For dilute self-propelled particles that align through rare binary collisions, a controlled coarse-graining from the particle dynamics to fluctuating hydrodynamics was achieved in 2022 only for polar particles with polar alignment (the Boltzmann–Vicsek model): the kinetic-scale fluctuations are non-Gaussian and interaction-dependent, while the surviving hydrodynamic noise is Gaussian with a variance proportional to the square of the density and explicitly dependent on the collision rules. Whether this structure persists when the alignment rule has nematic symmetry — polar particles with head-tail-symmetric alignment, as in self-propelled rods, and fully nematic (apolar) systems — remains an open derivation problem. Existing fluctuating theories for active nematics instead add a Dean-like noise whose validity in the dilute, collision-dominated regime is disputed. The problem is to derive the fluctuating kinetic theory and the corresponding fluctuating hydrodynamics for dilute nematic-aligning and apolar active particles, to determine the hydrodynamic noise statistics (Gaussianity, covariance, density scaling, interaction dependence), and to compare with the Dean-noise predictions previously used for active nematics.

Background

Active matter designates many-particle systems whose constituents convert stored energy into self-propulsion — motile bacteria, vibrated granular rods, synthetic colloidal swimmers. A minimal model is the Vicsek model: point particles move at constant speed while tending to align their heading with nearby particles, subject to angular noise. Alignment rules come in two symmetry classes. Polar alignment, as in the Vicsek model, makes two colliding particles adopt a common heading; nematic alignment makes them align head-to-tail, so that orientations θ\theta and θ+π\theta+\pi are equivalent — the rule appropriate to elongated self-propelled rods. Systems are called polar or apolar according to whether the particle itself has a head-tail asymmetry; apolar particles (for example vibrated rods) can only order nematically.

For dry active matter (no momentum-conserving ambient fluid), two coarse-graining regimes must be distinguished. In the weak-interaction regime each particle interacts weakly with very many others; the empirical density then obeys Dean's equation, and the resulting fluctuating hydrodynamics — hydrodynamic equations supplemented with a stochastic noise term — has a Gaussian noise of variance proportional to the density ρ\rho and independent of the interaction details (the Dean–Kawasaki framework). In the dilute regime relevant to most alignment simulations and experiments, collisions are rare binary events that abruptly reorient a particle; this is the analogue for active particles of the Boltzmann–Grad limit of kinetic gas theory, and the Dean–Kawasaki assumptions do not apply. What replaces them at the fluctuating level was established, for polar particles with polar alignment, by Feliachi, Besse, Nardini and Barré (2022). They derived a fluctuating kinetic theory for the Boltzmann–Vicsek model (the dilute binary-collision version of the Vicsek model) — a dynamical large-deviation principle for the empirical one-particle distribution, meaning that the probability of a fluctuating path decays exponentially in the particle number with a rate functional quantifying the cost of deviations; the non-quadratic rate function implies that the kinetic-scale noise is non-Gaussian. They then derived the corresponding fluctuating hydrodynamics, both deeply in the ordered phase and close to the order–disorder transition: the surviving hydrodynamic noise is Gaussian, but its variance is proportional to ρ2\rho^2 and depends explicitly on the collision rules, at variance with the Dean–Kawasaki result.

For nematic alignment the fluctuating level is much less settled. Bertin, Chaté, Ginelli, Mishra, Peshkov and Ramaswamy (2013) proposed a mesoscopic fluctuating theory for active nematics by adding a Dean-like noise to the Boltzmann equation; Feliachi et al. point out that this is "clearly out of the regime of applicability" of the Dean approach for dilute short-range aligning particles, since binary collisions should generate an additional interaction-dependent noise contribution. The deterministic kinetic and hydrodynamic theory of nematic alignment is well developed — Boltzmann-type equations with moment closures near the transition, and Landau-equation treatments beyond mean field — but no controlled derivation of the fluctuating kinetic theory and fluctuating hydrodynamics for nematic-aligning or apolar particles in the dilute limit exists. This matters because the noise term controls what fluctuating hydrodynamics predicts: equal-time structure factors (the static, Fourier-space variance of the fluctuating fields) and the giant number fluctuations (the anomalously strong growth of particle-number variance with the size of the observation window) characteristic of active nematics. An incorrectly structured noise (wrong density scaling or wrong interaction dependence) yields incorrect predictions for precisely the fluctuation observables by which these theories are tested against simulations and experiments.

Problem Statement

Derive, for dilute self-propelled particles whose only interactions are binary alignment collisions, the fluctuating kinetic theory and the corresponding fluctuating hydrodynamics when the alignment rule has nematic symmetry — covering (i) polar particles with nematic alignment (a head-tail-symmetric collision rule, as appropriate for self-propelled rods) and (ii) fully nematic (apolar) systems. The derivation should follow the controlled route established for the polar case: a dynamical large-deviation principle, or an equivalent fluctuating Boltzmann equation with explicit noise statistics, for the empirical one-particle distribution in the dilute (Boltzmann–Grad-type) limit under molecular chaos (the assumption that the pre-collision states of two colliding particles are statistically independent), followed by a hydrodynamic closure — deeply in the ordered nematic phase and/or close to the order–disorder transition — yielding explicit stochastic hydrodynamic equations. The answer must determine the structure of the hydrodynamic noise: whether it is Gaussian, its covariance as a functional of the hydrodynamic fields and of the collision rule, its density dependence (in particular whether the variance proportional to ρ2\rho^2 and the explicit interaction dependence found for polar alignment persist), and how it combines with any Dean-like angular-diffusion noise; and it must compare the result with the Dean-like noise assumed in earlier fluctuating theories of active nematics, stating the consequences for measurable fluctuation observables such as equal-time structure factors and number-fluctuation scaling.

References

  1. Ouassim Feliachi, Marc Besse, Cesare Nardini, Julien Barré, 'Fluctuating kinetic theory and fluctuating hydrodynamics of aligning active particles: the dilute limit', Journal of Statistical Mechanics (2022) 113207, DOI: 10.1088/1742-5468/ac9fc6, https://arxiv.org/abs/2206.11777
  2. Eric Bertin, Hugues Chaté, Francesco Ginelli, Shradha Mishra, Anton Peshkov, Sriram Ramaswamy, 'Mesoscopic theory for fluctuating active nematics', New Journal of Physics 15 (2013) 085032, DOI: 10.1088/1367-2630/15/8/085032, https://doi.org/10.1088/1367-2630/15/8/085032
  3. Horst-Holger Boltz, Benjamin Kohler, Thomas Ihle, 'Kinetic Theory of Self-Propelled Particles with Nematic Alignment', Entropy 26 (2024) 1054, DOI: 10.3390/e26121054, https://doi.org/10.3390/e26121054
  4. Raphaël Maire, Alessandro Petrini, Umberto Marini Bettolo Marconi, Lorenzo Caprini, 'Kinetic theory of chiral active disks: Odd transport and torque density', The Journal of Chemical Physics 165, 034501 (2026), DOI: 10.1063/5.0332479, https://doi.org/10.1063/5.0332479
  5. Julien Barré, Ouassim Feliachi, 'How general is the Jensen–Varadhan large deviation functional for 1D conservation laws?', Journal of Physics A: Mathematical and Theoretical 57, 325006 (2024), DOI: 10.1088/1751-8121/ad6226, https://arxiv.org/abs/2401.08371
  6. Pierre Illien, 'The Dean–Kawasaki equation and stochastic density functional theory', Reports on Progress in Physics (2025), DOI: 10.1088/1361-6633/adee2e, https://arxiv.org/abs/2411.13467
  7. Jay Armas, Akash Jain, Ruben Lier, 'Hydrodynamics of thermal active matter', Physical Review E 112, 055401 (2025), DOI: 10.1103/yclc-jmcj, arXiv:2405.11023, https://arxiv.org/abs/2405.11023