Active Brownian Dynamics from a Hamiltonian Model: Transitioning from Equilibrium to Activity

We introduce a Hamiltonian Active Brownian Particle (HABP) model that connects equilibrium dynamics with the non-equilibrium, two-dimensional overdamped behavior of standard Active Brownian Particles (ABPs). In equilibrium, the system follows overdamped Langevin equations that strictly satisfy the fluctuation-dissipation theorem. Coupling translational and rotational degrees of freedom to separate heat baths ($T_θ > T_{\textrm{tr}}$), drives the system out of equilibrium. In free space, matching the diffusion coefficients yields quantitative agreement with the ABP model in the limit $T_θ/T_{\textrm{tr}}\to \infty$, whereas in a harmonic potential, this matching is recovered even at finite temperature ratios. Entropy production analysis demonstrates that in the ABP limit, all energy injected by the swim force dissipates into the translational bath, leaving the rotational bath as a zero-cost entropy source. These insights provide the first steps toward equilibrium-inspired descriptions of active matter, facilitating the development of new theoretical frameworks to study activity-induced fluctuations, transport, and phase behavior in living and non-living soft matter systems far from equilibrium.