Universal bounds on entropy production from fluctuating coarse-grained trajectories

TL;DR

This work surveys model-free lower bounds on entropy production obtainable from coarse-grained trajectories within stochastic thermodynamics, focusing on systems with memoryless underlying dynamics. It delineates two main strategies: deriving bounds from coarse-grained data (e.g., state lumping, TUR, waiting times, and correlations) and deriving universal bounds by thermodynamically consistent coarse-graining of fine trajectories, yielding a master bound $\sigma \ge \sigma_{\rm cgr}$. Central results include the TUR bound $\sigma_{\mathrm{TUR}} \le \sigma$, bounds from observed ticks, and correlation-based bounds, as well as a general master relation for universal lower bounds and methods to handle coarse-graining via $k$-tuple and waiting-time formalisms. The review also addresses relaxation and time-dependent driving, experimental bridges, and the practical limitations and extensions of these bounds, including the distinction between thermodynamic and informatic entropy production in active systems. Overall, the framework provides principled, model-free tools to quantify irreversibility and energetic costs from partial observations, with wide relevance to colloidal, biomolecular, and active-matter systems.

Abstract

Entropy production is arguably the most universally applicable measure of non-equilibrium behavior, particularly for systems coupled to a heat bath. This setting encompasses driven soft matter as well as biomolecular, biochemical, and biophysical systems. Despite its central role, direct measurements of entropy production remain challenging - especially in small systems dominated by fluctuations. The main difficulty arises because not all degrees of freedom contributing to entropy production are experimentally accessible. A key question, therefore, is how to infer entropy production from coarse-grained observations, such as time series of experimentally measurable variables. Over the past decade, stochastic thermodynamics has provided several inequalities that yield model-free lower bounds on entropy production from such coarse-grained data. The major approaches rely on observations of coarse-grained states, fluctuating currents or ticks, correlation functions of coarse-grained observables, and waiting-time distributions between so-called Markovian events, which correspond to transitions between mesoscopic states. Here, we systematically review these techniques valid under the sole assumption of a Markovian, i.e., memoryless, dynamics on an underlying, not necessarily observable, network of states or following a possibly high-dimensional Langevin equation. We discuss in detail the large class of non-equilibrium steady states and highlight extensions of these methods to time-dependent and relaxing systems. While our focus is on mean entropy production, we also summarize recent progress in quantifying entropy production along individual coarse-grained trajectories.

Figures (3)

• Figure 1: Several strategies for data extraction from coarse-grained trajectories based on partial observations. Top: An underlying mesoscopic network of 11 states and corresponding trajectory $\gamma=[i(t)]$. Middle: When only the three lumped states ${\mathcal{I}},{\mathcal{J}},{\mathcal{K}}$ are observed, the coarse-grained trajectory $\Gamma=[{\mathcal{I}} (t)]$ yields fluxes $\nu_{{\mathcal{I}}{\mathcal{J}}}$ (blue double arrows) between these three states that enter the lower bound $\sigma_{\textrm{app}}$ in (\ref{['eq:sigma-app']}). Bottom: This observer can record only the transitions between states 4 and 10, denoted as $I$ and $\tilde{I}$, and those between 5 and 6, denoted as $J$ and $\tilde{J}$. Expressed as a time series of + and -- events, the mean and variance of the corresponding current $j$ (\ref{['eq:j']}), with $d_{4,10}=-d_{10,4}=d_{5,9}=-d_{9,5}=1$, enters the TUR-bound in $\sigma_{\textrm{TUR}}$ (\ref{['eq:tur']}). For evaluating the bound $\sigma_{\textrm{WTD}}$ in (\ref{['eq:sig-wtd']}), the waiting-time distributions $\psi_{IJ}(t)$ of two consecutive transitions are required. As an example, the time-intervals indicated by $t'$ and $t"$ contribute to $\psi_{\tilde{I}\tilde{I}}(t)$ and $\psi_{\tilde{I}\tilde{J}}(t)$, respectively.
• Figure 2: Markovian events, snippets and additional data for a continuous dynamics. Several fine-grained trajectories pass through the point $\alpha$ and, after the interval $t$, through the point $\beta$ which denotes two Markovian events. Some of these trajectories additionally cross the manifolds $M$ or $L$, or visit the region $A$, which enter as data ${\mathcal{O}}_{\alpha\beta}$ the lower bound $\sigma_{\textrm{WTD}}$ given in (\ref{['eq:sig-wtd-o']}). For a coarse-grained trajectory $\Gamma$, the part between $\alpha$ and $\beta$ makes up one snippet.
• Figure 3: Coarse-graining Markovian events. The observer records transitions from the set $A$ that leave the lumped state ${\mathcal{I}}$, and transitions from the set $\tilde{A}$ that enters this state, without resolving the underlying fine-grained events contributing to $A=\{4\to 10, 4\to 9, 2\to 6\}$ and $\tilde{A}=\{10\to 4,9\to 4, 6\to 2\}$. The observer also detects transitions between states 7 and 8 but cannot determine their direction, implying $B=\tilde{B}=\{7\to 8, 8 \to 7\}$. The non-vanishing contributions to the first sum in (\ref{['eq:sig-wtd-be']}) are the pairs $AB$ and $B\tilde{A}$. The second sum runs over $A$ and $\tilde{A}$ for this example.