Deep learning probability flows and entropy production rates in active matter

TL;DR

A broadly applicable deep learning framework is introduced that fuses recent advances in generative modeling with stochastic thermodynamics, yielding access to this canonically intractable density of nonequilibrium currents and the entropy production rate.

Abstract

Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. They involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework to estimate the score of this density. We show that the score, together with the microscopic equations of motion, gives access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles. To represent the score, we introduce a novel, spatially-local transformer network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of active particles undergoing motility-induced phase separation (MIPS). We show that a single network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

Figures (21)

• Figure 1: Method overview. (Green) The starting point for our approach is a microscopic dynamics describing the evolution of a set of interacting active particles. (Purple) The target is estimation of several definitions of the entropy production rate of the system, which we will accomplish by means of the probability flow. (Blue) Mathematically, our approach is built on viewing the system from the perspective of dynamical transport of measure. The microscopic stochastic dynamics induces a Fokker-Planck equation for a high-dimensional density describing the configuration of the system. This Fokker-Planck equation is equivalent to a transport equation that depends on the unknown "score" $\nabla\log\rho$ of the solution. The characteristics of this equation obey a probability flow ordinary differential equation, which gives immediate access to the entropy production rate. (Center) Illustration of nonequilibrium transport of measure at stationarity. (Orange) Algorithmically, our method approximates the unknown score by machine learning over a dataset of microscopic particle data. The learned approximation can be validated a-posteriori by checking invariants of the stationary Fokker-Planck equation, and can be plugged in directly to the definition of the entropy production rate to obtain an estimate.
• Figure 2: Stochastic dynamics and probability flows. (A) Individual stochastic trajectories of \ref{['eqn:interacting_particles']} for $N=2$ and $d=1$ in the variables $x_t = x^2_t - x^1_t$ and $g_t = g^2_t - g^1_t$, with periodic boundary conditions on $[0,L]$. The trajectories $(x_t,g_t)$ tend to accumulate in two clusters corresponding to situations where particle 1 is just in front of particle 2 or vice-versa. This occurs because one particle catches up to the other in a typical trajectory (since either $|g_t^1|>|g_t^2|$ or $|g_t^1|<|g_t^2|$), but does not pass over it due to the short-range repulsive force between them. Random transitions between these modes occur when the magnitudes of $|g_t^1|$ and $|g_t^2|$ change order. (B) Stationary probability density function $\rho$ of $(x_t, g_t)$ confirming the metastability observed in (A): $\rho$ is the solution of the stationary FPE \ref{['eqn:fpe']}. (C) Visualization of the (diffusion-weighted) norm of the probability current $j$, defined in \ref{['eqn:current']}, over the phase space. The current is concentrated in the two modes, but is also nonzero along transition pathways between them. (D) Phase portrait of the probability flow \ref{['eqn:pflow']}. Similar to the stochastic trajectories in (A), the flow lines preserve the density $\rho$ in (B), but are deterministic and interpretable, highlighting limit cycles within and between the two clusters. A movie of these limit cycles in a frame with one particle fixed is available https://www.dropbox.com/scl/fi/q2ryj5u9nfl0yk3ovw3u9/pflow_movie.mp4?rlkey=98cqnwl72xa6vvq2dbqyhcxy0&dl=0.
• Figure 3: System EPR. Visualization of $\nabla\cdot v(r)$ across the phase space for \ref{['eqn:interacting_particles']} with $N=2$ and $d=1$ in the variables $x_t = x_t^2 - x_t^1$ and $g_t = g_t^2 - g_t^1$. The system EPR along a trajectory $R_t(r)$ of the probability flow \ref{['eqn:pflow']} can be written as $\dot{s}_{\text{sys}}(t) = \nabla\cdot v(R_t(r))$, so that $\nabla\cdot v(r)$ gives insight into how entropy is generated locally by the system. Even though $\mathbb{E}_\rho[\nabla\cdot v]=0$ at stationarity, $\nabla\cdot v \neq 0$ pointwise when the system is out of equilibrium. Here, system entropy is locally produced when the two particles collide, and released when they separate.
• Figure 4: Network architecture. Depiction of the transformer architecture introduced in this work. The particle positions and orientations are fed into separate multi-layer perceptrons (MLPs) that embed the input into a latent space of higher dimensionality. The embeddings are concatenated particle-wise and fed into a transformer encoder block (SI appendix for further details), where multiple layers of multi-head attention modules learn relevant interactions between particles. The output of the encoder block is decoded by a shared MLP applied to each particle state to obtain the score.
• Figure 5: 64 swimmers in a harmonic trap: system EPR. (Top) The contribution of the per-particle orientational degrees of freedom to the system EPR $\nabla_{g^i}\cdot v_{g}^i$ as a function of the activity $v_0$, visualized directly on the particles. For $v_0 = 0$, the system is at equilibrium and the network learns that the system EPR vanishes. As $v_0$ increases, nonequilibrium effects emerge, and the particles on the boundary display the highest contribution to the EPR. (Bottom) A spatial map visualizing the typical contribution of a particle at position $(x, y)$ to the system EPR, obtained by averaging the data in the top row over many system snapshots. The map highlights the role of interfacial contributions, and displays a prominent ring at the boundary of the cluster.

Theorems & Definitions (9)

• Proposition A.1
• Proposition A.2
• proof : Proof of Proposition \ref{['prop:1']}
• proof : Proof of Proposition \ref{['prop:cond:expect']}
• Proposition B.1
• Proposition B.2
• proof : Proof of Proposition \ref{['prop:3']}
• Proposition B.3
• proof : Proof of Proposition \ref{['prop:2']}