | Compulsory | A weighted aggregate mark of 40% is required to pass the module |

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.