Transport properties of diffusive particles conditioned to survive in trapping environments

Abstract

We consider a one-dimensional Brownian motion with diffusion coefficient D in the presence of n partially absorbing traps with intensity β, separated by a distance L and evenly spaced around the initial position of the particle. We study the transport properties of the process conditioned to survive up to time t. We find that the surviving particle first diffuses normally, before it encounters the traps, then undergoes a period of transient anomalous diffusion, after which it reaches a final diffusive regime. The asymptotic regime is governed by an effective diffusion coefficient D eff, which is induced by the trapping environment and is typically different from the original one. We show that when the number of traps is finite, the environment enhances diffusion and induces an effective diffusion coefficient that is systematically equal to D eff = 2D, independently of the number of the traps, the trapping intensity β and the distance L. On the contrary, when the number of traps is infinite, we find that the environment inhibits diffusion with an effective diffusion coefficient that depends on the traps intensity β and the distance L through a non-trivial scaling function D eff = D F ( β L / D ) , for which we obtain a closed-form. Moreover, we provide a rejection-free algorithm to generate surviving trajectories by deriving an effective Langevin equation with an effective repulsive potential induced by the traps. Finally, we extend our results to other trapping environments.

Run-and-tumble motion in trapping environments

“…A first imple-mentation might be to consider reversible trapping, that is, the possibility for the particle to reactivate itself after being immobilized. Other possible extensions could be the analysis of planar motions [12,[26][27][28][29] or considering more complex environments, such as those described by a continuously variable trapping rate or by a periodic sequence of trapping intervals [4,5]. A final possible direction of investigation might be to consider the effect of trapping in fractional processes, which are known to produce anomalous diffusion in the free case [25,30,31] and whose properties in the presence of absorbing boundaries have been studied in the past [32][33][34] and recently extended to the so called g-fractional diffusion [35].…”

Run-and-tumble motion in trapping environments

“…A first implementation might be to consider reversible trapping, that is, the possibility for the particle to reactivate itself after trapping [43]. Other possible extensions could be the analysis of planar motions [12,[44][45][46][47] or considering more complex environments, such as those described by a continuously variable trapping rate, by a periodic sequence of trapping intervals [4,5] or by the presence of generic boundaries [48]. A final possible direction of investigation might be to consider different combinations of particle motion in the two phases, before and after trapping.…”

“…More recently interesting studies focused on trapping of photokinetic bacteria in structured light fields [1]. Modeling stochastic motion in trapping environments is then of great interest [2][3][4][5]. In particular, the study of active systems, composed by self-propelled particles, can give us a very general view of the process, which applies to many interesting physical and biological phenomena [6,7], allowing diffusive motion to be obtained as a limiting case.…”