Computational tools for active matter (and non-equilibrium physics in general)

Rodrigo Soto

Universidad de Chile, Chile

Abstract

In these lectures, we will discuss the main models used in the description of active matter, considering self-propelled particles, wet active matter with hydrodynamic interactions, and tissues. For each class of modes, we will study how they are implemented computationally, paying attention to efficiency and accuracy. Finally, we will discuss how simulations are analyzed and what information can be extracted from correlation functions, mean square displacements, structure factors, and response functions. Although the focus will be on active matter, many of the topics will be extended to other problems of non-equilibrium physics.

Outline

1) Self-propelled particles

- Models: ABP, run-and-tumble, active Ornstein-Uhlenbeck particles, elongated particles

- Excluded volume interactions

- Numerical integration of Langevin equations

- Sampling of directors and Poisson processes

- Efficiency tricks

- On-the-fly measurements (MSD, correlation functions, fields)

- Relevant fields: density, polarization, Q-tensor

2) Lattice models

- Why lattice models? speed, precision and universality classes

- Efficiency tricks

- Clusters y structures (connected components, structure factor)

3) Hydrodynamic interactions

- Singular solutions (stokeslet, stresslet, rotlet)

- Regularized solutions

- Boundary conditions (exact and IBM)

- Range of interactions (induced speed, torque)

- Brute force implementation

- Efficient solutions

4) Tissues

- Vertex models

- Agent-based simulations. Contact inhibition of locomotion

- Rheology. Linear and non-linear responses

5) General topics

- Phase transitions: order parameters, finite size effects

- Linear response: “size” of the perturbation, secular non-linear effects (e.g. heating)

- Tensorial analysis of correlation functions

- Computational order

Exercises (which one depends on the available time)

- Compute the density dependence of the diffusion coefficient of ABPs

- Determination of the size distribution functions of clusters in lattice models