Non-uniqueness of the steady state for run-and-tumble particles with a double-well interaction potential

TL;DR

The paper tackles the steady-state behavior of one-dimensional run-and-tumble particles with a double-well interaction that yields bound states. Using a self-consistent equation for the total density rho_s(x) in the large-N limit, the authors derive a renormalized interaction parameter k = k_0 - 3 m_2 and show a transition from a connected to a disconnected support, along with edge-scaling and convexity properties. A key finding is the non-uniqueness of stationary states: for sufficiently small tumbling rate gamma there exists bistability with two stable densities corresponding to distinct effective parameters k, and in the disconnected phase asymmetric steady states with nonzero m_3 can occur, depending on initial conditions. These results are corroborated by numerical simulations at N = 100 particles, highlighting qualitative differences from passive Brownian systems and offering insight into active matter with competing short-range repulsion and long-range attraction.

Abstract

We study $N$ run-and-tumble particles (RTPs) in one dimension interacting via a double-well potential $W(r)=-k_0 \, r^2/2+g \, r^4/4$, which is repulsive at short interparticle distance $r$ and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state $ρ_s(x)$, in the limit $N\to+\infty$. We obtain an explicit expression for $ρ_s(x)$ as a function of the ''renormalized" interaction parameter $k=k_0-3m_2$ where $m_2$ is the second moment of $ρ_s(x)$. Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of $k$, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around $x=0$. Furthermore, we find that the mapping $k_0\to k$ becomes multi-valued below a certain value of the tumbling rate $γ$ of the RTPs for some range of values of $k_0$ near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density $ρ_s(x)$ is not symmetric. All our analytical predictions are in good agreement with numerical simulations already for systems of $N = 100$ particles. The non-uniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part, but which decay at infinity.

Figures (11)

• Figure 1: Left: Plot of the absolute values of the real roots of \ref{['cubic_eq']}, $y_1$, $-y_2$ and $-y_3$, as a function of $k$. The largest root $y_1$ is real positive for any $k$ and is a continuous function of $k$. The roots $y_2$ and $y_3$ are real only for $k>k_c$ and are both negative, with $0<-y_2<-y_3<y_1$. Right: Diagram showing the different regimes in the behavior of the density $\rho_s(x)$, in the $(k,\gamma)$-plane. Each regime is illustrated by an inset plot of $\rho_s(x)$ obtained from the analytical expression for $N\to \infty$. The vertical red line at $k=k_c=3/2^{2/3}=1.88988...$ marks the transition from a connected support for $k<k_c$ to a disconnected support for $k>k_c$. The blue curves indicate a change in the edge behavior, from diverging to vanishing (at the exterior edges $\pm y_1$ for $\gamma_1(k)$ and at the interior edges $\pm y_3$ for $\gamma_3(k)$). The curve $\gamma_1(k)$ has a minimum at $k^*=-3/2^{4/3}=-1.19055...$, corresponding to $\gamma^*=3/2^{1/3}=2.38110...$, below which $\rho_s(x)$ always diverges at $\pm y_1$. Finally, the black lines $k=0$ and $k=-\gamma$ indicate a change of convexity of $\rho_s(x)$ around $x=0$. We also indicate the important intersection values $\gamma_1(0)=3$ and $\gamma_1(k_c)=9/2^{2/3}=5.66964...$ . The blue curve $\gamma_1(k)$ asymptotically coincides with the line $\gamma=-k$ as $k\to-\infty$, while for $k\to+\infty$, both $\gamma_1(k)$ and $\gamma_2(k)$ are asymptotically equal to $2k$. Note that this diagram is represented as a function of the renormalized parameter $k$, hence one should also take into account the mapping $k_0\to k$, which as discussed in the text is multi-valued in a small region (small $\gamma$ and $k$ near $k_c$).
• Figure 2: Plot of $\tilde{F}(x)$ (black curve) in the 3 different regimes: $\Delta<0$ (left), $\Delta>0$ (center) and $\Delta=0$ (right). The regions hatched in red are inaccessible to the particles in the stationary state. Particles in the $+$ state move to the right when the curve is above the line $-v_0$ (in blue), and to the left when it is below. Particles in the $-$ state move to the right when the curve is above the line $+v_0$ (in blue), and to the left when it is below. We recall that the particles switch between the states $+$ and $-$ with rate $\gamma$. The arrows on the bottom blue line indicate the direction of motion of $+$ particles, while the arrows on the top blue line indicate the direction of motion of $-$ particles. Crosses indicate points where the total velocity vanishes without changing sign, meaning that the particle would take an infinite time to reach this point (thus a tumbling event always occurs before it reaches it).
• Figure 3: Plots of the density $\rho_s(x)$ for different values of $k$ (corresponding to different regimes in the diagram of Fig. \ref{['phase_diagram_doublewell']} and for $\gamma=2$ (left) and $\gamma=6$ (right) (with $g=1$, $v_0=1$ and $x_{init}=1$). The plots in light colors show the theoretical prediction (given by \ref{['rhos_disjoint']} or \ref{['rhos_joint']}) while the darker lines were obtained by simulating the stochastic dynamics for $N=100$ particles and averaging the histogram of positions in the steady state. For each value of $k$, the corresponding value of $k_0$ to be used in the simulations was computed numerically using the relation \ref{['def_k']}.
• Figure 4: Top left: Plot of $m_2$ versus $k$ for different values of $\gamma$. In the limit $\gamma\to0^+$ (black curve), $m_2(k)$ has a simple expression given in \ref{['m2_gamma0']}. In this case, the function $m_2(k)$ is discontinuous at $k_c$, but it is smooth for any $\gamma>0$. Top right: Same plot zoomed around $k_c$, and for values of $\gamma$ close to $\gamma_c$. While there seems to be a cusp at $k_c$ for small $\gamma>0$ when looking at the large scales, the function $m_2(k)$ appears to be smooth when zooming sufficiently. Bottom left: Plot of $k$ versus $k_0$. When $\gamma$ is smaller than some critical value $\gamma_c= 0.1787369...$, the function $k_0(k)$ becomes non-monotonous close to $k_c$. The inverse function $k(k_0)$ thus becomes multi-valued, leading to the coexistence of 2 stable and one unstable steady states for the same value of $k_0$, corresponding to different values of $k$. The value $\gamma_c=0.1787369...$ was computed numerically, using as a criterion that the minimum of $\frac{dk_0}{dk}$ vs $k$ vanishes at $\gamma_c$ (it is negative for $\gamma<\gamma_c$ and positive for $\gamma>\gamma_c$). For $\gamma=0^+$, the bistability occurs in the interval $k_0\in [6.614585...,9.449407...]$. Bottom right: Same plot zoomed around $k_c$, and for values of $\gamma$ close to $\gamma_c$. The region where the function $k_0(k)$ is non-monotonous seems to be entirely located at $k<k_c$.
• Figure 5: Example where two steady states are observed in simulations for the same values of the parameters ($k_0=8$, $\gamma=0.01$, $g=1$, $v_0=1$ and $N=100$). The two densities were obtained using the same uniform initial condition with support $[-1,1]$. One of steady states (in blue) has a single support, while the other one (in orange) has a disjoint support. For each of the two steady states, the measured values of $m_2$ and $k$ ($k_{em}$ in the legend of the figure) coincide quite well with the two possible values predicted theoretically ($k_{\rm th}$ in the legend). The lighter lines show the two theoretical predictions for $N\to+\infty$ for the two possible values of $k$, which for such a small value of $\gamma$ are very to delta peaks at $x=\pm y_1$, and at $x= \pm y_3$ in the case of the disjoint support.