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Register a new userRun-and-tumble particles on a line with a fertile site
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We propose a model of run-and-tumble particles (RTPs) on a line with a fertile site at the origin. After going through the fertile site, a run-and-tumble particle gives rise to new particles until it flips direction. The process of creation of new particles is modelled by a fertility function (of the distance to the fertile site), multiplied by a fertility rate. If the initial conditions correspond to a single RTP with even probability density, the system is parity-invariant. The equations of motion can be solved in the Laplace domain, in terms of the density of right-movers at the origin. At large time, this density is shown to grow exponentially, at a rate that depends only on the fertility function and fertility rate. Moreover, the total density of RTPs (divided by the density of right-movers at the origin), reaches a stationary state that does not depend on the initial conditions, and presents a local minimum at the fertile site.
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Active Brownian particles (ABPs) and Run-and-Tumble particles (RTPs) both self-propel at fixed speed $v$ along a body-axis ${bf u}$ that reorients either through slow angular diffusion (ABPs) or sudden complete randomisation (RTPs). We compare the physics of these two model systems both at microscopic and macroscopic scales. Using exact results for their steady-state distribution in the presence of external potentials, we show that they both admit the same effective equilibrium regime perturbatively that breaks down for stronger external potentials, in a model-dependent way. In the presence of collisional repulsions such particles slow down at high density: their propulsive effort is unchanged, but their average speed along ${bf u}$ becomes $v(rho) < v$. A fruitful avenue is then to construct a mean-field description in which particles are ghost-like and have no collisions, but swim at a variable speed $v$ that is an explicit function or functional of the density $rho$. We give numerical evidence that the recently shown equivalence of the fluctuating hydrodynamics of ABPs and RTPs in this case, which we detail here, extends to microscopic models of ABPs and RTPs interacting with repulsive forces.
We simulate by lattice Boltzmann the nonequilibrium steady states of run-and-tumble particles (inspired by a minimal model of bacteria), interacting by far-field hydrodynamics, subject to confinement. Under gravity, hydrodynamic interactions barely perturb the steady state found without them, but for particles in a harmonic trap such a state is quite changed if the run length is larger than the confinement length: a self-assembled pump is formed. Particles likewise confined in a narrow channel show a generic upstream flux in Poiseuille flow: chiral swimming is not required.
We study two interacting identical run and tumble particles (RTPs) in one dimension. Each particle is driven by a telegraphic noise, and in some cases, also subjected to a thermal white noise with a corresponding diffusion constant $D$. We are interested in the stationary bound state formed by the two RTPs in the presence of a mutual attractive interaction. The distribution of the relative coordinate $y$ indeed reaches a steady state that we characterize in terms of the solution of a second-order differential equation. We obtain the explicit formula for the stationary probability $P(y)$ of $y$ for two examples of interaction potential $V(y)$. The first one corresponds to $V(y) sim |y|$. In this case, for $D=0$ we find that $P(y)$ contains a delta function part at $y=0$, signaling a strong clustering effect, together with a smooth exponential component. For $D>0$, the delta function part broadens, leading instead to weak clustering. The second example is the harmonic attraction $V(y) sim y^2$ in which case, for $D=0$, $P(y)$ is supported on a finite interval. We unveil an interesting relation between this two-RTP model with harmonic attraction and a three-state single RTP model in one dimension, as well as with a four-state single RTP model in two dimensions. We also provide a general discussion of the stationary bound state, including examples where it is not unique, e.g., when the particles cannot cross due to an additional short-range repulsion.
Run-and-tumble dynamics is a wide-spread mechanism of swimming bacteria. The accumulation of run-and-tumble microswimmers near impermeable surfaces is studied theoretically and numerically in the low-density limit in two and three spatial dimensions. Both uni-modal and exponential distributions of the run lengths are considered. Constant run lengths lead to {peaks and depletions regions} in the density distribution of particles near the surface, in contrast to {exponentially-distributed run lengths}. Finally, we present a universal accumulation law for large channel widths, which applies not only to run-and-tumble swimmers, but also to many other kinds of self-propelled particles.
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 space-time 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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