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.

翻译：活性物质系统，从自驱动胶体到运动细菌，其特点在于微观尺度上将自由能转化为有用功。这些系统涉及平衡统计力学无法触及的物理机制，而理解其非平衡态的本质一直是一个持续的挑战。熵产率和概率流通过测量时间反演对称性的破缺，为此提供了量化的研究途径。然而，由于它们依赖于系统未知且高维的概率密度，其高效计算一直难以实现。本文基于生成建模的最新进展，开发了一个深度学习框架来估计该密度的得分。我们证明，该得分与微观运动方程相结合，可用来计算熵产率、概率流，并将其分解为来自单个粒子的局域贡献。为了表示得分，我们引入了一种新颖的空间局部Transformer网络架构，该架构能够学习粒子间的高阶相互作用，同时尊重其底层的置换对称性。我们通过将该方法应用于多个经历运动诱导相分离的高维活性粒子系统，展示了其广泛的实用性和可扩展性。我们证明，在一个堆积分数下训练的4096个粒子系统上的单一网络，可以推广到相图的其他区域，包括多达32768个粒子的系统。我们利用这一观察结果，量化了MIPS中偏离平衡的空间结构，并将其作为粒子数和堆积分数的函数进行分析。