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Multidimensional Stationary Probability Distribution for Interacting Active Particles
Claudio Maggi, Umberto Marini Bettolo Marconi, Nicoletta Gnan, Roberto Di Leonardo
TL;DR
The paper addresses the lack of an explicit stationary probability distribution for multidimensional active-particle systems with colored noise and conservative interactions. It derives a multidimensional Unified Colored Noise Approximation and shows that the resulting distribution quantitatively captures several simulated non-equilibrium behaviors, including accumulation near repulsive obstacles and pressure-like effects.
Problem
Active-particle systems generally lack an explicit stationary probability distribution analogous to the Boltzmann distribution, limiting direct use of standard statistical mechanics.
Method
The paper derives the MUCNASP using a multidimensional Unified Colored Noise Approximation for Gaussian colored-noise dynamics under a generic conservative potential.
Results
The MUCNASP quantitatively describes simulated accumulation near repulsive obstacles and reproduces reported differences in contact behavior for interacting active particles.
Takeaways & Limitations
The theory provides an approximate stationary probability framework for analyzing several non-equilibrium properties of colored-noise-driven active particles, including pressure-like behavior.
Takeaways & Limitations
The dynamics assume independent Gaussian colored-noise processes with exponential time correlation, characterized by D and τ.
Abstract
from arXiv · showhide
We derive the stationary probability distribution for a non-equilibrium system composed by an arbitrary number of degrees of freedom that are subject to Gaussian colored noise and a conservative potential. This is based on a multidimensional version of the Unified Colored Noise Approximation. By comparing theory with numerical simulations we demonstrate that the theoretical probability density quantitatively describes the accumulation of active particles around repulsive obstacles. In particular, for two particles with repulsive interactions, the probability of close contact decreases when one of the two particle is pinned. Moreover, in the case of isotropic confining potentials, the radial density profile shows a non trivial scaling with radius. Finally we show that the theory well approximates the "pressure" generated by the active particles allowing to derive an equation of state for a system of non-interacting colored noise-driven particles.
Introduction
Active-particle systems generally lack an explicit stationary probability distribution analogous to the Boltzmann factor. This gap limits the direct use of standard statistical-mechanical methods for multidimensional, interacting systems.
- Active particles absorb environmental energy and convert it into persistent motion, producing stationary behavior unlike thermal equilibrium.
- The Boltzmann prescription assigns configuration weights for equilibrium systems with arbitrary fields and interactions, whereas active systems generally lack an equivalent distribution.
- Exact stationary probabilities are known only in rare active-particle cases, such as the one-dimensional run-and-tumble model in an external force field.
- Without an explicit stationary density, standard methods of statistical mechanics cannot be applied directly to active-particle systems.
Results
The paper models colored-noise-driven dynamics with a multidimensional Unified Colored Noise Approximation and derives a stationary probability for arbitrary conservative potentials. The resulting expression is obtained from the zero-current Fokker–Planck solution and incorporates the potential Hessian.
- The model uses N independent, zero-mean Gaussian processes with exponential time correlation characterized by diffusion coefficient D and relaxation time τ.
- The Unified Colored Noise Approximation converts the dynamics into a Stratonovich Langevin equation involving the identity matrix, potential Hessian, and white-noise sources.
- The stationary probability is proportional to a weight derived for the flow-free case of the approximated Langevin dynamics.
- The derivation uses the corresponding Fokker–Planck equation and solves it in the zero-current case with Jacobi’s formula.
One single degree of freedom
For a steep repulsive obstacle, the one-dimensional UCNA stationary distribution reproduces simulations and explains particle accumulation near the barrier. It also predicts the resulting force and pressure, recovering the ideal-gas law in the zero-correlation-time limit.
- Stationary probability: The normalized UCNA probability reproduces numerical distributions at two well-separated values of D and τ.The approximation is used because the exact GCN-driven stationary distribution is unknown.
- Average force: The UCNA prediction for ⟨|Φ′|⟩ is very close to numerical results across all investigated D and τ.This average follows from integrating the force magnitude against the predicted stationary probability.
- Barrier accumulation: The probability peaks where the steep external potential balances the particle’s root-mean-squared GCN-induced velocity, with |Φ′(x∗)| ≈ D/τ.In the hard-wall limit, the peak region has width set by the active-motion correlation length √(Dτ).
- Pressure: For N independent particles with τ → 0 and D = µkBT, the pressure becomes NkBT/L∗ and is an upper bound for the active-system pressure.The same pressure expression is also obtained exactly for the RnT model with two hard walls.
Two interacting particles in one dimension.
For two interacting GCN-driven particles, MUCNASP predicts the separation distribution and interaction-force averages, revealing that mobility changes contact statistics relative to equilibrium.
- Model and prediction: MUCNASP computes the probability of finding two particles at separation ∆x from the multidimensional stochastic dynamics.The pair potential depends on ∆x = |x1 − x2|, with periodic boundaries in one dimension.
- Agreement with simulations: The predicted separation distribution and average interaction force agree well with simulations for steep repulsive interactions.The same approximation describes both the probability density and ⟨|Φ′|⟩.
- Contrast with equilibrium: The two-particle result differs from equilibrium statistical mechanics, where pinning one particle does not change the probability of a given separation.The comparison concerns the equilibrium prediction for two particles at fixed distance.
- Physical interpretation: When both particles move freely, coherent motion increases contact frequency without particles pushing against each other, lowering the average interaction force.This differs from the pinned-particle case and provides a physical interpretation of the predicted contact statistics.
Radially symmetric potentials.
For spherically symmetric potentials, the stationary distribution has a nontrivial radial dependence beyond the usual geometric factor, and MUCNASP reproduces simulated accumulation and force trends.
- Radial stationary distribution: For a spherically symmetric potential Φ(x) = Φ(r), the stationary distribution includes radial geometry and colored-noise-dependent terms.Unlike the Boltzmann distribution, whose dimensional dependence enters only through r^(d−1), the GCN case has more complicated dimensional dependence.
- Circular repulsive potential: In two dimensions, particles accumulate near the circular repulsive boundary at r = R, and the theoretical radial probability reproduces the simulation.The studied potential is Φ(r) = (r − R)^−12 with R = 5 µm.
- Force and velocity prediction: The predicted radial velocity component ⟨|Φ′|⟩ agrees well with simulations across the investigated parameters.The steep-potential approximation follows the numerical trend through the corresponding dashed lines.
- Equation of state: In the white-noise limit, the radial force yields the ideal-gas pressure p = NkBT/(πR*^2) for N independent particles.This follows after setting τ = 0 and using D = µkBT.
Discussion
The MUCNASP accurately captures several non-equilibrium behaviors of active particles, including accumulation near repulsive boundaries and altered contact statistics. Its predictions closely match numerical results and approximate results for run-and-tumble particles.
- The MUCNASP predicts that a single active particle concentrates on the repulsive portion of a steep potential, unlike a Brownian particle.
- Two repulsively interacting active particles contact more often when both are mobile than when one particle is fixed.
- For particles confined by a repulsive ring, probability peaks at the boundary, with peak area increasing with radius and persistence length.
- MUCNASP results closely match run-and-tumble-particle results, despite the lack of an analytical solution for the latter.