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

Abstract

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, $Δs_\mathrm{tot}$, satisfies a detailed fluctuation theorem, $ρ_{\mathcal{R}}(Σ)/ρ_{\mathcal{R}}(-Σ)=e^Σ$, which holds for the distribution $ρ_{\mathcal{R}}(Σ)$ defined as the probability of observing a value $Σ$ of the quantity $\mathcal{R}\equiv Δs_\mathrm{tot}-B_\mathrm{act}$, 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, $\langle Δs_\mathrm{tot} \rangle \ge \langle B_\mathrm{act} \rangle$, 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 Lévy 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.

Figures (19)

• Figure 1: EPRs of a Brownian particle in an active bath. Shown are $\dot{S}_\mathrm{tot}$ (blue) and its three components: $\dot{S}_\mathrm{m}$ (orange), $\dot{S}_\mathrm{act}$ (green), and $\dot{S}_\mathrm{sys}$ (red) in both plots. (a) $\mu=0$. All EPRs decay to 0, showing equilibrium-like behavior. (b) $\mu=0.1$. $\dot{S}_\mathrm{tot}$ saturates at a positive value due to active fluctuation, showing a nonequilibrium steady state.
• Figure 2: (a) Schematic of the ABP system: A central ABP (red) is connected to $m$ chains (illustrated with $m=4$) consisting of $n$ BPs (white) each (with $n=3$ shown) and a fixed BP (black) at the end of each chain. (b) EPRs of the ABP under unbiased jump noise with $m = 4$ and $n = 3$. Shown are $\dot{S}_\mathrm{tot}$ (blue) and its three components: $\dot{S}_\mathrm{m}$ (orange), $\dot{S}_\mathrm{act}$ (green), and $\dot{S}_\mathrm{sys}$ (red). (c) Active EPR $\dot{S}_\mathrm{act}$ of the ABP under biased jump noise for various combinations of $(m, n)$. Solid and dashed lines correspond to $m = 3$ and $m = 4$, respectively, with colors indicating different $n$ values: green ($n = 1$), blue ($n = 3$), and beige ($n = 7$).
• Figure 3: The verification of DFT for the first example in the main text is shown in (a) for $\mu = 0$, and in (b) for $\mu = 0.1$. The blue dots represent the numerical results, i.e., the empirical probability ratios $\log\!\left[\rho_{\mathcal{R}}(\Sigma) / \rho_{\mathcal{R}}(-\Sigma)\right]$ evaluated at the corresponding values of $\Sigma$. The red dashed line indicates the theoretical prediction $y = \Sigma$. The fitted slopes of the data points show excellent agreement with the theoretical value.
• Figure 4: Probability flows of the Brownian particle immersed in a periodic active bath. The top panel illustrates the temporal evolution of the probability distribution as a heat map, overlaid with two selective stochastic trajectories based on the Monte Carlo simulation (red) and the two deterministic trajectories based on the transport map \ref{['appeqn:interactingODE']} (white). The bottom panels compare the probability distributions $P(\bm{r},t)$ from the Monte Carlo simulation and the proposed method in the time-state space.
• Figure 5: Total variation distances between $P^\mathrm{MC}$ and $P^\mathrm{NN}$ for the Brownian particle immersed in an active bath. Top panel: $\mu=0$; bottom panel: $\mu=0.1$.