Jamming of multiple persistent random walkers in arbitrary spatial dimension

Abstract

We consider the persistent exclusion process in which a set of persistent random walkers interact via hard-core exclusion on a hypercubic lattice in d dimensions. We work within the ballistic regime whereby particles continue to hop in the same direction over many lattice sites before reorienting. In the case of two particles, we find the mean first-passage time to a jammed state where the particles occupy adjacent sites and face each other. This is achieved within an approximation that amounts to embedding the one-dimensional system in a higher-dimensional reservoir. Numerical results demonstrate the validity of this approximation, even for small lattices. The results admit a straightforward generalization to dilute systems comprising more than two particles. A self-consistency condition on the validity of these results suggest that clusters may form at arbitrarily low densities in the ballistic regime, in contrast to what has been found in the diffusive limit.

From a microscopic solution to a continuum description of active particles with a recoil interaction

“…Perhaps the simplest interaction is hard-core exclusion, which in passive (equilibrium) systems serves only to reduce the volume available to particles to explore: the statistical weights of the accessible microstates remain unchanged. By contrast, a hard-core repulsion between persistent particles induces an effective attraction [36] which can lead to particles clustering [37][38][39][40][41], consistent with the prediction of motility-induced phase separation [42].…”

“…We now solve (41) for an arbitrary combination of the probabilities u, v, w and shock distribution ρ(x) in (38) and (39). Suppose first of all that we have obtained a solution u p (x) of the equation…”

Dispersal and organization of polarized cells: non-linear diffusion and cluster formation without adhesion

“…We limit cell-cell interactions to exclusion effects in order to examine the interplay between polarization and cell density and their effects on cellular organization. Similar premises were also considered before: in absence of depolarization of cells before direction changes in [32], and in the ballistic regime for diluted systems [33]. Migration of cancer cells with adhesion has been addressed before and compared with experiments on gap junctions [34] and on formation of deformable aggregates with proliferation [35].…”

Time to reach the maximum for a stationary stochastic process

“…Substituting the expressions for P st (x 0 ), GM (M − , s|x 0 ), and QM (M − , s), respectively given in Eqs. ( 83), (86), and (89), into the formula for P (t m |T ) in Eq. ( 41), we get…”

“…More recently, this process was exploited to describe the persistent motion of a class of bacteria, including E. coli [81], which move along a fixed direction (they "run"), randomizing their orientation (they "tumble") at random times. Quite remarkably, such a simple model displays several nontrivial features, including clustering at the boundaries in a confining domain [82], non-Boltzmann steady-state distributions [83,84], and jamming [85,86].…”