This is a teaching + research position. The candidate will be expected to build their own teaching portfolio and lead an independent research programme, which complements the research activities at the School of Mathematics and Statistics.
A high percentage of our students are part-time students, mature students, students with disabilities, caring responsibilities, and other circumstances. As such, there is no formal face-to-face lectures, but instead, academics are expected to produce high quality course books, which are then sent out to students. A pdf example of a unit/chapter of a course book can be found in: https://www.open.edu/openlearn/science-maths-technology/introduction-the-calculus-variations/content-section-1
Students will also attend either face-to-face tutorials (usually at major cities across the UK) or online tutorials in the evenings/weekends. Note that as a central academic, you are not required to give tutorials, however you may apply to become a tutor (and you will get paid extra on a separate contract).
Candidate will be expected to develop their own independent research programme, which complements the research activities at the School of Mathematics and Statistics, see here: https://www.open.ac.uk/stem/mathematics-and-statistics/research/research-groups/applied-mathematics-and-theoretical-physics
The candidate will also receive excellent mentorship and support from the school to apply for an external funding to further develop their research leadership. Recently, the school has had few successes in securing external grants (despite its relatively small size), including 2 EPSRC Research Fellowships, several EPSRC and LMS grants. Some of our staff are also members of the Strategic Advisory Team for the UKRI.
The academic workload for a lecturer is similar to other UK universities, i.e. 1/3 teaching, 1/3 research, and 1/3 other things. The `other things’ may include administration, writing grants, and contributing to outreach activities. The OU has a long history of partnership with the BBC, see full list of portfolio here: https://www.bbc.co.uk/commissioning/open-university
Normally, you will work together with the media team to deliver a high quality video/radio content.
Click link below to apply NOW! https://www.open.ac.uk/about/employment/vacancies/lecturer-applied-mathematics-20979
You are only required to submit your CV and personal statement at this stage. Make sure you explain how you fulfil each criteria in the job specification and give evidence to support each criteria. (possible date for interview + research + teaching presentation: first half of September)
]]>The aim of this project is to combine microscopic modelling and simulations (e.g. Brownian dynamics) with continuum theory (e.g. soft glass rheology and statistical field theory) to study the macroscopic properties of growing and active living matter such as biological tissues and biofilms
Candidate will also be expected to develop their own professional skills, such as supervising undergraduate research project, delivering graduate lectures, participating in outreach activities, and/or applying external/internal funding under the guidance from the supervisor.
The post is funded by my EPSRC grant, `Thermodynamics of Growing Active and Living Matter’ for 30 months and is available with a flexible start date from 1st August 2023. (extension is possible under extraordinary circumstances)
For non UK-residents: We will reimburse the UK Worker’s Visa fee for the candidate, but unfortunately we will not be able to reimburse Immigration Health Surcharge (IHS), or the Visa and IHS costs for their partner/family. However, the University can offer interest-free loan if they wish to.
About the School: We are a relatively small but friendly department and provide an inclusive and supportive environment for everyone to achieve their best potentials. In recent years, we also had few successes in securing external funding such as two EPSRC Research Fellowships and several EPSRC standard grants.
To apply click (deadline 15th March 2023): https://www.open.ac.uk/about/employment/vacancies/pdra-applied-mathematicstheoretical-physics-20757
]]>Here we will consider the active model B+, which is a model for phase-separation in active matter (see previous post). We consider a scalar order parameter $\phi(\mathbf{r},t)$. The free energy is given by:
\[F[\phi] = \int dV\left\{ - \frac{A}{2}\phi^{2} + \frac{A}{4}\phi^{4} + \frac{K}{2}|\nabla\phi|^{2} \right\}\]The dynamics for active model B+, is given by (see Tjhung, Nardini, Cates, PRX, (2018)):
\[\frac{\partial\phi}{\partial t} = \underbrace{\nabla^{2}}_{(3)}(-A\phi+A\phi^{3}-K\underbrace{\nabla^{2}\phi}_{(2)})+\sqrt{2D}\underbrace{\nabla\cdot}_{(1)}\boldsymbol{\Lambda} \underbrace{\nabla\cdot}_{(4)}(\zeta\underbrace{(\nabla^{2}\phi)}_{(2)}\underbrace{\nabla}_{(4)}\phi - \underbrace{\nabla}_{(4)}(\lambda|\underbrace{\nabla}_{(4)}\phi|^{2}))\]where $\lambda$ and $\zeta$ are the activity parameters. ($\lambda=\zeta=0$) corresponds to the passive/equilibrium limit.) The last two terms (proportional to $\lambda$ and $\zeta$) above are the non-equilibrium terms. $\boldsymbol{\Lambda}$ is Gaussian white noise with zero mean and delta-function correlation:
\[\left<\Lambda_\alpha(\mathbf{r},t)\Lambda_\beta(\mathbf{r}',t')\right> = \delta_{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')\]Numerically, we have to discretize the Laplacian and gradient operator in the $\phi$-dynamics. First, for derivatives in (1), we must use higher order derivatives. This is because the noise is of order $\sqrt{\Delta t/\Delta x\Delta y}$ so we need to be accurate in the derivatives:
\[\partial_{x}\phi = \frac{\frac{1}{280}\phi_{i-4,j} - \frac{4}{105}\phi_{i-3,j}+\frac{1}{5}\phi_{i-2,j} - \frac{4}{5}\phi_{i-1,j} +\frac{4}{5}\phi_{i+1,j} - \frac{1}{5}\phi_{i+2,j} + \frac{4}{105}\phi_{i+3,j} - \frac{1}{280}\phi_{i+4,j}}{\Delta x}\]For derivative in (3), we apply (1) twice to ensure detailed balance exactly on the lattice.
For Laplacian in (2) and the gradients in (4), we use the isotropic form of the numerical Laplacian and gradients because we nucleate small bubbles, which is bad for $\nabla^2\phi$ (see Pooley, Furtado, PRE, (2007)):
\[\begin{align} \nabla^{2}\phi &= \frac{1}{\Delta x\Delta y} \left[\begin{array}{ccc} -\frac{1}{2} & 2 & -\frac{1}{2}\\ 2 & -6 & 2\\ -\frac{1}{2} & 2 & -\frac{1}{2} \end{array}\right]\phi_{ij} \\ \partial_x\phi &= \frac{1}{\Delta x} \left[\begin{array}{ccc} -\frac{1}{10} & 0 & \frac{1}{10}\\ -\frac{3}{10} & 0 & \frac{3}{10}\\ -\frac{1}{10} & 0 & \frac{1}{10} \end{array}\right]\phi_{ij} \end{align}\]Source code is available here source code.
]]>Let us consider some trajectory $\phi(\mathbf{r},t)$ from time $t = t_i$ to $t = t_f$. The rate of entropy production $\mathcal{S}$ is defined to be the logarithm of the ratio of forward probability to the backwards probability:
\[\mathcal{S} = \lim_{(t_f-t_i)\rightarrow\infty} \frac{1}{t_f-t_i} \left<\ln\left(\frac{P_F[\phi]}{P_B[\phi]}\right)\right>\]where $P_F[\phi]$ is the probability functional of obtaining a particular trajectory $\{\phi(\mathbf{r},t)|t\in[t_i,t_f]\}$ and $P_B[\phi]$ is the probability functional of obtaining the same trajectory going backwards in time (see image below). In another word, entropy production is a measure of time-irreversibility.
The angle brackets above $\left<\dots\right>$ indicates averaging over all noise realizations.
Active model B is a model for phase separation in active matter, which is a derivative of the active model B+ (see previous post). The active model B dynamics is given by:
\[\frac{\partial\phi}{\partial t} + \nabla\cdot\left(\mathbf{J}_{d} + \boldsymbol{\Lambda}\right) = 0\]$\mathbf{J}_d$ is the deterministic current and $\boldsymbol{\Lambda}$ is a Gaussian white noise with zero mean and delta function correlation:
\[\left<\Lambda_\alpha(\mathbf{r},t)\Lambda_\beta(\mathbf{r}',t')\right> = 2D\delta_{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')\]The deterministic current is given by:
\[\mathbf{J}_{d} = -\nabla\underbrace{\left(\frac{\delta F}{\delta\phi} + \lambda|\nabla\phi|^{2}\right)}_{\mu}\]where $\mu$ is the chemical potential, which consists of the passive/equilibrium part: $\delta F/\delta\phi$ and the active part: $\lambda|\nabla\phi|^2$ The rate of entropy production for this dynamics can be shown to be given by (Nardini, Fodor, Tjhung, van Wijland, Tailleur, Cates, PRX, (2017)):
\[\begin{align} \mathcal{S} & =\frac{1}{\tau}\int_{0}^{\tau}dt\,\frac{\partial\phi}{\partial t}\circ\mu\nonumber \\ & =\frac{1}{\tau}\sum_{t}\Delta t\frac{\phi_{t+\Delta t}-\phi_{t-\Delta t}}{2\Delta t}\mu_{t},\nonumber \end{align}\]where the time integral is interpreted as Stratonovich. $\phi_t$ is the density field at time $t$ and $\mu_t$ is the chemical potential at time $t$. The reason for the Strotonovich choice is follows:
When we are computing entropy production, basically, we are comparing the trajectory going forward in time and the trajectory going backwards in time.
Now the time is discretized into $t = 0, dt, 2dt, 3dt, \dots$. That means the trajectory is also split into segments. The way we split the trajectory is done in the Stratonovich way. That means the value of $\phi$ at time $t$ means the value of $\phi$ mid-point of $[t, t+dt]$. i.e. $\phi(\mathbf{r},t+dt/2)$. If we discretize the trajectory in `Ito’ way, the time-reversed of that trajectory is not identical (see drawing below).
However, when we perform the simulations, we do it in `Ito’ way. i.e. we start from $\phi(\mathbf{r},0)$, then we update the time to get $\phi(\mathbf{r},0), \phi(\mathbf{r},dt), \phi(\mathbf{r},2dt), \phi(\mathbf{r},3dt), \dots$.
]]>In statistical physics we often say this system is in equilibrium and that system is out-of-equilibrium et cetera. But what is equilibrium? and what is out-of-equilibrium?
We say that a system is in equilibrium if it respects time reversal symmetry (TRS) at steady state and out-of-equilibrium if it violates TRS at steady state. For example, consider the pedestrations on Champs Elysee. They clearly constitute a non-equilibrium system because if we watch the movie backwards they will appear to walk backwards.
But now suppose we pay half the people 5€ each to walk backwards. They now become an equilibrium system because if we watch the movie backwards they look the same (statistically).
Probably we are familiar with the Cahn-Hiliard equation:
\[\frac{\partial\phi}{\partial t} + \nabla\cdot\left(-\nabla\frac{\delta F}{\delta\phi} + \boldsymbol{\Lambda}\right) = 0\]In the equation above, $\phi$ is the density of the fluid. If we start from a homogenous density $\phi = -0.6$, the system will phase-separate into high density $\phi = +1$ (liquid) and low density $\phi = -1$ (vapour) phase. In steady state $t = \infty$, we end up with a single big blob of liquid.
Introducing the non-equilibrium Cahn-Hiliard equation (also called active model B+)….
\[\frac{\partial\phi}{\partial t} + \nabla\cdot\left( -\nabla\frac{\delta F}{\delta\phi} + \boldsymbol{\Lambda} + \lambda\nabla|\nabla\phi|^2 + \zeta(\nabla^2\phi)\nabla\phi \right) = 0\]If we start from the same intial configuration, we see similar phenomena where the fluid phase-separate into high density and low density. However in the steady state, we get a boiling droplet. The droplet is in perpetual boiling state (see movie at the end), which is not possible in equilibrium because all the liquid will be evaporated.
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