Comparison between Smoluchowski and Boltzmann approaches for self-propelled rods

TL;DR

This paper compares continuum equations derived from Smoluchowski and Boltzmann frameworks for self-propelled rods, highlighting their similarities, differences, and the need for advanced closure schemes in active matter modeling.

Contribution

It provides a detailed comparison of continuum equations from both approaches and clarifies the origin of additional terms when interaction forces are included.

Findings

Continuum equations are structurally similar for point-like particles with alignment.

Adding interaction forces introduces new terms in the Smoluchowski framework.

Standard closure schemes may be insufficient for active systems, requiring more complex approaches.

Abstract

Considering systems of self-propelled polar particles with nematic interactions ("rods"), we compare the continuum equations describing the evolution of polar and nematic order parameters, derived either from Smoluchowski or Boltzmann equations. Our main goal is to understand the discrepancies between the continuum equations obtained so far in both frameworks. We first show that in the simple case of point-like particles with only alignment interactions, the continuum equations obtained have the same structure in both cases. We further study, in the Smoluchowski framework, the case where an interaction force is added on top of the aligning torque. This clarifies the origin of the additional terms obtained in previous works. Our observations lead us to emphasize the need for a more involved closure scheme than the standard normal form of the distribution when dealing with active systems.