# MSc Year

Below follows a summary of the work I undertook in my first year of MasDoc (the MSc year). This is split into three main areas

During my first year I undertook a total of six modules.

Two in analysis:

- A1: Analysis in Linear PDEs Lecturer: Professor James Robinson
- A2:Topics in Pdes Lecturer: Professor Charles Elliot.

Two in computing:

- C1:Scientific Computing Lecturers: Dr Andreas Dedner and Dr Björn Stinner
- C2:Advanced Topics in Numerical Analysis Lecturers: Dr Andreas Dedner and Dr Björn Stinner.

Two in probability:

- P1:An Invitation to Graduate Probability Lecturer: Professor Wilfrid Kendall
- P2:Probability Theory and Examples Lecturer:Professor Neil O'Connell.

During my first year of MasDoc I undertook two group work projects as part of the research study group module. This was split into stages a proposal stage and a implementation stage, where a different problem was considered in each stage

My first group project (the proposal stage) was on multi-scale materials. During this project I worked with Amal Alphonse MMath MSc, Chin Lun MMath MSc, Abhishek Shukla MSc MSc and Matthew Thorpe MSc MSc we were supervised by Dr Stefan Adams and Dr Christoph Ortner. Our proposal focused on four different ways of calculating the free-energy of a particle system,thes four different methods are Cauchy-Born, Statistical Mechanics, Coarse-Graining and Molecular dynamics. A major part of our proposl was the idea of attempting to create a finite temperature Cauchy-Born rule and test its validity. Our proposal is given here and a brief summary is given in our presentation. The second stage of this project was implemented with major modifications by Maha Al-Hajri MSc MSc , Tessa Colledge MMath MSc , Owen Daniel MSc MSc, Barnaby Garrod MMath MSc , and Alexander Kister MSc MSc again supervised by Dr Adams and Dr Ortner, a summary of their work can be found on their website (warwick access required).

My second group project (the implementation stage) called media and motion and concentrated on simulating spiral waves on different curved surfaces as a first approximation of the heart. During this project I worked with Amal Alphonse and Yuchen ("Jeff") Pei MSc MSc under the supervision of Dr Andreas Dedner and Professor Dwight Barkley . We took inspiration from the work of Maha Al-Hajri, Tessa Colledge and Alexander Kister again supervised by Dr Dedner and Professor Barkley (see their proposal ). In this project we focus on the set of equations given by the Barkley Model .

\begin{align*} \dot {u} +u\nabla_\Gamma\cdot\mathbf v -a \Delta_\Gamma u &=f(u,v) \textrm{ in } \mathcal G_T:=\bigcup_{t\in[0,T]}\{t\}\times \Gamma_t \\ \dot {v}+v\nabla_\Gamma\cdot\mathbf v &=g(u,v) \textrm{ in } \mathcal G_T \end{align*}

where

\begin{align*} f(u,v) &= {1 \over \epsilon } u \left (1-u \right) \left( u - {{v+b } \over c} \right) \\ g(u,v) &= u-v \end{align*}

with appropriate initial conditions and zero Neumann boundary conditions applied at any boundary.

Our simulations are carried out the Distributed and Unified Numerics Environment (DUNE).

We will principally concentrate on simulating waves on spherical surfaces. In one section (see here requires warwick acess) we take a unit sphere and apply a deformation along the \(y\)-axis of magnitude:

\begin{equation*} 1+ \alpha \sin \left( 2 \pi \beta t \right) \end{equation*}

Below we show spiral waves on the sphere with deformation given by \( \alpha=0.5\) and \( \beta=0.1 \)

We also focus on different ways of adding inhomogeneities (see here requires warwick access). The simplest method we use is to filter out elements from our sphere to create a new physical domain. The results are shown below for two different sizes of hole:

A summary of all our work in an interactive format is given on our website (requies warwick access) and in a more condensed form in both our poster and our presentation. A full summary of our work is given here.

For my dissertation this summer I worked on a project in Density Functional Theory, My first supervisor was Dr Ortner and my second supervisor was Dr Adams, a brief summary of my proposed work is given here.

My main MSc research project considered a density functional theory method of looking at the 1-dimensional hard-core gas in a classical situation. The idea of density functional theory is to express the free energy as a functional of the one-particle density ( \( \rho^{(1)} _{\Lambda^N}(x) \)) and another term

\begin{equation*}A_{\beta} ^ {\Lambda^N} [V] = F_{HK} [ \rho^{(1)} _ {\Lambda^N} (x)] + \int_{\Lambda^N} V(x) \rho^{(1)} _ {\Lambda^N} (x) dx \end{equation*}

where \(N\) is the number of particles \(\Lambda \subset \mathbb R^ {dN} \) is a finite box to which our particles are restricted, where \(d\) is the dimension of the space, \(\beta= {k_b T} ^{-1}\) is inversely proportional to the temperature \(T\) and \(V(x)\) is the external potential.

After developing the theory required for density functional theory we considered as an introductory example the ideal gas (the case where the internal potential vanishes). Having obtained some results for this example we considered a hard-core interaction. In this case the external potential is given by

\begin{equation*} U= \displaystyle \sum_{1 \le i < j \le N } W_a ( x_i -x_j) \end{equation*}

where

\begin{equation*} \begin{cases} W_a (x_i -x_j) &=0 ~~~~ \|x_i - x_j \| >a \\ W_a (x_i -x_j ) &= \infty ~~~ \| x_i -x_j \| \le a \end{cases} \end{equation*}

and \(a\) is the radius of the hard core. Using this we develop a methodology for constructing a functional that is unfortunately only applicable in the one-dimensional case and for a limited number of external potentials. We then started with a discrete approximation of the problem and used a continuum limit again this was only viable in one dimension. In attempt to make further progress we developed an approximation based on the Mayer expansion, for the purposes of comparison we computed this in one dimension although it should be applicable in higher dimensions.

For the interested reader my thesis can be found here , a summary poster of the ideal gas is found here. A summary talk on my MSc dissertation can be found here and information about my 1st year PhD which attempts to extend some of the work of my MSc is given here.