Entropy Production in Non-Gaussian Active Matter: A Unified Fluctuation Theorem and Deep Learning Framework

We present a general framework for deriving entropy production rates (EPRs) in active matter systems driven by non-Gaussian active fluctuations. Employing the probability-flow equivalence technique, we rigorously obtain an entropy production (EP) decomposition formula. We demonstrate that the EP, $\Delta s_\mathrm{tot}$, satisfies a detailed fluctuation theorem, $\rho_{\mathcal{R}}(\Sigma)/\rho_{\mathcal{R}}(-\Sigma)=e^{\Sigma}$, which holds for the distribution $\rho_{\mathcal{R}}(\Sigma)$ defined as the probability of observing a value $\Sigma$ of the quantity $\mathcal{R}\equiv \Delta s_\mathrm{tot}-B_\mathrm{act}$, where $B_\mathrm{act}$ is a path-dependent random variable associated with active fluctuations. Moreover, an integral fluctuation theorem, $\langle e^{- \mathcal{R} } \rangle = 1$, and the generalized second law of thermodynamics, $\langle \Delta s_\mathrm{tot} \rangle \ge \langle B_\mathrm{act} \rangle$, follow directly. Our results hold under steady-state conditions and can be straightforwardly extended to arbitrary initial states. In the limiting case where active fluctuations vanish, these theorems reduce to the established results of stochastic thermodynamics. Building on this theoretical foundation, we introduce a deep-learning-based methodology for efficiently computing the EP, utilizing the L\'{e}vy score we propose. To illustrate the validity of our approach, we apply it to two representative systems: a Brownian particle in a periodic active bath and an active polymer composed of an active Brownian cross-linker interacting with passive Brownian beads. Our work provides a unified framework for analyzing EP in active matter and offers practical computational tools for investigating complex nonequilibrium behavior.