Non-equilibrium Statistical Mechanics of Self-Propelled Particles

Abstract

Self-propelled particles (SPPs) are particles who, by themselves, are able to generate persistent motion by converting energy from an ambient reservoir into work. Collections of such particles form a class of intrinsically out-of-equilibrium fluids called active fluids which have energy input and dissipation at the scale of the particle constituents. Despite their non-equilibrium nature, large scale, cohesive structures will often spontaneously emerge. These structures can manifest in microscopic realizations such as collective cell motility but also in much larger objects like flocks of birds. In this work we apply the powerful tools of non-equilibrium statistical mechanics to study SPPs both at the single particle level and for collections of interacting particles.

The primary non-equilibrium characteristic of a SPP is the persistent correlation in its direction of motion. In the first theme, we address the following question: What is the effect of the details of the decorrelation process on long time properties of SPPs? This question is addressed in 2 parts. First, we compare the response of active Brownian particles and run-and-tumble particles when subject to external torques. Second, we investigate the nature of the non-equilibrium steady state by constructing the Smoluchowski equation. The second topic comes with the added feature that it allows us to address the validity of different approximation techniques available to deal with correlated stochastic processes.

In the second theme we construct a theoretical framework to characterize the non-equilibrium steady states of interacting SPPs. Starting from a microscopic model of self-propelled hard spheres we use tools of non-equilibrium statistical mechanics and the kinetic theory of hard spheres to derive a Smoluchowski equation for interacting active Brownian particles. We illustrate the utility of the statistical mechanics framework developed with two applications. First, we derive the steady state pressure of the hard sphere active fluid in terms of the microscopic parameters and second, we identify the critical density for the onset of motility-induced phase separation in this system. We show that both these quantities agree well with overdamped simulations of active Brownian particles with excluded volume interactions given by steeply repulsive potentials. The results presented in this section can be used to incorporate excluded volume effects in diverse models of self-propelled particles.

The final theme is an application of the self-propelled particle model to systems of motile cells. Some cells are able to deform the substrate they are adhered to and at the same time are able to sense and respond to their mechanical environment. For a collection of cells this can lead to a elastic interaction between them. In this study the cells are modeled as self-propelled “force dipoles” that deform the surface. We find that a combination of only activity and the medium mediated elastic interaction is enough to form the collective swarming, clustering, and streaming seen in some experiments. The numerical phenomenology is then rationalized using a mean-field hydrodynamic theory.