Motility Induced Phase Separation (ongoing)

---
layout: post
read_time: true
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title:  Motility Induced Phase Separation
date:   2022-10-18 13:32:20 -0000
description: Motility Induced Phase Separation in Active Brownian Particle System.
img: posts/20221018/detail.png
tags: [Stochastic Process, Scientific Computation]
author: Hanchun Wang
category: project
github:  
mathjax: yes
---

Summary

In this Active Brownian Particle system, particles will aggregate without any attracting force inbetween.

Vidoes of my results in Motility Induced Phase Separation:

MIPS cluster formation YouTube
MIPS cluster merging YouTube
MIPS cluster dissolution YouTube

Voronoi tessellation of a MIPS cluster:

Adjacency graph of a MIPS cluster:

Introduction

From cytoskeletal filaments and bacterial aggregation to the birds’ flock and fish school, biological agents consume energy from the environment to sustain different kinds of activities. The activity provides agents the capacity to have steady states away from the equilibrium. This capacity against the maximum entropy.) is the key to understanding the mystery of why biological creatures can live for decades.

One of the minimal models to study active matter is the active Brownian particles model (ABP model). In the ABP model, particles are colloidal spheres. Particles are governed by Langevin’s equation that particles can self-propel themselves by absorbing and converting energy from the environment. This is one of the simplest forms of activity, however, previous research has shown that simple activeness is enough to achieve steady non-equilibrium states in the system. Particles will aggregate into clusters without any presence of attracting mechanisms and this phenomenon is called motility-induced phase separation (MIPS).

Dynamics

The over-damped Langevin’s dynamics is given as

$$ \left\{\begin{split} \boldsymbol{\dot r_i} &= \frac{1}{\gamma }\boldsymbol{F_i} + {v_p}\boldsymbol{\hat n_i} + \sqrt {2D} {\eta _i}\\ {\dot \theta }_i &= \sqrt {2{D_R}} {\xi _i} \end{split}\right. $$

Where $F_i=-{\nabla _r}\sum_j {V_j}\left( r_i \right) $ is the total force given by all the particles to the particle $i$. $\vec{\hat n}$ is the self-propelling direction of the particle. $D$ is the transitional diffusion constant, $D_R$ is the rotational diffusion constant. $\xi, \eta \sim \mathcal{N}(0,1)$ are independent stochastic variables under the multidimensional Gaussian white noise.

The potential is

$$ V(r)=4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6+\frac{1}{4}\right] \Theta\left(\sigma_*-|r|\right) $$

The under-damped dynamics is given as

$$ \left\{\begin{aligned} \boldsymbol{\ddot r_i} &=-\gamma_T \boldsymbol{\dot{r}_i} +\boldsymbol{F_i} +\gamma_T v_p \boldsymbol{\hat{n}_i} +\sqrt{2 D_T} \gamma_T \eta_i \\ \ddot{\theta}_i &=-\gamma_R \dot{\theta}_i+\sqrt{2 D_R} \gamma_R \xi_i \end{aligned}\right. $$

Voronoi Tessellation

The local density in the system can be studied by the Voronoi Tessellation