Escape rate of active particles in the effective equilibrium approach

Abstract

The escape rate of a Brownian particle over a potential barrier is accurately described by the Kramers theory. A quantitative theory explicitly taking the activity of Brownian particles into account has been lacking due to the inherently out-of-equilibrium nature of these particles. Using an effective equilibrium approach [Farage et al., Phys. Rev. E 91, 042310 (2015)] we study the escape rate of active particles over a potential barrier and compare our analytical results with data from direct numerical simulation of the colored noise Langevin equation. The effective equilibrium approach generates an effective potential that, when used as input to Kramers rate theory, provides results in excellent agreement with the simulation data. DOI: 10.1103/PhysRevE.95.012115 The escape of a Brownian particle over a potential barrier is a thermally activated process. Kramers theory accurately describes the escape process by taking into account the force acting on a particle due to the confining potential and solvent-induced Brownian motion. Kramers showed that in the limit of vanishing particle flux across the barrier, the escape rate decreases exponentially with increasing barrier height [1]. In contrast to Brownian particles, active particles undergo both Brownian motion and a self-propulsion, which requires a continual consumption of energy from the local environment [2][3][4][5]. Due to self-propulsion, active particles are expected to escape a potential barrier at a higher rate than their passive counterparts. However, a quantitative description of their escape rate, explicitly taking the activity into account has been lacking. Active particles, in general, have coupled orientational and positional degrees of freedom [6,7]. This makes the theoretical treatment of escaping active particles over a potential barrier a difficult problem as shown in Ref. [8], in which the authors explicitly considered the orientational diffusion of self-propelled particles.In this paper we show that a Kramers-like rate expression can be obtained for a closely related model system of active particles in which the velocities are represented by a stochastic variable and the orientations are not considered explicitly. For small activity the steady-state properties obtained from this model exhibit intriguing similarities with an equilibrium system [9] and several sedimentation and trapping problems are analytically tractable on the single-particle level [10]. As a starting point for a theoretical treatment of the nonstationary case we will employ this model in the form of a coarse-grained Langevin equation for the particle position [6,7,11] with activity of particles appearing as a colored-noise term. It is important to note that the colored-noise Langevin equation describes a non-Markovian process and thus cannot yield an exact Fokker-Planck equation.The colored-noise Langevin equation serves as the basis for effective equilibrium approaches that map an active system to a passive equilibrium system with modified interaction potential and an...

Effective equilibrium states in mixtures of active particles driven by colored noise

“…This procedure yields an approximate Smoluchowski equation, which, in the steady state, admits an analytic solution for the configurational probability distribution [45] and closed formulas for active pressure and interfacial tension [46][47][48]. The former allows us to define effective interaction potentials, which can be directly used to determine density profiles [44,49,50] and rate equations [51] of individual ideal particles and, when implemented in equilibrium liquid-state theory, the structure and phase behavior of interacting systems [11,52,53]. The described effective equilibrium approach is most accurate in one spatial dimension and for small persistence time, which can be explicitly verified by studying exactly solvable models [48].…”

Escape rate of transiently active Brownian particle in one dimension

“…The effective equilibrium approach [2,16] describes ABPs in an external potential. In this approach, one obtains an approximate Fokker-Planck equation with an effective external potential φ eff (x) and an effective position-dependent diffusion constant D(x):…”

“…The expressions in Eqs. (5) and (6) were obtained under the assumption [2,16,18] that the stochastic process corresponding to time evolution of the orientation vector can be considered as a Gaussian noise process with a finite correlation time. However, the mapping is only approximate because the process is not Gaussian.…”

“…As we show below the approach also yields accurate description in the case of transient activity. For the sake of completeness, the expression for the escape rate, r act , for a constant self-propelled velocity, obtained in the effective equilibrium approach is given as [2] r act = r pass exp…”

Learning dynamical information from static protein and sequencing data

“…Such extensions will require replacing Eq. (2) through suitable generalized rate formulas, as have been derived for correlated noise 1, 81 . Conversely, the present framework provides a means to test for diffusive dynamics: if the MFPTs of an observed system differ markedly from those inferred by the above protocol, then either important degrees of freedom have not been measured; the system is out of equilibrium on measurement time scales; or the system does not have Brownian transition statistics, necessitating further careful investigation of its time dependence.…”