🔬 condensed matter

Crystal to liquid cross-over for active particles with inverse-square power-law interaction

This paper investigates a one-dimensional system of run-and-tumble particles with inverse-square repulsion in a harmonic trap, revealing a distinct activity-driven crossover from a crystal-like state with sharp density peaks to a liquid-like Wigner semi-circle profile, supported by both analytical covariance calculations and numerical simulations.

Original authors: Saikat Santra, Leo Touzo, Chandan Dasgupta, Abhishek Dhar, Suman Dutta, Anupam Kundu, Pierre Le Doussal, Gregory Schehr, Prashant Singh

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Saikat Santra, Leo Touzo, Chandan Dasgupta, Abhishek Dhar, Suman Dutta, Anupam Kundu, Pierre Le Doussal, Gregory Schehr, Prashant Singh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where tiny objects are not pushed by invisible thermal jitters, but by their own internal engines. These are active particles, a class of matter found in everything from swimming bacteria to self-propelled robots. Unlike ordinary dust motes that drift aimlessly, these particles consume energy to move in a specific direction for a while before randomly changing course. This constant, self-driven motion breaks the usual rules of equilibrium, creating strange and complex behaviors that scientists are only beginning to understand. When many of these active particles are crowded together and confined, they do not simply mix into a uniform soup. Instead, they can organize into distinct patterns, shifting between solid-like structures and fluid-like flows in ways that passive matter never does.

A team of researchers has now mapped out exactly how this transformation happens in a specific, controlled setting. They studied a one-dimensional line of particles trapped inside a bowl-shaped force field, where each particle pushes away from its neighbors with a force that grows stronger the closer they get. Crucially, these particles were modeled as "run-and-tumble" agents, meaning they sprint forward at a steady speed and then randomly tumble to pick a new direction. By running extensive computer simulations and developing new mathematical tools, the team discovered that as the speed of these particles increases, the entire system undergoes a dramatic and predictable evolution. It starts as a rigid crystal, melts into a smooth liquid, and finally stretches out into a broad, bell-shaped cloud.

In the beginning, when the particles move very slowly, they behave like a frozen crystal. Because the repulsive force between them is so strong, they lock into precise, fixed positions relative to one another. If you were to take a snapshot of where they are over time, you would see sharp, distinct peaks in the density, with each peak representing a particle holding its ground. The researchers found that these positions correspond to the mathematical zeros of a specific type of polynomial, a pattern that has been known for decades in passive physics but here emerges from active motion. As long as the particles' internal energy is low, they merely wiggle slightly around these fixed spots, maintaining their orderly, solid-like structure.

As the researchers increased the speed of the particles, the system began to melt. The particles started to wander further from their original spots, and their movements began to overlap. The sharp peaks in the density profile smoothed out, blurring together until the distinct positions of individual particles could no longer be distinguished. In this intermediate stage, the system looked like a liquid. Remarkably, the overall shape of this liquid cloud matched a famous mathematical curve known as the Wigner semi-circle. This shape is typically associated with passive systems in thermal equilibrium, yet here it appeared in a system driven far from equilibrium by active motion. The researchers confirmed this transition by measuring how much the particles fluctuated compared to the distance between them. When these fluctuations became large enough to blur the gaps between neighbors, the crystal had effectively turned into a liquid.

However, the story did not end there. When the particles were pushed to move even faster, the smooth semi-circle began to distort. The density profile stretched out, losing its curved top and taking on a broad, bell-shaped form that extended much further than the liquid phase had. In this highly active state, the particles were no longer just jiggling in place; they were exploring the entire trap with such vigor that the edges of the cloud became defined by their maximum speed rather than the balance of forces. The researchers observed that the density at the very edges of this cloud did not drop off sharply but instead faded away following a specific power law, a pattern that appears to be a universal feature of such systems when particles experience infinite repulsion upon contact.

To understand why these changes occurred, the team developed a theoretical framework based on the idea that the particles were making small deviations from their ideal, frozen positions. By calculating how these deviations correlated with one another, they could predict the variance, or the spread, of the particles' positions. Their calculations showed that the transition from the crystal to the liquid happens when the particle speed reaches a certain threshold, independent of how many particles are in the system. The second transition, from the liquid to the bell-shaped cloud, occurs only when the speed becomes so high that it scales with the total number of particles. This analytical approach matched their computer simulations perfectly, confirming that the shift from order to disorder, and then to a new kind of active disorder, is governed by simple, predictable rules.

The findings offer a clear picture of how active matter organizes itself under confinement. The researchers showed that even in a system driven far from equilibrium, the particles can settle into a state that mimics the smooth, liquid-like behavior of passive systems, provided the activity is not too extreme. But once the activity crosses a critical limit, the system breaks free from these familiar shapes, adopting a new form dictated by the sheer momentum of the particles. This work not only clarifies the behavior of active particles in a trap but also suggests that similar transitions might be observable in real-world experiments with bacteria or synthetic micro-robots, provided they can be confined and their interactions controlled. The study stands as a testament to the power of combining precise simulation with rigorous theory to reveal the hidden order within complex, self-driven systems.

Technical Summary: Crystal to Liquid Cross-over for Active Particles with Inverse-Square Power-Law Interaction

Problem Statement
This study investigates the non-equilibrium steady-state properties of a one-dimensional system comprising $N$ run-and-tumble particles (RTPs) confined within a harmonic trap. The particles interact via a repulsive inverse-square power-law potential ( $V(x) \propto 1/x^2$ ), a model known in Hamiltonian dynamics as the Calogero-Moser model or the Inverse-Square Model (ISM). The primary objective is to understand how the introduction of activity (self-propulsion) alters the system's density profile and structural order, specifically examining the crossover from a crystalline state to a liquid-like state and eventually to a distinct strongly active regime. This work extends previous analytical studies on the active Dyson Brownian motion (DBM), which features logarithmic interactions, to the active ISM to explore connections between these two integrable systems in the presence of non-equilibrium driving.

Methodology
The authors employ a dual approach combining extensive numerical simulations and analytical derivations in the weak noise limit.

Numerical Simulations: The system is evolved according to the Langevin-like equation of motion for RTPs with inverse-square repulsion and a harmonic confining potential. The steady-state density profile $\rho(x)$ is computed by averaging particle positions over time and multiple realizations. Key observables include:
- The global density profile and its scaling with system size $N$ .
- The Lindemann's ratio ( $\eta_i$ ), defined as the ratio of the standard deviation of particle positions to the average inter-particle separation, used to quantify the crystal-to-liquid transition.
- A distance measure ( $\chi$ ), quantifying the deviation of the steady-state density from the Wigner semi-circle (WSc) profile, used to identify the transition to a strongly active regime.
- The variance of inter-particle gaps.
Analytical Approach: In the limit of small activity ( $v_0 \to 0$ ) and large persistence time ( $\gamma \to 0$ ), the authors linearize the equations of motion around the equilibrium configuration (the zeros of Hermite polynomials). They utilize the Hessian matrix of the ISM potential, denoted $H_{ISM}$ . A key theoretical insight utilized is the relationship $H_{ISM} = H_L^2$ , where $H_L$ is the Hessian of the DBM. By diagonalizing $H_L$ using Hermite polynomials and their associated eigenvectors, the authors derive closed-form expressions for the covariance and variance of particle displacements and inter-particle gaps. These results are subsequently extended to finite tumbling rates $\gamma$ using Fourier space techniques.

Key Results

Three Distinct Activity Regimes: The study identifies three distinct regimes based on the activity parameter $v_0$ :
- Weakly Active Regime ( $v_0 \ll 1$ ): The system resembles a crystal. The density profile exhibits sharp peaks centered at the equilibrium positions (Hermite zeros). The fluctuations are small compared to the inter-particle spacing.
- Intermediate Activity Regime ( $v_0 \sim O(1)$ ): As activity increases, particle fluctuations become comparable to the inter-particle distance. The sharp peaks vanish, and the density profile smoothens into a Wigner semi-circle (WSc) form in the bulk. This "liquid" phase persists over a broad range of activity.
- Strongly Active Regime ( $v_0 \sim O(N)$ ): At very high activity, the density profile deviates significantly from the WSc, transforming into a bell-shaped distribution with a support extending to $\sim \pm v_0$ . Notably, the edges of this distribution exhibit a universal power-law decay $\sim 1/x^3$ .
Quantitative Crossover Criteria:
- The transition from the crystalline (peaked) to the liquid (smooth WSc) phase occurs when the Lindemann's ratio $\eta_i$ exceeds a threshold (approx. 0.5). Analytical results suggest this crossover happens when $v_0 \sim O(1)$ , independent of system size $N$ .
- The transition from the liquid (WSc) to the strongly active (bell-shaped) phase is marked by the distance measure $\chi$ exceeding 0.1. This occurs when $v_0 \sim O(N)$ .
Analytical Predictions for Fluctuations:
- The authors derive exact expressions for the variance of particle positions $\langle \delta x_i^2 \rangle$ and the variance of inter-particle gaps $g_{i,n}^2$ in the large $N$ limit.
- Unlike the active DBM, where edge particles exhibit different scaling behaviors than bulk particles, the active ISM shows that the variance decays as $1/N$ for both bulk and edge particles. This is attributed to the faster convergence of the series expansion involving the $1/k^4$ factor (derived from the $H_{ISM} = H_L^2$ relation) compared to the $1/k^2$ factor in the DBM.
- The analytical predictions for the variance of gaps near the center of the trap agree well with numerical simulations even for finite $\gamma$ and up to $v_0 \sim O(N)$ .
Edge Behavior and Wings:
- In the intermediate regime, the active ISM does not exhibit the "wings" (extended tails) observed in the active DBM. The authors attribute this to the fact that in the ISM, the interaction force on edge particles is dominated by nearest neighbors (scaling as $\sqrt{N}$ ), effectively pinning them, whereas in the DBM, the force is dominated by the collective bulk (scaling as $\sqrt{N}$ but with different coefficients), allowing for larger fluctuations.
- However, in the strongly active regime with small $\gamma$ , "wing-like" structures can emerge near the edges due to the dominance of the driving velocity over the harmonic trap.

Significance and Claims
The paper claims to provide a comprehensive characterization of the active ISM, establishing a direct parallel with the active DBM while highlighting crucial differences in fluctuation scaling and edge behavior.

Universality of the Wigner Semi-Circle: The study reinforces the finding that the Wigner semi-circle is a robust density profile for interacting active particles in a harmonic trap, surviving even at high activity levels before a second transition to a bell-shaped profile occurs.
Analytical Tractability: By leveraging the connection between the ISM and DBM Hessians, the authors successfully derive analytical formulas for position and gap variances that remain accurate across a wide range of parameters, including the "liquid" phase.
Phase Diagram: The construction of a phase diagram in the $(a, 1/\gamma)$ plane (where $v_0 = aN$ ) quantitatively delineates the boundaries between crystal, liquid, and strongly active phases.
Experimental Relevance: The authors suggest that the predicted crystal-to-liquid melting transition and the specific density profiles could be experimentally observable in current systems of trapped active particles (e.g., bacteria, Janus particles) if effective power-law interactions can be engineered, though they note that interaction effects have not yet been fully probed in such experiments.

The work concludes that while the active ISM and active DBM share qualitative features (three regimes, WSc intermediate phase), the specific power-law interaction in the ISM leads to distinct quantitative behaviors in fluctuations and edge stability, offering a deeper understanding of non-equilibrium statistical mechanics in integrable systems.

More like this