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

This paper demonstrates that run-and-tumble particles interacting via a double-well potential exhibit a non-unique steady state with multiple stable solutions and symmetry-breaking behaviors in the large-$N$ limit, a phenomenon absent in equilibrium Brownian systems due to the specific interplay between active noise and the interaction potential.

The Gist

Imagine a crowded dance floor where everyone is trying to move to their own rhythm, but they are also bumping into each other. This is the story of Run-and-Tumble Particles (RTPs)—a model used by physicists to understand how tiny, self-propelled things (like bacteria or synthetic micro-robots) behave when they interact.

In this paper, the authors set up a specific scenario to see what happens when these "dancers" have a very specific relationship with one another: they want to be close, but not too close.

Here is the breakdown of their discovery, explained through simple analogies.

1. The Setup: The "Goldilocks" Dance Floor

The particles are moving in a straight line (one dimension). They have a special interaction potential (a force field) that acts like a double-well:

• Too close? They repel each other (like magnets with the same pole facing).
• Too far? They attract each other (like magnets with opposite poles).
• Just right? They settle into a comfortable distance.

In the world of normal, passive particles (like dust motes in a sunbeam), if you wait long enough, they will eventually settle into one single, predictable, symmetrical pattern. There is only one "correct" final state.

2. The Twist: Active Noise Makes Things Weird

These particles, however, are active. They don't just drift; they "run" in a straight line, then randomly "tumble" and change direction. This is like a drunk person walking in a straight line until they stumble and spin around.

The authors discovered that because of this active "tumbling," the system behaves in ways that are impossible for passive particles. Specifically, the final state is not unique.

3. The Big Discovery: The "Split Personality" of the System

The most surprising finding is that depending on how the system starts, it can end up in two completely different stable states, even if the rules of the game (the temperature, the speed, the interaction strength) are exactly the same.

Think of it like this:

• Scenario A: You start with the dancers evenly spread out. They settle into a single, large group in the middle of the room.
• Scenario B: You start with the dancers already split into two groups on opposite sides. They stay split forever, forming two distinct clusters.

In a normal system, Scenario B would eventually melt back into Scenario A. But here, both states are stable. The system has "memory" of how it started. This is called bistability.

4. The "Broken Symmetry" Surprise

Usually, physics loves symmetry. If you have a group of particles, you expect them to split evenly: 50% on the left, 50% on the right.

The authors found that in the "split" state, the particles can do something shocking: they can split unevenly.

• You could have 60% of the particles on the left and 40% on the right.
• Or 70/30.
• Or any ratio in between!

It's like a group of friends deciding to split up for dinner, but instead of splitting into two equal tables, they spontaneously decide to sit at one table with 6 people and another with 4, and they are perfectly happy staying that way forever. The system doesn't care about the "fair" 50/50 split; it cares about the history of how they got there.

5. The "Tumbling Rate" Key

Why does this happen? It depends on how often the particles "tumble" (change direction).

• If they tumble too fast, they act like normal, passive particles. They forget their history and settle into one unique, symmetrical state.
• If they tumble slowly (they have high "persistence"), they get stuck in their current configuration. This persistence allows the "split" and "uneven" states to become permanent.

6. The "Edge" Behavior

The authors also looked at the edges of these particle groups.

• In normal physics, density usually fades away smoothly at the edges.
• Here, the density can spike (go to infinity) or vanish abruptly at the edges, depending on the tumbling rate. It's like the crowd at the edge of the dance floor either suddenly getting crushed together or suddenly disappearing, rather than thinning out gradually.

Summary: Why Does This Matter?

This paper shows that active matter (living things, robots, bacteria) follows different rules than dead matter.

1. History Matters: The final state depends on how you started.
2. Symmetry Breaking: Nature doesn't always choose the "fair" or "even" option.
3. Multiple Realities: The same set of rules can lead to two different stable worlds.

The authors argue that while their math uses a simplified "toy model" (a specific mathematical curve), these weird behaviors likely happen in real life with bacteria or active colloids that have both attractive and repulsive forces. It suggests that in the world of the very small and the very active, uniqueness is a luxury, and chaos (or multiple stable states) is the rule.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium statistical mechanics of a system of $N$ interacting Run-and-Tumble Particles (RTPs) in one dimension. RTPs are active particles that move at a constant speed $v_0$ and randomly switch direction (tumble) at a rate $\gamma$.

The specific system studied consists of $N$ RTPs interacting via a pairwise double-well potential:
$$W(r) = -\frac{k_0}{2}r^2 + \frac{g}{4}r^4$$
where $k_0, g > 0$. This potential is repulsive at short distances and attractive at large distances. The study focuses on the stationary state (steady state) of the system in the limit of large $N$ ($N \to \infty$) and zero thermal noise ($T=0$).

The central problem is to determine the total particle density $\rho_s(x)$ and understand how the nature of the steady state (connected vs. disconnected support, uniqueness vs. multiplicity) depends on the interaction parameters and the tumbling rate. The authors specifically contrast these active systems with passive Brownian particles, where the Gibbs equilibrium is unique.

2. Methodology

The authors employ a combination of analytical self-consistent field theory and numerical simulations.

• Dean-Kawasaki Method Extension: Since the interaction potential satisfies $W'(0)=0$ (allowing particles to cross), the authors utilize the extension of the Dean-Kawasaki method to RTPs. This allows them to derive a closed equation for the time-dependent densities of particles moving right ($\rho_+$) and left ($\rho_-$).
• Self-Consistent Equation: In the limit $N \to \infty$ and $T=0$, the stationary total density $\rho_s(x)$ satisfies a non-linear integro-differential equation:
  $$\rho_s(x) = \frac{K}{v_0^2 - \tilde{F}(x)^2} \exp\left( 2\gamma \int^x dz \frac{\tilde{F}(z)}{v_0^2 - \tilde{F}(z)^2} \right)$$
  where $\tilde{F}(x)$ is an effective force field generated by the density itself: $\tilde{F}(x) = -\int dy W'(x-y)\rho_s(y)$.
• Renormalization: For the specific double-well potential, the effective force simplifies to a cubic polynomial dependent on the second moment $m_2$ of the density. The authors introduce a renormalized interaction parameter $k = k_0 - 3m_2$, which decouples the force calculation from the full density profile initially.
• Symmetry Breaking Analysis: The authors first solve for symmetric solutions ($\rho_s(x) = \rho_s(-x)$) and then investigate the possibility of non-symmetric steady states where the center of mass is fixed ($m_1=0$) but the third moment $m_3 \neq 0$.
• Numerical Validation: Direct stochastic simulations of the equations of motion for $N=100$ particles are performed to validate the analytical predictions, particularly regarding bistability and symmetry breaking.

3. Key Contributions and Results

A. Phase Transition in Support Topology

The stationary density exhibits a transition based on the renormalized parameter $k$:

• Connected Support ($k < k_c$): The density is supported on a single interval $[-y_1, y_1]$.
• Disconnected Support ($k > k_c$): The density splits into two disjoint intervals $[-y_1, y_3] \cup [-y_3, y_1]$. This corresponds to the spontaneous formation of two "active super-molecules" or clusters.
• Critical Point: The transition occurs at $k_c = 3/2^{2/3}$. At this point, the density at the center $x=0$ vanishes with an essential singularity (exponential decay of the form $e^{-1/\sqrt{k_c-k}}$).

B. Non-Uniqueness and Bistability

A major finding is that the mapping between the bare interaction parameter $k_0$ and the renormalized parameter $k$ is not always one-to-one.

• For low tumbling rates ($\gamma < \gamma_c \approx 0.1787$), the function $k_0(k)$ becomes non-monotonic near the transition.
• Consequence: For a specific range of $k_0$, there exist two distinct stable stationary states (bistability): one with a connected support and one with a disconnected support. The system's final state depends on the initial conditions and the realization of the noise.
• This contrasts sharply with passive Brownian systems, which possess a unique Gibbs equilibrium state.

C. Edge Behavior and Convexity

The authors characterize the behavior of the density at the edges of the support ($\pm y_1, \pm y_3$):

• The density can either diverge or vanish algebraically at the edges depending on the relationship between the tumbling rate $\gamma$ and the curvature of the effective force.
• A critical curve $\gamma_1(k)$ separates these regimes. Below this curve, the density diverges at the outer edges; above it, it vanishes.
• Around $x=0$, the density can transition from unimodal to bimodal (local minimum at 0) depending on the sign of $k$ and the value of $\gamma$.

D. Non-Symmetric Steady States (Symmetry Breaking)

In the regime of disconnected support ($k > k_c$), the authors demonstrate that parity symmetry can be spontaneously broken.

• If the initial condition is asymmetric (e.g., more particles on the left than the right), the system can settle into a steady state where the two clusters have unequal populations.
• This state is characterized by a continuous range of values for the third moment $m_3$ (or the fraction of particles $\alpha$ in the right cluster).
• The fraction $\alpha$ can vary continuously within a range $[1/3, 2/3]$ (in the limit $\gamma \to 0$), implying an infinite number of possible steady states for a single set of parameters, determined by the history of the system.

E. Generalization to Realistic Potentials

The authors argue in Appendix C that while their model uses a potential that diverges at infinity (unrealistic), the phenomenology (bound states, transitions to disjoint support, and non-uniqueness) persists for more realistic potentials that decay to zero at infinity, provided the force profile has a similar "repulsive-short/attractive-long" structure.

4. Significance

• Active Matter Physics: This work provides one of the few exact analytical results for interacting active particles in the continuum limit. It highlights that active noise (telegraphic noise) fundamentally alters the phase behavior compared to thermal noise.
• Non-Equilibrium Uniqueness: The paper establishes that non-uniqueness of the steady state is a generic feature of active systems with specific interaction ranges. Unlike equilibrium systems where the state is determined solely by thermodynamic parameters, active systems retain memory of initial conditions and noise history.
• Symmetry Breaking: The demonstration of spontaneous symmetry breaking in 1D active systems (which is forbidden in equilibrium 1D systems with short-range interactions due to the Mermin-Wagner theorem) underscores the unique nature of active matter.
• Methodological Advance: The successful application of the Dean-Kawasaki method to RTPs with crossing particles and the derivation of explicit density profiles for double-well potentials offer a robust framework for studying other active interaction models.

In summary, the paper reveals that a simple model of interacting active particles exhibits rich, complex behavior including bistability, discontinuous transitions, and symmetry breaking, none of which are present in their passive counterparts. These findings suggest that "active super-molecules" can form and persist in multiple stable configurations depending on the system's history.