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

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.

💡 Research Summary

The manuscript presents a comprehensive microscopic derivation of a continuum field theory for active Brownian particles that possess both translational and rotational inertia. Starting from the under‑damped Langevin equations for a two‑dimensional ensemble of N particles, the authors include mass m, moment of inertia I, translational friction γ, rotational friction γ_R, a self‑propulsion force of magnitude v₀ directed along the unit vector û(φ), and Gaussian translational and rotational noises. By constructing the corresponding Fokker‑Planck operator, they integrate out all but one particle to obtain a one‑body kinetic equation for the probability density P₁(r,p,û,l).

A central methodological step is the factorization approximation for the two‑body distribution, P₂≈P₁P₁ g(r,r′,û,û′), together with a generalized local equilibrium ansatz for P₁. The latter assumes that the one‑particle distribution can be written as a Maxwell‑Boltzmann form with locally defined temperature T(r,û), velocity v(r,û) and angular velocity w(r,û). Importantly, the authors demonstrate that even though inertial active particles exhibit non‑trivial velocity correlations, these correlations are fully captured by the product of the local velocity field and the pair‑distribution function, thereby justifying the factorization in the inertial regime.

Using the local equilibrium form, the authors derive the hierarchy of macroscopic balance equations: mass conservation, momentum balance, angular momentum balance, and an energy equation. Each equation contains interaction contributions that involve the two‑body pair distribution g and the interaction potential U₂. To make these terms tractable, the authors perform a combined Fourier series expansion in the relative orientation angles (θ₁, θ₂) and a gradient expansion in the inter‑particle distance r. This yields explicit expressions for the translational interaction term I_trans and the rotational interaction term I_rot as series with coefficients A_i and B_i, respectively, which are integrals over the radial dependence of the pair correlation functions weighted by derivatives of the interaction potential.

A novel aspect of the work is the systematic inclusion of additional hydrodynamic fields beyond density and velocity. By expanding the one‑particle distribution in orientational harmonics, the authors introduce the polarization P(r), velocity polarization v_P(r), and angular‑velocity polarization W(r). These fields are essential for describing phenomena unique to inertial active matter, such as the coexistence of temperature gradients between coexisting phases and the coupling between particle orientation and kinetic energy. The final set of continuum equations (68)–(70) therefore contains, besides the usual advective and pressure terms, contributions from the newly defined polarizations, non‑symmetric stress tensors, and the A_i/B_i interaction coefficients.

The paper concludes by discussing the applicability of the factorization and local equilibrium approximations in inertial systems, showing that they remain mathematically consistent and physically meaningful. The derived theory provides a unified framework that can be directly compared with experiments on macroscopic robotic swarms, ultracold atomic gases with engineered self‑propulsion, and other systems where inertia cannot be neglected. By offering explicit expressions for the interaction coefficients, the work opens the door to quantitative validation against particle‑based simulations and to the systematic exploration of novel nonequilibrium thermodynamic effects in active matter with inertia.