The asymptotic speed of reaction fronts in active reaction–diffusion systems

Demaerel, Thibaut; Maes, Christian

doi:10.1088/1751-8121/ab1d8d

Cited by 7 publications

(13 citation statements)

References 29 publications

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“…For later purposes, we also provide an explicit expression for the generating function q(x, r) defined in (25). Using the relation (26) and the result in (41) we get, e.g. for x = 0…”

Section: Survival Probability Of a Two-state Persistent Random Walkmentioning

confidence: 99%

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Survival probability of a run-and-tumble particle in the presence of a drift

De Bruyne,

Majumdar,

Schehr

2021

Preprint

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We consider a one-dimensional run-and-tumble particle, or persistent random walk, in the presence of an absorbing boundary located at the origin. After each tumbling event, which occurs at a constant rate γ, the (new) velocity of the particle is drawn randomly from a distribution W (v). We study the survival probability S(x, t) of a particle starting from x ≥ 0 up to time t and obtain an explicit expression for its double Laplace transform (with respect to both x and t) for an arbitrary velocity distribution W (v), not necessarily symmetric. This result is obtained as a consequence of Spitzer's formula, which is well known in the theory of random walks and can be viewed as a generalization of the Sparre Andersen theorem. We then apply this general result to the specific case of a two-state particle with velocity ±v 0 , the so-called persistent random walk (PRW), and in the presence of a constant drift µ and obtain an explicit expression for S(x, t), for which we present more detailed results. Depending on the drift µ, we find a rich variety of behaviours for S(x, t), leading to three distinct cases: (i) subcritical drift −v 0 < µ < v 0 , (ii) supercritical drift µ < −v 0 and (iii) critical drift µ = −v 0 . In these three cases, we obtain exact analytical expressions for the survival probability S(x, t) and establish connections with existing formulae in the mathematics literature. Finally, we discuss some applications of these results to record statistics and to the statistics of last-passage times.

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“…For later purposes, we also provide an explicit expression for the generating function q(x, r) defined in (25). Using the relation (26) and the result in (41) we get, e.g. for x = 0…”

Section: Survival Probability Of a Two-state Persistent Random Walkmentioning

confidence: 99%

“…Interestingly, active particles also display quite rich behaviors, already at the level of a single particle or of noninteracting RTP's. These include non-trivial density profiles [31][32][33][34][35][36][37], dynamical phase transitions [38,39] or anomalous transport properties [38,[40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Survival probability of a run-and-tumble particle in the presence of a drift

De Bruyne,

Majumdar,

Schehr

2021

Preprint

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“…In the stationary phase, for δ ≥ 1, all particles eventually perish, there is no TF solution and the system reaches a stationary state. For δ < 1 there are two distinct phases [51,52]: A persistent phase for δ + γ < 1/2 where the velocity is constant and given by v0 and an intermittent phase for δ + γ > 1/2 where the velocity vi is given in Eq. ( 6).…”

Section: A Model and Observablesmentioning

confidence: 99%

“…In this article we investigate the EVS of an evolving colony of active matter by revisiting the 1D branching run-andtumble particle model considered in [51,52]. A phase transition has been uncovered in the velocity of the front describing the EVS of the process by studying the evolution of the density of particles [52]. However, the physical origin of this phase transition and its impact on the overall shape of the travelling front have not been analysed.…”

Section: Introductionmentioning

confidence: 99%

Extreme value statistics for branching run-and-tumble particles

Lacroix-A-Chez-Toine,

Miron

2020

Preprint

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The extreme value statistics of active matter offer significant insight into their unique properties. A phase transition has recently been reported in a model of branching run-and-tumble particles, describing the spatial spreading of an evolving colony of active matter in one-dimension. In a "persistent" phase, the particles form macroscopic robust clusters that ballistically propagate as a whole while in an "intermittent" phase, particles are isolated instead. We focus our study on the fluctuations of the rightmost position xmax(t) reached by time t for this model. At long time, as the colony progressively invades the unexplored region, the cumulative probability of xmax(t) is described by a travelling front. The transition has a remarkable impact on this front. In the intermittent phase it is qualitatively similar to the front satisfying the Fisher-KPP equation, which famously describes the extreme value statistics of the non-active branching Brownian motion. A dramatically different behaviour appears in the persistent phase, where activity imparts the front with unexpected and unusual features which we compute exactly.

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“…Even for such noninteracting systems, a plethora of interesting phenomena have been observed, arising purely from the "active nature" of the driving noise. Such phenomena include, e.g., non-trivial density profiles [39][40][41][42][43][44][45][46], dynamical phase transitions [47][48][49], anomalous transport properties [48,[50][51][52], or interesting first-passage and extremal statistics [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71].…”

Section: Introductionmentioning

confidence: 99%

Generating constrained run-and-tumble trajectories

De Bruyne,

Majumdar,

Schehr

2021

Preprint

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We propose a method to exactly generate bridge run-and-tumble trajectories that are constrained to start at the origin with a given velocity and to return to the origin after a fixed time with another given velocity. The method extends the concept of effective Langevin equations, valid for Markovian stochastic processes such as Brownian motion, to a non-Markovian stochastic process driven by a telegraphic noise, with exponentially decaying correlations. We obtain effective spacetime dependent tumbling rates that implicitly accounts for the bridge constraint. We extend the method to other types of constrained run-and-tumble particles such as excursions and meanders. The method is implemented numerically and is shown to be very efficient.

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The asymptotic speed of reaction fronts in active reaction–diffusion systems

Cited by 7 publications

References 29 publications

Survival probability of a run-and-tumble particle in the presence of a drift

Survival probability of a run-and-tumble particle in the presence of a drift

Extreme value statistics for branching run-and-tumble particles

Generating constrained run-and-tumble trajectories

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