Microscopic field theory for active Brownian particles with translational and rotational inertia

TL;DR

The paper addresses the need for a microscopic continuum theory of inertial active matter that includes translational inertia, rotational inertia, temperature, and multiple orientational fields. It builds a continuum description from microscopic Langevin dynamics using the interaction-expansion method and a generalized local-equilibrium ansatz, yielding coupled evolution equations for fields such as the density $\rho$, velocity $\mathbf{v}$, angular velocity $\omega$, temperature $T$, polarization $\mathbf{P}$, velocity polarization $\underline{\boldsymbol{v}}_{\vec{P}}$, and angular-velocity polarization $\vec{W}$. A key result is that a factorization approximation remains viable in the inertial regime when velocity correlations are incorporated via local fields and a time-independent pair distribution $g$, with translational and rotational interactions captured by coefficients $A_i$ and $B_i$ derived from pair correlations. The framework thus provides a comprehensive, albeit complex, set of hydrodynamic-like equations for inertial active systems and clarifies when common approximations are permissible, offering pathways to practical limiting-case models for applications in robotics, dusty plasmas, and quantum active matter.

Abstract

While active matter physics has traditionally focused on particles with overdamped dynamics, recent years have seen an increase of experimental and theoretical work on active systems with inertia. This also leads to an increased need for theoretical models that describe inertial active dynamics. Here, we present a microscopic derivation for a general continuum model describing the nonequilibrium thermodynamics of inertial active matter that generalizes several previously existing works. It applies to particles with translational and rotational inertia and contains particle density, velocity, angular velocity, temperature, polarization, velocity polarization, and angular velocity polarization as dynamical variables. We moreover discuss to which extend commonly used approximations (factorization and local equilibrium) used in the derivation of hydrodynamic models are applicable to inertial active matter.