# ENERGY, ENTROPY AND NUMERICAL PHYSICS - 2019/0

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 by 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

Physics

ERKAL Denis (Physics)

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

Independent Learning Hours: 79

Lecture Hours: 32

Tutorial Hours: 6

Laboratory Hours: 33

Semester 1

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.

• Introduction to the use of Bayes' theorem.

## Assessment pattern

Assessment type Unit of assessment Weighting
Examination END OF SEMESTER 2HR EXAMINATION 70
Coursework COMPUTATIONAL PHYSICS COURSEWORK 30

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:

• Two computing coursework assignments: In the first the computer program (only) is assessed (10% contribution to module mark), while for the second (20% of module mark) a page-limited report plus a program is assessed.

• 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 in tutorials and in the supervised computational classes.  Written feedback is given on the computational assignments, with feedback on each 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 Recall both statistical and thermodynamic descriptions of entropy and be able to assess how entropy is related to uncertainty as to the state of the system, the direction of time, and heat flow. KC 002 Compare the statistical and thermodynamic definitions of entropy. K 003 Solve problems by applying the thermodynamic method. C 004 Explain how the state variables (pressure, volume and temperature) and bulk properties (modulus and thermal expansivity) are inter-related. K 005 State Gibb's expression for the entropy and the partition function at constant temperature, K 006 Derive both the Boltzmann weight of a state at constant temperature, and also the weight of a state at constant chemical potential/Fermi level. KC 007 Assess why a statistical approach is required in the study of matter such as gases, liquids and solids. C 008 Explain the role of fluctuations, and estimate their size in a range of contexts. C 009 Calculate the properties of the two-level system KC 010 Explain how fluctuations are related to thermodynamic functions, such as the heat capacities. C 011 Recall both the partition function for a simple classical particle and the equipartition theorem, and judge which one is required for a given system. K 012 Analyse phase transitions such as the ferromagnetic phase transition using statistical physics methods KC 013 Solve ordinary differential equations numerically using simple finite difference algorithms KCT 014 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 understanding of a simple physical system. KCT 015 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 skills in writing computer programs to solve problems in mathematics and physics

The learning and teaching methods include:

• 32h of lectures and 6h of tutorials

• 33h of supervised computational laboratory  as 3h/week x 11 weeks

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.

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

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics with Nuclear Astrophysics MPhys 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 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 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 with Quantum Technologies MPhys 1 Compulsory A weighted aggregate mark of 40% is required to pass the module
Physics with Quantum Technologies BSc (Hons) 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 2019/0 academic year.