Active Ornstein-Uhlenbeck particles

Abstract

Active Ornstein-Uhlenbeck particles (AOUPs) are overdamped particles in an interaction potential subject to external Ornstein-Uhlenbeck noises. They can be transformed into a system of underdamped particles under additional velocity dependent forces and subject to white noise forces. There has been some discussion in the literature on whether AOUPs can be in equilibrium for particular interaction potentials and how far from equilibrium they are in the limit of small persistence time. By using a theorem on the time reversed form of the AOUP Langevin-Ito equations, I prove that they have an equilibrium probability density invariant under time reversal if and only if their smooth interaction potential has zero third derivatives. In the limit of small persistence Ornstein-Uhlenbeck time τ , a Chapman-Enskog expansion of the Fokker-Planck equation shows that the probability density has a local equilibrium solution in the particle momenta modulated by a reduced probability density that varies slowly with the position. The reduced probability density satisfies a continuity equation in which the probability current has an asymptotic expansion in powers of τ . Keeping up to O(τ ) terms, this equation is a diffusion equation, which has an equilibrium stationary solution with zero current. However, O(τ 2 ) terms contain fifth and sixth order spatial derivatives and the continuity equation no longer has a zero current stationary solution. The expansion of the overall stationary solution now contains odd terms in the momenta, which clearly shows that it is not an equilibrium.

Fluctuation–Dissipation Relations in Active Matter Systems

“…We consider a well-known scheme to describe the behavior of self-propelled particles, the Active Ornstein-Uhlenbeck (AOUP) model [39][40][41][42][43][44][45][46] (also known as Gaussian Colored Noise (GCN)). The AOUP has been employed to reproduce the phenomenology of passive colloids immersed in a bath of active particles [47][48][49][50] but also-perhaps at a more approximate level-the dynamics of self-propelled particles themselves.…”

“…Except for these special cases, the probability distribution, p(x, f a ), is unknown. An approximation can be obtained, valid in the proximity of equilibrium (τ = 0) [25,43,61,63]. Therefore, in general, this model of self-propelled particles cannot be treated within the FDR of class A, because the FDR remains implicit:…”

Time-dependent properties of interacting active matter: Dynamical behavior of one-dimensional systems of self-propelled particles

“…We study dense systems of N interacting self-propelled particles at density ρ 0 , employing the active Ornstein-Uhlenbeck particle (AOUP) model [50][51][52][53][54][55][56][57]. The AOUP is a versatile and popular model of active matter that can reproduce many aspects of the phenomenology of self-propelled particles, including the accumulation near rigid boundaries [58][59][60][61][62] or obstacles and the motility induced phase separation (MIPS) [63].…”

Linear Response Theory and Green-Kubo Relations for Active Matter

“…[In contrast to Active Ornstein-Uhlenbeck Particles (AOUP) which fulfil detailed balance in the small drive limit [44,49]].…”

“…For instance, in the dilute limit, ABP behave as an equilibrium ideal gas at a higher T ef f [41][42][43]57]]. Indeed, genuine non-equilibrium behaviour (with no effective equilibrium description) results from the combined effect of interactions and activity, as observed in active colloidal suspensions [58] and proven for AOUP [44,49,59] (among other examples). Incidentally, AOUP behave as an effective equilibrium system when the potential has zero third derivatives, while in our case equilibrium-like response is recovered if the potential has zero second derivatives.…”