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Register a new userFirst-order condensation transition in the position distribution of a run-and-tumble particle in one dimension
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We consider a single run-and-tumble particle (RTP) moving in one dimension. We assume that the velocity of the particle is drawn independently at each tumbling from a zero-mean Gaussian distribution and that the run times are exponentially distributed. We investigate the probability distribution $P(X,N)$ of the position $X$ of the particle after $N$ runs, with $Ngg 1$. We show that in the regime $ X sim N^{3/4}$ the distribution $P(X,N)$ has a large deviation form with a rate function characterized by a discontinuous derivative at the critical value $X=X_c>0$. The same is true for $X=-X_c$ due to the symmetry of $P(X,N)$. We show that this singularity corresponds to a first-order condensation transition: for $X>X_c$ a single large jump dominates the RTP trajectory. We consider the participation ratio of the single-run displacements as the order parameter of the system, showing that this quantity is discontinuous at $X=X_c$. Our results are supported by numerical simulations performed with a constrained Markov chain Monte Carlo algorithm.
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We study the probability distribution $P(X_N=X,N)$ of the total displacement $X_N$ of an $N$-step run and tumble particle on a line, in presence of a constant nonzero drive $E$. While the central limit theorem predicts a standard Gaussian form for $P(X,N)$ near its peak, we show that for large positive and negative $X$, the distribution exhibits anomalous large deviation forms. For large positive $X$, the associated rate function is nonanalytic at a critical value of the scaled distance from the peak where its first derivative is discontinuous. This signals a first-order dynamical phase transition from a homogeneous `fluid phase to a `condensed phase that is dominated by a single large run. A similar first-order transition occurs for negative large fluctuations as well. Numerical simulations are in excellent agreement with our analytical predictions.
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 consider self-propelled particles undergoing run-and-tumble dynamics (as exhibited by E. coli) in one dimension. Building on previous analyses at drift-diffusion level for the one-particle density, we add both interactions and noise, enabling discussion of domain formation by self-trapping, and other collective phenomena. Mapping onto detailed-balance systems is possible in certain cases.
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 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.
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