Active Ornstein-Uhlenbeck particles.

Area of Science:

• Statistical Mechanics
• Non-equilibrium Thermodynamics

Background:

• Active Ornstein-Uhlenbeck particles (AOUPs) are overdamped systems subjected to interaction potentials and external noises.
• The equilibrium properties of AOUPs, particularly in the limit of small persistence times, are a subject of ongoing research.

Purpose of the Study:

• To determine the conditions under which AOUPs can reach an equilibrium state.
• To analyze the deviation from equilibrium for AOUPs as a function of persistence time.

Main Methods:

• Utilized a theorem on the time-reversed form of Langevin-Ito equations for AOUPs.
• Applied Chapman-Enskog expansion to the Fokker-Planck equation in the limit of small persistence time (τ).

Main Results:

• Proved that equilibrium probability density invariant under time reversal exists if and only if the interaction potential's third derivative is zero.
• Showed that for small τ, the probability density exhibits local equilibrium in momenta, modulated by a position-dependent reduced density.
• Demonstrated that terms up to O(τ) in the probability current expansion lead to a diffusion equation with a zero-current stationary solution.
• Identified that O(τ²) terms introduce higher-order spatial derivatives, resulting in a non-zero current and odd momentum terms, indicating a non-equilibrium state.

Conclusions:

• AOUPs can achieve a specific type of equilibrium only under strict conditions on their interaction potential.
• Deviations from equilibrium become significant at higher orders of the small persistence time expansion, characterized by complex spatial derivatives and non-zero probability currents.