Model-free learning of probability flows: Elucidating the nonequilibrium dynamics of flocking

Abstract

Active systems comprise a class of nonequilibrium dynamics in which individual components autonomously dissipate energy. Efforts towards understanding the role played by activity have centered on computation of the entropy production rate (EPR), which quantifies the breakdown of time reversal symmetry. A fundamental difficulty in this program is that high dimensionality of the phase space renders traditional computational techniques infeasible for estimating the EPR. Here, we overcome this challenge with a novel deep learning approach that estimates probability currents directly from stochastic system trajectories. We derive a new physical connection between the probability current and two local definitions of the EPR for inertial systems, which we apply to characterize the departure from equilibrium in a canonical model of flocking. Our results highlight that entropy is produced and consumed on the spatial interface of a flock as the interplay between alignment and fluctuation dynamically creates and annihilates order. By enabling the direct visualization of when and where a given system is out of equilibrium, we anticipate that our methodology will advance the understanding of a broad class of complex nonequilibrium dynamics.

Figures (7)

• Figure 1: The nonequilibrium dynamics of flocking. System dynamics for \ref{['eqn:vicsek']}, with trail indicating past motion. The particles spontaneously self-organize into polar flocks that irregularly split and combine in a manner that breaks time reversal symmetry. In this work, we study how these TRS breaking events are distributed spatially.
• Figure 2: Two particles: Phase portrait and EPR. (A) Flow lines of the probability current on the NESS. Pendulum-like limit cycles of alignment and anti-alignment can be seen. Comparison with (B) and (C) highlights alternating regions of $\dot{s}_{\text{tot}} \neq 0$ and $\dot{s}_{\text{tot}} = 0$, as well as alternating regions of system entropy production ($\dot{s}_{\text{sys}} > 0$) and system entropy consumption ($\dot{s}_{\text{sys}} < 0$) as the particles oscillate in and out of the aligned state. (B) Phase space distribution of $\dot{s}_{\text{tot}}$, which concentrates during the particle-particle interaction ($|g^{\mathsf{R}}|$ is visualized rather than $|g^{\mathsf{R}}|^{2}$ to reduce the influence of outliers). (C) Phase space distribution of $\dot{s}_{\text{sys}}$, which concentrates in alternating, asymmetric lobes of entropy production and entropy consumption. Entropy production occurs during anti-alignment -- regions where $\mathsf{sign}(x) = \mathsf{sign}(v)$ -- and indicates the annihilation of order. Entropy consumption occurs during alignment, which happens when $\mathsf{sign}(x) = -\mathsf{sign}(v)$, and signifies the creation of order.
• Figure 3: Sixty-four particles in two dimensions. The reader is strongly encouraged to view the accompanying movie https://www.dropbox.com/scl/fi/edh2nuzxgozd4z1wxby2e/sde_entropy_movie_plot.mp4?rlkey=fh156967p9msfp6fg85ktpq41&st=ljvqq6yo&dl=0, as well as the longer movie https://www.dropbox.com/scl/fi/wttsp4rhpb6kgfiwesnxi/sde_entropy_movie_plot_long.mp4?rlkey=9chq2pru34csiz0joonj0g57d&st=ywv2k0lw&dl=0. (A) Uncolored reference depiction of the particle trajectories. Frames chosen based on large spikes in $\dot{s}_{\text{tot}}$, shown by vertical lines in (E) and (F). (B) Particle contributions to $\dot{s}_{\text{sys}}$. Particles exhibit negative system EPR during alignment and positive system EPR during anti-alignment. Positive values occur on the boundary of flocks, while negative values occur both on the boundary and during collisions. (C) Particle contributions to $\dot{s}_{\text{tot}}$. The signal is similar to $\dot{s}_{\text{sys}}$, but is more dominated by outliers, and does not display signed information about the creation or annihilation of order. Particles primarily contribute to $\dot{s}_{\text{tot}}$ during collisions and during flock breakup. (E) Time series of $\dot{s}_{\text{sys}}(x_t, v_t)$. Large negative spikes indicate an increase in system order, such as during a merger between two flocks. Positive spikes indicate a decrease in system order, corresponding to particles leaving alignment, as driven by random fluctuation on flock boundaries. (F) Time series of $\dot{s}_{\text{tot}}(x_t, v_t)$. Large spikes typically correspond to flock breakup or flock formation.
• Figure 4: EPR statistics. (A) Density of the per-particle system EPR. The distribution exhibits asymmetry between entropy production and entropy consumption. (B) Density of the per-particle total EPR. The distribution has a heavy tail, but does not distinguish between production and consumption.
• Figure 5: Power spectral density. Time series and PSD for the system EPR (A) and the total EPR (B). Dashed line on PSD indicates nonlinear fit to $c/\omega^{p}$.