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Advanced Theoretical Techniques and Modelling Matter - PHY00074H

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  • Department: Physics
  • Credit value: 20 credits
  • Credit level: H
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

Advanced Theoretical Techniques

The Advanced Theoretical Techniques part of this module introduces mathematical ideas and tools which underpin modern theoretical physics. For example, the variational principle and Lagrangian mechanics are essential to classical and quantum field theories. Tensors are a mathematical language that allow us to write physical laws that are independent of the frame of reference – the fundamental principle of special and general relativity. This course will teach you how to use these mathematical tools to analyse challenging theoretical physics problems from a range of subject areas and applications.

Modelling Matter

In this part of the module, you will be introduced to several major techniques for modelling matter using both classical and quantum approaches. These approaches (e.g. molecular dynamics, monte carlo and particle-in-cell) are widely used in many different areas of physics, and are also active research topics in their own right. This course will teach you the key ideas behind these techniques, and illustrate with practical computing sessions.

Related modules

Pre-requisites: Statistical & Solid State Physics (PHY00054I) and Mathematics, Professional Skills & Introduction to Laboratories or Mathematics, Professional Skills & Experimental Laboratories or Mathematics, Professional Skills & Computational Laboratories or equivalent.

Module will run

Occurrence Teaching period
A Semester 1 2024-25

Module aims

This module will introduce a range of computational and analytic methods that can be used to model the properties and dynamics of physical systems. The module is divided into two parts: Advanced Theoretical Techniques and Modelling Matter.

Modelling Matter

In Molecular Dynamics we will simulate the dynamic behaviour of systems at the atomic level. Particular attention will be paid to the physical basis of the algorithms used and their efficient and reliable implementation. We will then focus on how to extract physical properties from the results of the simulation and assess the errors in them. A range of applications will be introduced.

In Monte Carlo and Spin dynamics we will simulate the thermodynamic behaviour of magnetic materials, starting with the prototypical Ising model and the Monte Carlo method and onto the 3D Heisenberg model and atomistic spin dynamics. We will consider a range of topical applications including permanent magnetic materials, data storage and ultrafast magnetism.

In Density Functional Theory, we will explore computation methods for modelling systems at the quantum mechanical level. Whilst simple problems can be solved analytically, numerical/computational methods have to be used for anything more complex than a few electrons.

Finally, the Particle in Cell method will be introduced, to illustrate the application of modelling matter at longer length scales.

In all cases, the application of the theoretical ideas and consequence of approximations used will be explored by lectures and supported by practical sessions

Advanced Theoretical Techniques

The aim of the Advanced Theoretical Techniques part of this module is to develop your mathematical and theoretical skills to solve advanced problems in physics. We start by studying the Fourier transform and its extension, the Laplace transform, which can be used to solve differential equations and analyse physical situations. Next we study the ‘calculus of variations’ – a powerful concept which can be used to solve geometric and mechanics problems, from the shape a bubble takes to minimise tension to the behaviour of electromagnetic fields. Lastly, we study how laws of physics can be defined mathematically so that they are the same in all frames of reference – a cornerstone of modern physics and the foundation for the general theory of relativity.

Module learning outcomes

Modelling Matter

  • Describe the physical principles of computational material simulation approaches and assess their benefits and limitations

  • Use available materials simulation software packages to model the physical properties of matter

  • Apply simulations to study and evaluate the physical properties of a variety of materials

  • Analyse the results of a materials simulation to extract physical properties

Advanced Theoretical Techniques

  • Apply integral transforms, such as Fourier and Laplace, to solve differential equations and interpret physical processes

  • Extend solutions to differential equations to include nonlinear behaviour

  • Use the calculus of variations to find extremal solutions in fields such as geometry (shortest path), statistics (maximum entropy) and mechanics (principle of least action)

  • Transform physical quantities between general coordinate systems (including reference frames).

Module content

Advanced Theoretical Techniques

  • Integral transforms (Fourier, Laplace and convolutions) are used to interpret physical phenomena and solve differential equations

  • The calculus of variations is defined via the functional derivative and used to find extremal solutions to various problems, for example Lagrange’s equations for fields

  • Tensors are defined via a general coordinate transformation and used to express laws of physics which are independent of the reference frame.

Modelling Matter

  • Molecular Dynamics simulations

    • Equations of motion for atomic systems; phase space and trajectories

    • Numerical integration and the Velocity-Verlet scheme

    • Interatomic potentials (Lennard-Jones, MEAM), statistical ensembles and Langevin dynamics

    • Computational techniques for efficient calculation of forces,periodic boundary conditions and the minimum image criterion; potential truncation and neighbour lists

  • Monte Carlo (MC) and Spin Dynamics simulations

    • The Ising model and Metropolis algorithm, underlying principles and basic statistical mechanics

    • Atomistic spin dynamics and magnetic materials simulation

    • Calculations of long ranged interactions

    • Applications of spin dynamics

  • Density Functional Theory

    • Many electron problems: example of electrons+nuclei with key approximations (Born-Oppenheimer, independent electron approx, self-consistent field)

    • Kohn-Sham DFT

    • Basis sets & pseudopotentials: atomic basis sets and plane waves

    • Calculation of material properties.

  • Particle in Cell techniques

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 40
Essay/coursework 10
Essay/coursework 40
Essay/coursework 10

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 40
Essay/coursework 40

Module feedback

'Feedback’ at a university level can be understood as any part of the learning process which is designed to guide your progress through your degree programme. We aim to help you reflect on your own learning and help you feel more clear about your progress through clarifying what is expected of you in both formative and summative assessments.

A comprehensive guide to feedback and to forms of feedback is available in the Guide to Assessment Standards, Marking and Feedback. This can be found at:

https://www.york.ac.uk/students/studying/assessment-and-examination/guide-to-assessment/

The School of Physics, Engineering & Technology aims to provide some form of feedback on all formative and summative assessments that are carried out during the degree programme. In general, feedback on any written work/assignments undertaken will be sufficient so as to indicate the nature of the changes needed in order to improve the work. Students are provided with their examination results within 25 working days of the end of any given examination period. The School will also endeavour to return all coursework feedback within 25 working days of the submission deadline. The School would normally expect to adhere to the times given, however, it is possible that exceptional circumstances may delay feedback. The School will endeavour to keep such delays to a minimum. Please note that any marks released are subject to ratification by the Board of Examiners and Senate. Meetings at the start/end of each semester provide you with an opportunity to discuss and reflect with your supervisor on your overall performance to date.

Our policy on how you receive feedback for formative and summative purposes is contained in our Physics at York Taught Student Handbook.

Indicative reading

Modelling Matter:

  • Haile J M: Molecular dynamics simulation (Wiley)
  • Rapaport D C: The art of molecular dynamics simulations (CUP)
  • Allen M P and Tildesley D J: Computer simulation of liquids (OUP)
  • Frenkel D and Smit B: Understanding molecular simulation (Academic Press)
  • Computational Physics, 2nd Ed by JM Thijssen (Cambridge University Press, 2007)

Advanced Theoretical Techniques

  • Integral transforms and their applications by Davies, B
  • Calculus of variations by Gel'fand, I. M
  • Introduction to vectors and tensors by Bowen, Ray M.
  • Introduction to tensor calculus, relativity and cosmology by Lawden, D.F
  • Special relativity and classical field theory by Susskind, Leonard.

Richard Fitzpatrick: Classical Electromagnetism lecture notes: http://farside.ph.utexas.edu/teaching/em/lectures/node106.html


The information on this page is indicative of the module that is currently on offer. The University constantly explores ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary. In some instances it may be appropriate for the University to notify and consult with affected students about module changes in accordance with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.