Localizing entropy production along non-equilibrium trajectories

TL;DR

This work addresses the quantification and spatiotemporal localisation of entropy production in complex processes from experimental data through a data-driven approach that combines the recently developed short-time thermodynamic uncertainty relation based inference scheme with machine learning techniques.

Abstract

An important open problem in nonequilibrium thermodynamics is the quantification and spatiotemporal localisation of entropy production in complex processes from experimental data. Here we address this issue through a data-driven approach that combines the recently developed short-time thermodynamic uncertainty relation based inference scheme with machine learning techniques. Our approach leverages the flexible function representation provided by deep neural networks to achieve accurate reconstruction of high-dimensional, potentially time-dependent dissipative force fields as well as the localization of fluctuating entropy production in both space and time along nonequilibrium trajectories. We demonstrate the versatility of the framework through applications to diverse systems of fundamental interest and experimental significance, where it successfully addresses distinct challenges in localising entropy production.

Figures (13)

• Figure 1: Schematic of entropy production inference in an active biological network model. Input: The method processes experimentally measurable trajectory data without requiring prior knowledge of system parameters. Outputs: Using short-time Thermodynamic Uncertainty Relations and neural networks, we simultaneously infer (i) the dissipative (thermodynamic) force field ${\bm F}({\bm x},t)$ driving the nonequilibrium dynamics and (ii) the corresponding fluctuating entropy production (color scale: $\pm 0.015 k_B/s$), localized in both space and time.
• Figure 2: Brownian gyrator model: (a) 2D-dimensional trajectories of a Brownian gyrator system with harmonic confining potential. [Parameters: $k_1 = 1$, $k_2 = 2$, $\gamma = 1$, $\theta = \pi/4$, $D_1 = 1$, $D_2 = 0.1$]. (b) local entropy production rate and (c) thermodynamic force field for the system with harmonic confinement - estimated using the neural network representation. (d) 2D-dimensional trajectories of a Brownian gyrator system with a bi-stable confining potential. [Parameters: $k = 1$, $b = 1$, $\gamma = 1$, $\theta = \pi/4$, $D_1 = 1$, $D_2 = 0.1$] (e) local entropy production rate and (f) thermodynamic force field estimated using the neural network representation. (g) 2D-dimensional trajectories of a Brownian gyrator system with a quartic confining potential. [Parameters: $k_1 = k_2 = 10$, $\gamma = 1$, $\theta = \pi/4$, $D_1 = 10$, $D_2 = 1$] (h) local entropy production rate and (i) thermodynamic force field estimated using the neural network representation. The colours corresponding to the local entropy production rate (in units of $k_B/s$) of the gyrators are thresholded between $[-\alpha\,\mathrm{median},\, \alpha\,\mathrm{median}]$ , where $\alpha$ (typically $20 -50$) multiplies the median of the corresponding local entropy production dataset. Values outside these ranges are clipped for visualisation purposes to prevent rare large fluctuations from dominating the colour mapping. Similarly, the thermodynamic force field values for the gyrators are thresholded within $[0,\, \mathrm{median}]$. The numerical trajectories are usually generated for $2000s$ with a sampling rate of $1\ kHz$ - from which trajectory traces of $500 s$ are shown in the plots. The colorbars in panels (a), (d), and (e) indicate the progression along the trajectory.
• Figure 3: Local tests of fluctuation theorem: (a) The boxed regions highlight three representative areas characterized by low (black), intermediate (red), and high (goldenrod) local dissipation (average values shown in (b)), chosen to probe distinct dynamical environments. (c) Probability distributions $P(dS_{tot})$ conditioned on these regions, illustrating pronounced region-dependent differences in the statistics of entropy production. The low-dissipation region exhibit narrow, nearly symmetric distributions, while higher-dissipation regions display broader, strongly skewed distributions with extended tails. The inset shows the corresponding fluctuation ratios $\ln[P(dS_{tot})/P(-dS_{tot})]$ as a function of $dS_{tot}$, demonstrating that each region independently satisfies a local fluctuation theorem with unit slope.
• Figure 4: (a) Temperature profile of the $N$-dimensional gyrator setup. (b) Analytically estimated local entropy production rate (in units of $k_B/s$) of the system. (c) Local entropy production rate (in units of $k_B/s$) inferred from the numerical trajectories using a neural network representation. The colors do not indicate the true values of the fluctuating entropy current, but they are thresholded for better visualisation. (d) Convergence test ($R^2$ test) of the neural network–based estimation of the fluctuating entropy production rate for an $N$-dimensional Brownian gyrator with $N=100$. The inferred and analytical local entropy production rates, averaged over all beads, exhibit a finite spread around the linear fit. (e) Comparison of the inferred average entropy production for each bead with the corresponding theoretical estimate. (Inset) The same data shown on a logarithmic (y-) scale reveals that dissipation of beads associated with low irreversible signature (entropy production) are challenging for the neural network to capture, resulting in a mismatch with the theoretical prediction.
• Figure 5: Entropy production and finite-time fluctuations in active--bistable mechanical networks. (a) Example disordered two-dimensional spring network with a fraction of hot nodes (red, temperature $T_{\mathrm{hot}}$) and cold nodes (blue, $T_{\mathrm{cold}}$). Thin green bonds denote linear springs, while thick green bonds indicate bistable springs. (b,c) Average entropy production rate as a function of the fraction of hot nodes $h$ at fixed bistable bond fraction $b=0.5$ (b), and as a function of the bistable bond fraction $b$ at fixed $h=0.2$ (c). Symbols show inferred values from training and test data, while solid lines indicate theoretical predictions for the total entropy production rate. (d) Spatial maps of the inferred local entropy production rate for increasing hot-node fraction $h$ at fixed $b=0.5$. (e) Spatial maps of the inferred local entropy production rate for increasing bistable bond fraction $b$ at fixed $h=0.2$. In (d) and (e), the network bonds are colored by the mean value of dissipation of the two nodes in the bond. Additionally, the time-series corresponds to 1000 consecutive steady state configurations. (f) Skewness of the time-integrated entropy production $\Delta S_{\rm tot}$ as a function of the integration time $t$ for different bistable bond fractions $b$, showing a pronounced nonmonotonic dependence. (g) Fraction of time-integrated entropy production fluctuations lying above the mean, $\langle T_+(t)\rangle = \mathbb{P}(\Delta S_{\rm tot}>\langle \Delta S_{\rm tot}\rangle)$, demonstrating a finite-time bias with $\langle T_+(t)\rangle<1/2$. (h) Characteristic integration time $t_\ast$ at which the skewness is maximal, as a function of the bistable bond fraction $b$. Increasing the fraction of bistable bonds systematically amplifies finite-time asymmetries in cumulative entropy production and shifts the characteristic timescale to shorter values, demonstrating that mechanical nonlinearity enhances emergent non-Gaussian entropy production statistics at experimentally relevant finite times.