Boundary layers, transport and universal distribution in boundary driven active systems

TL;DR

This work analyzes a one-dimensional boundary-driven RTP in contact with particle reservoirs, uncovering kinetic boundary layers, current without a density gradient, and current reversal induced by tuning activity and diffusion. By formulating the steady-state and time-dependent problems, the authors derive explicit profiles $P(x)$ and $Q(x)$, a diffusion-dominated bulk with an activity-induced magnetisation $Q_b$, and a Milne length $l_M$ that renormalizes the effective system size. The time-dependent problem is solved via a two-band eigenspectrum separated by the tumble rate $\\omega$, revealing a crossover from diffusive to tumble-dominated relaxation as $L$ decreases, and showing that the large-time distribution in the bulk retains a strong active contribution, especially at high persistence. A proposed universality for the large-time distribution in absorbing-boundary problems with short-range colored noise is presented, connecting RTP, AOUP, and ABP dynamics and suggesting a common framework for diverse nonequilibrium transport phenomena in active matter.

Abstract

We discuss analytical results for a run-and-tumble particle (RTP) in one dimension in presence of boundary reservoirs. It exhibits `kinetic boundary layers', nonmonotonous distribution, current without density gradient, diffusion facilitated current reversal and optimisation on tuning dynamical parameters, and a new transport effect in the steady state. The spatial and internal degrees of freedom together possess a symmetry, using which we find the eigenspectrum for large systems. The eigenvalues are arranged in two bands which can mix in certain conditions resulting in a crossover in the relaxation. The late time distribution for large systems is obtained analytically; it retains a strong and often dominant `active' contribution in the bulk rendering an effective passive-like description inadequate. A nontrivial `Milne length' also emerges in the dynamics. Finally, a novel universality is proposed in the absorbing boundary problem for dynamics with short-range colored noise. Active processes driven by active reservoirs may thus provide a common physical ground for diverse and new nonequilibrium phenomena.

Figures (10)

• Figure 1: Schematic of the boundary driven active system. An RTP moves on a one-dimensional line bounded between $x=0$ and $L$. The system is connected to particle reservoirs at both the ends maintaining fixed particle densities $P_0$ and $P_1$ and magnetisation $Q_0$ and $Q_1$ at $x=0,\,L$ respectively.
• Figure 2: Left panel: Density (solid line) and magnetisation (dashed line) profiles in the steady state of the boundary driven RTP. Here $v=1.0,D=1.0,w=0.3,L=10.0$ and $P_0=0.075,P_1=0.125,Q_0=Q_1=0.05$. The distribution is nonmonotonous and the slope in the bulk is negative, that corresponds to a current from the left to the right. Right panel: Density and magnetisation plots with the same boundary conditions and system size, but now with particle parameters $v=1.0,D=1.0,w=5.0$. The density profile is monotonous in this case and the slope in the bulk is positive corresponding to a current in the reverse direction. The boundary layers can be identified in both the cases.
• Figure 3: Nonmonotonicity of the steady state current. Left panel: Diffusivity $D$ is tuned. Here $v=1.0,w=0.5,L=200.0$, $P_0=Q_0=0.002,P_1=Q_1=0.004$. A current reversal also occurs at higher values of $D$. Right panel: Tumble rate $\omega$ is tuned. Here $v=1.0,D=1.0,L=200.0$, $P_0=0.002,P_1=0.006,Q_0=Q_1=0.001$. (Note that in both panels the $y$-axis is amplified $10^4$ times for better visibility.)
• Figure 4: Steady state for $Q_0=Q_1=0 :$ (a) the probability density; (b) magnetisation for different tumble rate $\omega$. Here $L=10,v=1.0,D=1.0$ and the boundary densities are $P_0=0.15,P_1=0.02$. Points are obtained by simulating the exit probabilities and using Eq. \ref{['den-with-exit']}.
• Figure 5: The symmetry of the dynamics. The master equations and the boundary conditions remain invariant under $\sigma \rightarrow - \sigma$ along with $x\rightarrow L-x$.