Emergent universal statistics in nonequilibrium systems with dynamical scale selection

Abstract

Pattern-forming nonequilibrium systems are ubiquitous in nature, from driven quantum matter and biological life forms to atmospheric and interstellar gases. Identifying universal aspects of their far-from-equilibrium dynamics and statistics poses major conceptual and practical challenges due to the absence of energy and momentum conservation laws. Here, we experimentally and theoretically investigate the statistics of prototypical nonequilibrium systems in which inherent length-scale selection confines the dynamics near a mean energy hypersurface. Guided by spectral analysis of the field modes and scaling arguments, we derive a universal nonequilibrium distribution for kinetic field observables. We confirm the predicted energy distributions in experimental observations of Faraday surface waves, and in random scattering and active turbulence simulations. Our results indicate that pattern dynamics and transport in driven physical and biological matter can often be described through monochromatic random fields, suggesting a path towards a unified statistical field theory of nonequilibrium systems with length-scale selection.

Figures (15)

• Figure 1: Weakly chaotic Faraday surface waves exhibiting scar patterns. Faraday waves emerging on a vertically oscillating bath of water (Movies 1 and 2, Sec. \ref{['sec:methods']}) are representatives of a broader class of non-equilibrium systems with spontaneous scale selection. The photograph shows an oblique view of the fluid surface 2017Harris. The shadows cast by larger wave crests give rise to dark scar-like patterns corresponding to regions of higher surface-gradient energy. In our experiments, the dynamically evolving surface height fields were reconstructed using a free-surface Schlieren technique Wildeman2018 (Sec. \ref{['sec:methods']}). Faraday wavelength: $\lambda_\text{F}\approx 4.6$ mm (Sec. \ref{['sec:methods']}).
• Figure 2: Non-equilibrium field dynamics with length scale selection in experiment and simulations.a, Snapshots from our Faraday wave experiments (Fig. \ref{['fig:faraday_waves']} and Movie 2), random scattering simulations in a smooth isotropic random potential (Eq. \ref{['eq:schrodinger1']}, Movie 3), and active turbulence simulations (Eq. \ref{['activeNS-a']}, Movie 4). Colors indicate the normalized height field $h$, number density $|\psi|^2$ and vorticity field, respectively. b, Associated real space energy densities (Eq. \ref{['energy']}) reveal qualitatively similar structures across the different systems: the surface gradient energy of the Faraday waves, and the kinetic energies of the quantum and active fluid systems are characterized by scars extending throughout the system. c, Spectral energy $e_{\bf k}$ (see Eq.\ref{['Fenergy']}) at modes ${\bf k}$ shows that the energy of the system is concentrated within a narrow shell of a fixed wave-number radius. Each panel represents a typical snapshot of the dynamical system at a time much larger than the initial relaxation period. See Sec. \ref{['sec:methods']} for details and parameters of experiments, simulations and colorbar limits.
• Figure 3: Emergent universal statistics in experiment and simulations.a, Energy distribution functions for a representative subset of individual energy modes follow exponential Boltzmann distributions with different mode temperatures. The figures show the statistics for individual modes normalized by the mode temperature $T_{\bf k} = \langle {e_{\bf k}} \rangle_t$. Insets: all systems show uniform statistics for phases of Fourier modes. See Sec. \ref{['sec:methods']} for details and parameters of experiments and simulations. b, Probability density functions (PDFs) of the Fourier mode energies ${e_{\bf k}}$ measured in the experiment (blue) and simulations (orange, green) follow the predicted superstatistics (solid lines) given by Eq. \ref{['eq:superstatistics']} with system specific global temperatures $T_\text{g}$ (Sec. \ref{['sec:methods']}). We introduce a low-energy cutoff allowing for direct normalization of $\mathcal{N}$ (see Sec. \ref{['sec:methods']}). Here $\mathcal{N} \propto \text{PDF}(e/T_\text{g} = \varepsilon)$. Insets: blow-ups of the energy statistics at low energies shown on a linear scale. See also Figs. \ref{['fig:autocorrelation']}-\ref{['fig:divergence']} for additional analysis of the energy distributions.
• Figure 4: Estimating active transport by sampling from monochromatic random fields.a, Example trajectories of passive tracer particles advected by active turbulent flow solutions (Fig. \ref{['fig:main_figure']}a) of the linearly forced Navier-Stokes equations \ref{['activeNS']}. Trajectories are calculated for total time 50$\tau$, where $\tau$ is the typical time scale of pattern growth in Eq. \ref{['e:tau_GNS']}. b, Velocity autocorrelations in the solutions of Eqs. \ref{['activeNS']} decay on the order of the pattern growth scale $\tau$ (Sec. \ref{['sec:methods']}). c, Tracer particles advected by active turbulence move ballistically on time-scales $t \ll \tau$ and diffusively for $t>\tau$. d, Sample trajectories of tracer particles in monochromatic random flow fields (Sec. \ref{['sec:methods']}) that were periodically updated after time $\tau_c$. e, Vorticity fields $\omega = -\nabla^2 \psi$ corresponding to four stream functions $\psi$ as used in panels (d-f). Stream functions were sampled from superstatistical distributions (Sec. \ref{['sec:methods']}) with same system parameters as in Movie 4, Fig. \ref{['fig:main_figure']}a and panels (a-c). f, Mean squared displacements for tracer particles in monochromatic superstatistics random flow fields agree with those for active turbulence system in panel (c). Panels (c) and (f) show PDFs of the normalized tracer particle displacement at times indicated by the solid circles. Mean squared displacement $\langle [{\bf X}_n(t)-{\bf X}_n(0)]^2 \rangle_n$ in (c, f) are based on 100,000 trajectories, respectively.
• Figure 5: Schematic of the experimental setup. (a) The fluid bath was vibrated with an electromagnetic shaker connected by a thin rod coupled with a linear air bearing. The forcing acceleration was monitored and maintained through two piezoelectric accelerometers and a PID feedback loop. (b) Schematic of the test cell where the liquid was confined to a circular bath. A free-surface Schlieren technique was used to reconstruct the interface height by demodulating the optical distortion induced by the Faraday waves in a checkerboard pattern located at the bottom of the bath. (c) Reconstructed surface height shows weakly chaotic Faraday wave patterns. (d) Bath geometries with different rotational symmetries investigated experimentally.