Reduced density fluctuations via anti-aligning in active matter

TL;DR

The paper addresses whether anti-aligning interactions in active matter can reduce long-range density fluctuations even without global order. It develops a Poisson-representation-based fluctuating hydrodynamics for a 1D lattice gas and derives a Langevin equation with imaginary noise, enabling analytic calculation of the structure factor $S(k)$. The key contributions include an exact 1D solution showing a finite offset $S_0$ and reduced fluctuations, plus 2D agent-based Vicsek-like simulations demonstrating similar trends with apparent hyperuniformity though with non-universal exponents and finite-size caveats. The work provides a mechanism for tunable density statistics in active matter and cautions against over-interpretation of apparent hyperuniformity in finite systems, highlighting implications for non-equilibrium steady states.

Abstract

We highlight the importance of long-range correlations in active matter systems of self-propelling particles even in the absence of global order or steric interactions by demonstrating that long-range density fluctuations are reduced. We show this analytically for a one-dimensional lattice process employing a Poisson representation. Within this framework, we are able to derive the fluctuating hydrodynamics for the Poisson fields. The emergent imaginary noise indicates the non-Poissonian nature of the number fluctuations and manifests in a non-trivial structure factor $S(k)$ which we are computing analytically. Numerically, we corroborate the relevance of these findings for off-lattice Vicsek-type models with anti-aligning interactions for which we observe apparent non-universal hyperuniformity which we suggest to interpret as a reduction with integer power-law to a finite value.

Figures (6)

• Figure 1: Graphical representation of the one-dimensional model. The agent-based model consists of particles on a lattice that move respective to their internal orientation to an adjacent lattice site (left part, possible motion indicated by shaded particles). In the analytical treatment, we move from the integer occupation numbers $n_i^\pm$ to complex-valued Poisson fields (right part), $\alpha_j$ and $\beta_j$, which control the local number statistics. The Poisson fields themselves are random quantities from a distribution $f$. Any observable moment is to be inferred by averaging over all realisations of the Poisson fields and, thus, real.
• Figure 2: Structure factor as a function of the wave number (in units of inverse lattice constants) from a numerical realization of the one-dimensional active lattice gas model. Here, we operated directly in the limit $r_1 \gg r_0$ as the conversion rate is directly proportional to $m$. We provide guides to the eye to show the dominant power-law behaviors for small $k$. Shown is the measured structure factor rescaled to account for the difference between binomial and Poissonian statistics in a finite system and reduced by predicted offset $S_0$.
• Figure 3: Raw data of the structure factor (symbols and colors being identical to the ones used in Fig. \ref{['fig:sf1d']}) with a direct power-law line as a fictitious inference of anomalous hyperuniformity. In this case, we would find $k^\alpha$ with $\alpha\approx 0.25$. For the simple, one-dimensional model, this works only on a small-range of $k$-values spanning roughly half a decade.
• Figure 4: Results of agent-based simulations of an anti-aligning Vicsek-like model. Following the nomenclature of ref. ihle2023, the simulation parameters are $M=\rho \pi R^2=1$ for the reduced density (with $\rho$ the number density) and $\mathrm{Sc}=\Gamma R/v=1$ for the dimensionless coupling strength. The particle number $N$ varies $N=2^{5},\ldots,2^{12}$ which is also indicated by symbol brightness and size (smaller, darker circles corresponding to larger systems). Large: The structure factor $S(k)$ as a function of the wavenumber $k$ in units of the inverse of the interaction length $R$. Inset left: The apparent anomalous scaling exponent $k^{\approx 0.5}$ found from a direct fit can be explained by an exponent of two and an offset as in the 1D model, here $\Delta S=S(k)-S(0)$ with inferred $S(0)$. Squares correspond to results from half the running time to establish convergence. Inset right: The number fluctuations seen by comparison of windows of size $r$; here, we rescaled in such a way that the apparent slope is expected to be the negative of the one seen in the structure factor torquato2018. For $r\approx L$, the number fluctuations are inevitably suppressed. We provide a guide to the eye for the slope in both cases.
• Figure 5: Demonstration of the diffusive breakdown of the long-range suppression of density-fluctuations by comparing the structure factor for a system with no orientational noise (full circles). The noise is implemented as additional white random forces $\mu \eta_i$ with $\langle \eta_i \rangle=0$ and $\langle \eta_i(t)\eta_j(t')\rangle=\delta_{ij}\delta(t-t')$. Here, we used $\mu=1$ as relevant noise (empty squares) and $\mu=0.1$ as weak noise (empty triangles) and compare them to the deterministic case (full circles).