Hyperuniformity and conservation laws in non-equilibrium systems

TL;DR

The work shows that hyperuniformity in non-equilibrium systems fundamentally relies on conservation laws interacting with driving, providing a unifying framework that connects hyperuniform density fluctuations to self-organized criticality-like mechanisms. Starting from hydrodynamic arguments, it demonstrates that energy injection at small scales combined with global damping and momentum-conserving noise produces suppressed long-wavelength fluctuations, yielding $S(k) o 0$ as $k o 0$, typically with $S(k) \,\sim\, k^2$. The authors then generalize to systems that conservatively carry higher-order mass multipoles, showing that non-equilibrium dynamics can realize arbitrarily strong hyperuniformity with $S(k) \,\sim\, k^m$, where $m$ is set by the highest conserved multipole, a result supported by lattice models and fracton-inspired hydrodynamics. They also demonstrate that hyperuniformity is fragile without conservation, as nonlinear damping or nonlinear couplings reintroduce effective noise that destroys the large-scale suppression. Overall, the paper provides a cohesive theoretical framework and concrete models for achieving robust hyperuniformity in non-equilibrium matter, with potential implications for material design and the study of fracton-like dynamics.

Abstract

We demonstrate that hyperuniformity, the suppression of density fluctuations at large length scales, emerges generically from the interplay between conservation laws and non-equilibrium driving. The underlying mechanism for this emergence is analogous to self-organized criticality. Based on this understanding, we introduce four non-equilibrium models that consistently demonstrate hyperuniformity. Furthermore, we show that systems with an arbitrary number of conserved mass multipole moments exhibit an arbitrary strong tunable hyperuniform scaling, with the structure factor following $S(k) \sim k^m$, where $m$ is set by the number of conserved multipoles. Finally, we find that hyperuniformity arising from a combination of conserved noise and partially conserved average motion is not robust against non-linear perturbations. Notably, non-linear damping destroys hyperuniformity in hyperuniform fluids. These results highlight the central role of conservation laws in stabilizing hyperuniformity and reveal a unifying mechanism for its emergence in non-equilibrium systems.

Figures (6)

• Figure 1: Simulations of four models in 2D periodic square boxes displaying hyperuniformity. Top panels: radially averaged structure factor $S(k)$. Bottom panels: corresponding snapshots of the systems. Details of the models and simulation parameters are given in Appendix \ref{['app: chiral active']}, \ref{['app: dpd']}, \ref{['app: rods']} and \ref{['app: oscillating radius']} for a), b), c) and d) respectively.
• Figure 2: Relation between detailed balance breaking and hyperuniformity. The main panel shows the structure factor $S(k)$ for a 1D lattice model with center of mass conservation, for different values of the jump-rate ratio $\beta/\alpha$. The case $\beta/\alpha=1$ corresponds to equilibrium and yields a flat, non-hyperuniform spectrum. Inset: largest wavevector $k=k_*$ such that $S(k_*)/S(k_+)<S_*=0.5$, with $k_+$ the maximal lattice wavevector, as a function of the non-equilibrium control parameter $\beta/\alpha$. The dashed curve is the prediction $k_*\propto \sqrt{1-\beta/\alpha}$
• Figure 3: Steady-state structure factor $S(k)$ for one-dimensional lattice models with conservation laws up to different multipole orders. (a) Dipole (center of mass) conservation; (b) conservation up to the third multipole; (c) conservation up to the fifth multipole. For each case we simulate an equilibrium system ($\alpha=\beta$) and a non-equilibrium system ($\beta=0$). The cartoons above each panel depict the underlying non-equilibrium hopping process.
• Figure 4: Structure factor of the non-equilibrium $\phi^4$ model (Eq. \ref{['eq: non eq phi^4']}) in a $d=2$ periodic box of size $L\times L$ driven by a conserved noise (divergence noise) but non-conserved deterministic relaxation, for various values of the non-linearity. Hyperuniformity is present only in the linear case and is destroyed by arbitrarily small non-linearities. Simulations are performed using a pseudo-spectral method caballero2024cupss with $\sqrt{\tau}dx=\sqrt{\tau}dy=1$, $\tau L^2=1024^2$ and $\tau dt= 0.025$.
• Figure 5: Structure factor of a 1D lattice simulation with kernels $\bm{\mathcal{L}}=[-1, 1, 3, -5, 2]$ and $\bm{\mathcal{L}}=[ 1, -3, 0, 10, -15, 9, -2]$, conserving up to the second and fourth multipole moments, respectively. The dynamics is non-equilibrium with $\beta=0$.