Persistent motion of a Brownian particle subject to repulsive feedback with time delay

Spontaneous velocity alignment of Brownian particles with feedback-induced propulsion (a)

“…As shown in [24] for the singleparticle case, feedback parameters fulfilling the condition Aτ /γb 2 > 1 lead to a constant, long-time velocity vector v i,∞ of the particle in the deterministic limit. The magnitude of this velocity can be obtained analytically, it is given by v ∞ = ( √ 2b/τ ) − ln (γb 2 /(Aτ )) [24]. In the presence of noise, the direction of the single-particle velocity randomizes on the time scale of a "persistence time" τ eff while the finite magnitude v eff remains, on average, constant in the long-time limit (with a value close to v ∞ ).…”

“…Such a potential could, in principle, be created by optical forces [6,8,9,29,30]; similar time-delayed potentials occur, e.g., when cells move in a viscoelastic medium [3]. As shown in [24] for the singleparticle case, feedback parameters fulfilling the condition Aτ /γb 2 > 1 lead to a constant, long-time velocity vector v i,∞ of the particle in the deterministic limit. The magnitude of this velocity can be obtained analytically, it is given by v ∞ = ( √ 2b/τ ) − ln (γb 2 /(Aτ )) [24].…”

“…Specifically, our model system is composed of repulsively interacting (isotropic) particles subject to nonlinear, time-delayed repulsive feedback. In a previous study [24] on the singleparticle dynamics we have shown that the time delay can actually generate persistent motion: at appropriate feedback parameters, the particle develops a constant effective velocity v eff whose direction randomizes after some persistence time τ eff . Overall, the single-particle dynamics reveals crucial similarities to that of active Brownian particles (ABP) [25], suggesting that time-delayed feedback could indeed be an alternative mechanism to induce selfpropulsion (consistent with very recent experiments [9]).…”

“…The feedback strength A and feedback range b are set to positive constants. Physically, the feedback potential can then be interpreted as a source of a "nudge" following the particle with some delay [24]. Such a potential could, in principle, be created by optical forces [6,8,9,29,30]; similar time-delayed potentials occur, e.g., when cells move in a viscoelastic medium [3].…”

“…In the presence of noise, the direction of the single-particle velocity randomizes on the time scale of a "persistence time" τ eff while the finite magnitude v eff remains, on average, constant in the long-time limit (with a value close to v ∞ ). The resulting mean-squared displacement (MSD) resembles [24] that of an ABP (see the Supplementary Material Supplementarymaterial.pdf (SM) for an example). The particles interact via the (instantaneous) repulsive forces…”