# 1st Year PhD

Term One

I began my PhD with my first supervisor is Dr Stefan Adams and my second supervisor is Dr Christoph Ortner. Following from my MSc dissertation (see here) we looked again at density functional theory however in this case emphasising the quantum mechanical nature of the problem. A summary of my first term work is found here

Term Two

Having struggled to find any interesting tractable problems concerning quantum mechanical DFT. We looked again at classical DFT, we discovered an interesting model Phase Field Crystal (PFC) that can be derived as a special case of DFT. This change of approach means my first supervisor is now Dr Christoph Ortner and my second supervisor is Professer Charles Elliott. A summary of my second term work can be found here.

I am supplementing my research by attending some lecture courses. In the first term attended

- Large Deviations in Statistical Mechanics: Lecturer Dr Stefan Adams
- Stochastic Analysis : Lecturer Dr Xue-Mei Li
- Differential Geometry: Lecturer Professor Peter Topping

I have given several talks in 1st year more information can be found here

- 17th October 2012: MasDoc Semainer Talk: Summary of MSc work, approximately 40 minutes
- 26th March 2013: MasDoc-CCA Conference Talk: Summary of 1st year term two work, approximately 30 minutes
- 22nd May 2013: Post-Graduate Semainar Talk: Summary of 1st year work up to mid term 3, approximately 1 hour
- 4th June 2013: Applied PDEs Semainar Talk: Summary of 1st year work up to mid term 3, approximately 1 hour

**PFC Functional**

We consider the methodology of Phase Field Crystals. In this model the energy of the system is given by the PFC functional

\[ \mathcal F [u] = \int_{\Omega} \frac{u}{2} \left ( \Delta +1 \right) ^2 u - \delta \frac{u^2}{2} + \frac{u^4}{4} \textrm{d} x \]

The equilibrium state of the system is given by minimising this functional whilst conserving the integral of \(u \)

\[ \bar u = \frac{1}{| \Omega |} \int_{\Omega} u \textrm{d} x \].

It can be shown that this functional is minimised by three different states that can be switched between using \( \delta \) and \( \bar u \)

- striped phase
- a hexagonal phase (representing a lattice)
- a constant phase (representing a liquid)

During the third term PhD progess is assessed by a PAC meeting several documents relevant to PAC are here

For our numerical implementation we wish to consider the PFC equation in Fourier space . We do this we wish to consider the discrete Lagrangian in Fourier space. Specifically we claim

\[ \widehat{\Delta_h {U_j}^n} = F[k] \widehat {U^n} [k] \label{ForF} \]

where

\[ F[k]=\displaystyle \sum_{i=1}^d \frac{2} {h_i^2} \left( \cos \left[ \frac{ 2 \pi k_i }{m_i} \right]-1 \right )\]

This is shown here.

PAC Report

The summary of my current work as presented in my PAC meeting are given in my PAC report here.

**Animation**

Below is shown an animation of the algorithm starting from random initial condition. The domain is \[ 2.2 \pi \times 2.2 \sqrt{3} \pi \] and \[ \delta=0.9 , \bar u =0.5 \]