Jekyll2021-09-10T09:42:17+00:00/feed.xmlElsen TjhungThe website is created with Jekyll. Durham-Oxford-Strathclyde Meeting in Anisotropic Materials, 13th September 20212021-08-09T00:00:00+00:002021-08-09T00:00:00+00:00/2021/08/09/dos<h2 id="anisotropic-materials">Anisotropic materials</h2> <p>We are pleased to announce an upcoming meeting on anisotropic materials. The timetable is below.</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/DOS.jpg" alt="drawing" width="800" /></p>Anisotropic materialsCalculating entropy production of active field theory2020-12-27T00:00:00+00:002020-12-27T00:00:00+00:00/2020/12/27/entropy<h2 id="entropy-production">Entropy production</h2> <p>Let us consider some trajectory <em>ϕ(<strong>r</strong>,t)</em> from time <em>t = t<sub>i</sub></em> to <em>t = t<sub>f</sub></em>. The rate of entropy production <em>S</em> is defined to be the logarithm of the ratio of forward probability to the backwards probability:</p> <p><img src="http://latex.codecogs.com/svg.latex?\mathcal{S}=\lim_{(t_f-t_i)\rightarrow\infty}\frac{1}{t_f-t_i}\left&lt;\ln\left(\frac{P_F[\phi]}{P_B[\phi]}\right)\right&gt;" border="0" />,</p> <p>where <em>P<sub>F</sub>[ϕ]</em> is the probability functional of obtaining a particular trajectory <img src="http://latex.codecogs.com/svg.latex?\left\{\phi(\mathbf{r},t)|t\in[t_i,t_f]\right\}" border="0" /> and <em>P<sub>B</sub>[ϕ]</em> is the probability functional of obtaining the same trajectory going backwards in time (see image below). <em>In another word, entropy production is a measure of time-irreversibility</em>.</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/irreversibility2.jpg" alt="drawing" width="400" /></p> <p>The angle brackets above <img src="http://latex.codecogs.com/svg.latex?\left&lt;\dots\right&gt;" border="0" /> indicates averaging over all noise realizations.</p> <h2 id="entropy-production-for-active-model-b">Entropy production for active model B</h2> <p>Active model B is a model for phase separation in active matter, which is a derivative of the active model B+ (see <a href="https://elsentjhung.github.io/2019/04/07/active.html">previous post</a>). The active model B dynamics is given by:</p> <p><img src="http://latex.codecogs.com/svg.latex?\frac{\partial\phi}{\partial t}+\nabla\cdot\left(\mathbf{J}^{d}+\boldsymbol{\Lambda}\right)=0" border="0" />,</p> <p><em><strong>J</strong><sub>d</sub></em> is the deterministic current and <em><strong>Λ</strong></em> is a Gaussian white noise with zero mean and delta function correlation:</p> <p><img src="http://latex.codecogs.com/svg.latex?\left&lt;\Lambda_\alpha(\mathbf{r},t)\Lambda_\beta(\mathbf{r}',t')\right&gt;=2D\delta_{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')" border="0" /></p> <p>The deterministic current is given by:</p> <p><img src="http://latex.codecogs.com/svg.latex?\mathbf{J}_{d} =-\nabla\underbrace{\left(\frac{\delta F}{\delta\phi}+\lambda|\nabla\phi|^{2}\right)}_{\mu}" border="0" /></p> <p>where <em>μ</em> is the chemical potential, which consists of the passive/equilibrium part: <em>δF/δϕ</em> and the active part: <img src="http://latex.codecogs.com/svg.latex?\lambda|\nabla\phi|^2" border="0" />. The rate of entropy production for this dynamics can be shown to be given by (<a href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.7.021007">Nardini, Fodor, Tjhung, van Wijland, Tailleur, Cates, <em>PRX</em>, (2017)</a>):</p> <p><img src="http://latex.codecogs.com/svg.latex?\begin{align} \mathcal{S} &amp; =\frac{1}{\tau}\int_{0}^{\tau}dt\,\frac{\partial\phi}{\partial t}\circ\mu\nonumber\\ &amp; =\frac{1}{\tau}\sum_{t}\Delta t\frac{\phi_{t+\Delta t}-\phi_{t-\Delta t}}{2\Delta t}\mu_{t},\nonumber \end{align}" border="0" /></p> <p>where the time integral is interpreted as Stratonovich. <em>ϕ<sub>t</sub></em> is the density field at time <em>t</em> and <em>μ<sub>t</sub></em> is the chemical potential at time <em>t</em>. The reason for the Strotonovich choice is follows:</p> <p>When we are computing entropy production, basically, we are comparing the trajectory going forward in time and the trajectory going backwards in time.</p> <p>Now the time is discretized into <em>t = 0, dt, 2dt, 3dt, …</em>. 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 <em>ϕ</em> at time <em>t</em> means the value of <em>ϕ</em> mid-point of <em>[t, t+dt]</em>. i.e. <em>ϕ(<strong>r</strong>,t+dt/2)</em>. If we discretize the trajectory in Ito’ way, the time-reversed of that trajectory is not identical (see drawing below).</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/ito-trajectory.png" alt="drawing" width="800" /></p> <p>However, when we perform the simulations, we do it in Ito’ way. i.e. we start from <em>ϕ(<strong>r</strong>,0)</em>, then we update the time to get <em>ϕ(<strong>r</strong>,0), ϕ(<strong>r</strong>,dt), ϕ(<strong>r</strong>,2dt), ϕ(<strong>r</strong>,3dt), …</em>.</p>Entropy productionNumerical discretization for active scalar field theory2020-12-26T00:00:00+00:002020-12-26T00:00:00+00:00/2020/12/26/discretization<h2 id="active-model-b">Active model B+</h2> <p>Here we will consider the active model B+, which is a model for phase-separation in active matter (see <a href="https://elsentjhung.github.io/2019/04/07/active.html">previous post</a>). We consider a scalar order parameter <em>ϕ(<strong>r</strong>,t)</em>. The free energy is given by:</p> <p><img src="http://latex.codecogs.com/svg.latex?F[\phi]=\int dV\left\{ -\frac{A}{2}\phi^{2}+\frac{A}{4}\phi^{4}+\frac{K}{2}|\nabla\phi|^{2}\right\}" border="0" /></p> <p>The dynamics for active model B+, is given by (see <a href="https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.031080">Tjhung, Nardini, Cates, <em>PRX</em>, (2018)</a>):</p> <p><img src="http://latex.codecogs.com/svg.latex?\frac{\partial\phi}{\partial t}=\underbrace{\nabla^{2}}_{(3)}(-A\phi+A\phi^{3}-K\underbrace{\nabla^{2}\phi}_{(2)}+\lambda|\underbrace{\nabla}_{(1)}\phi|^{2})" border="0" /> <img src="http://latex.codecogs.com/svg.latex?-\underbrace{\nabla\cdot}_{(1)}(\zeta\underbrace{(\nabla^{2}\phi)}_{(2)}\underbrace{\nabla}_{(1)}\phi)+\sqrt{2D}\underbrace{\nabla\cdot}_{(1)}\boldsymbol{\Lambda}" border="0" />,</p> <p>where <em>λ</em> and <em>ζ</em> are the activity parameters. (<em>λ = ζ = 0</em> corresponds to the passive/equilibrium limit.) <strong>Λ</strong> is Gaussian white noise with zero mean and delta-function correlation:</p> <p><img src="http://latex.codecogs.com/svg.latex?\left&lt;\Lambda_\alpha(\mathbf{r},t)\Lambda_\beta(\mathbf{r}',t')\right&gt;=\delta_{\alpha\beta}\delta(\mathbf{r}-\mathbf{r}')\delta(t-t')" border="0" />.</p> <p>Numerically, we have to discretize the Laplacian and gradient operator in the <em>ϕ</em>-dynamics. First, for derivatives in (1), we must use higher order derivatives. This is because the noise is of order <img src="http://latex.codecogs.com/svg.latex?\sqrt{\Delta t/\Delta x\Delta y}" border="0" /> so we need to be accurate in the derivatives:</p> <p><img src="http://latex.codecogs.com/svg.latex?\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}" border="0" />.</p> <p>For Laplacian in (2), we use the isotropic form of the numerical Laplacian because we nucleate small bubbles, which is bad for <em>∇<sup>2</sup>ϕ</em> (see <a href="https://journals.aps.org/pre/abstract/10.1103/PhysRevE.77.046702">Pooley, Furtado, <em>PRE</em>, (2007)</a>):</p> <p><img src="http://latex.codecogs.com/svg.latex?\nabla^{2}\phi=\frac{1}{\Delta x\Delta y}\left[\begin{array}{ccc} -\frac{1}{2} &amp; 2 &amp; -\frac{1}{2}\\ 2 &amp; -6 &amp; 2\\ -\frac{1}{2} &amp; 2 &amp; -\frac{1}{2} \end{array}\right]\phi_{ij}" border="0" />.</p> <p>For derivative in (3), we apply (1) twice to ensure detailed balance exactly on the lattice.</p>Active model B+What is Active Matter?2019-04-07T00:00:00+00:002019-04-07T00:00:00+00:00/2019/04/07/active<h2 id="time-reversal-symmetry-breaking">Time reversal symmetry breaking</h2> <p>In statistical physics we often say this system is in equilibrium and that system is out-of-equilibrium <em>et cetera</em>. But what is equilibrium? and what is out-of-equilibrium?</p> <p>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.</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/people1.png" alt="drawing" width="800" /></p> <p>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).</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/people2.png" alt="drawing" width="800" /></p> <h2 id="active-field-theories">Active field theories</h2> <p>Probably we are familiar with the Cahn-Hiliard equation:</p> <p><img src="http://latex.codecogs.com/svg.latex?\frac{\partial\phi}{\partial t}+\nabla\cdot\left(-\nabla\frac{\delta F}{\delta\phi}+\boldsymbol{\Lambda}\right)=0" border="0" /></p> <p>In the equation above, <em>ϕ</em> is the density of the fluid. If we start from a homogenous density <em>ϕ = -0.6</em>, the system will phase-separate into high density <em>ϕ = +1</em> (liquid) and low density <em>ϕ = -1</em> (vapour) phase. In steady state <em>t = ∞</em>, we end up with a single big blob of liquid.</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/MB.png" alt="drawing" width="1200" /></p> <p>Introducing the non-equilibrium Cahn-Hiliard equation (also called <a href="https://link.aps.org/doi/10.1103/PhysRevX.8.031080">active model B+</a>)….</p> <p><img src="http://latex.codecogs.com/svg.latex?\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" border="0" /></p> <p>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 <em>perpetual</em> boiling state (see movie at the end), which is not possible in equilibrium because all the liquid will be evaporated.</p> <p><img src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_figures/AMB+.png" alt="drawing" width="1200" /></p> <video src="https://raw.githubusercontent.com/elsentjhung/elsentjhung.github.io/master/_movies/boiling-droplet.mov" controls="controls" width="480" height="480" />Time reversal symmetry breaking