Consistent time reversal and reliable and accurate inference in the presence of memory

TL;DR

The paper addresses the fundamental challenge of inferring dissipation from coarse-grained, memory-bearing observations. It introduces a measure-theoretic framework and a family of $k$-th order estimators $\dot{S}^{\rm est}_{k}$ that provide guaranteed lower bounds on the true entropy production rate when memory is present, saturating at the actual memory order. By applying these estimators to diverse network topologies (cycle graphs, trees, Sierpinski fractals, Brusselator, and tree-diamond graphs), the authors demonstrate that neglecting memory yields artefacts such as spurious power-law scalings, while properly accounting for memory recovers the correct dissipation and reveals the presence or absence of hidden dissipative cycles. The work establishes rigorous conditions for estimator robustness to overestimation and clarifies how memory controls physically meaningful time-reversal, with significant implications for multi-scale thermodynamic inference and experimental design. Overall, the framework enables reliable, memory-consistent thermodynamic inference from coarse observations and cautions against interpreting apparent scale-dependent dissipation without saturating the memory-aware estimator.

Abstract

Thermodynamic inference from coarse observations remains a key challenge. Memory, in particular correlations between consecutively observed mesostates, blur signatures of irreversibility and must be accounted for in defining physical time-reversal, which remains an open problem. We derive an experimentally accessible k-th order estimator for the entropy production rate. Using novel measure-theoretic techniques we prove necessary and sufficient conditions for guaranteed lower bounds on the dissipation even in the strongly non-Markovian setting. The proof reveals that estimators saturated in the order unravel the duration of memory which needs to be considered in defining physically consistent time-reversal. We show that Markovian estimators in absence of a time-scale separation lead to artifacts, which convey no physical meaning. Similarly, estimators not saturated in the order may overestimate the dissipation. The necessity of correctly accounting for memory in thermodynamic inference from strongly non-Markovian observations underscores the still underappreciated challenges and intricacies in defining and understanding irreversibility in presence of memory. Our results will hopefully stimulate experiments systematically considering thermodynamic inference on multiple scales consistently accounting for memory.

Figures (16)

• Figure 1: Coarse graining (left) and estimators (right) for $u=v=2$: $\dot{S}^{\mathrm{est}}_{k}$ allows for good inference of $\dot{S}$ already for $k=2$ and saturates in the order for $k\geq 3$.
• Figure 2: Model systems: (a) Cycle graph with transition rates $\omega$ and $\kappa$ in counter-clockwise ($+$) and clockwise ($-$) direction, respectively. Lumps $L_1, L_2, \ldots, L_{n/\lambda}$ are highlighted in blue.(b) Tree of depth $d=5$ with lumps of depth $l=1$ (shaded in blue) and depth $l=2$ (circled in dashed red). Transition rates down $+$ and up $-$ (cyclic with respect to the thick edge) are set ${\propto \omega}$ and ${\propto \kappa}$, respectively, chosen such that the cumulative transition rate between each level is ${\omega - \kappa}$. Self-similarity becomes obvious upon noticing that each tree consists of multiple smaller trees. Inset: Four fundamental cycles passing through the highlighted subgraph on levels $U_3$, $U_4$ and corresponding transition rates. (c) Self-similar Sierpinski-type graph with lumps of size $\lambda=12$ indicated by the respective vertex coloring. A stationary current $\omega$ runs through the outer side of each polygon, whereby the size of said polygons depends on the respective recursive depth. The depicted graph has recursion depth $5$ and $n=768$ vertices. Inset:Furthermore, a corresponding sequence of increasing jumptimes in $[0,t]$ must exist.lement of recursion-depth $2$ with all irreversible currents indicated by arrows. (d) Brusselator depicted as a grid graph as in yuInversePowerLaw2021. A portion of the total $n = 450 \times 450$ states is shown with lumps of size $\lambda = 2 \times 2$ (shaded in blue) and $\lambda = 4 \times 4$ (circled in dashed red).
• Figure 3: Entropy-production estimates for different coarse-graining scales: (a) $\dot S^{\rm est}_k$ for the ring graph with $n = 60$ states. Higher-order estimators ($k \geq 2$) allows for correct inference of $\dot{S}$ (green, blue), while omitting memory leads to an artefactual power law (black and red lines). Analytical results are fully corroborated by simulations. (b) $\dot S^{\rm est}_k$ for the tree of depth $d=23$, i.e. $n = 16777215$. Higher-order estimators allow for correct inference of $\dot{S}$ (green, blue), whereas ignoring memory leads to a spurious inverse energy cascade (black line), which may be easily mistaken for a power law (red dashed fit). In both cases, we considered $100$ independent stationary trajectories each visiting $10^9$ microscopic states.
• Figure 4: Entropy-production estimates for different coarse-graining scales: (a) $\dot{S}^{\rm est}$ for Sierpinski-type graph for recursion depth 7 (${n = 12288}$ vertices). We find $\dot{S}_{\rm M}\propto\lambda^{-1}$ while higher-order estimators $\dot{S}^{\rm est}_{k\ge 2}$ yield a power law with ${\alpha=0.93}$, corroborating numerically that the lumped process is a $2^{\rm nd}$ order semi-Markov process. (b) $\dot{S}^{\rm est}$ for the Brusselator with $n = 202500$ vertices. The virtual power law for the Markov estimate ${\dot{S}_1^{\rm est}\propto \lambda^{-0.52}}$ is faster decaying than $\dot{S}^{\rm est}_{k\ge 2}\propto\lambda^{-0.31}$ obtained by accounting for memory. For details, see SM.
• Figure 5: (a) 6-state Markov process coarse-grained to process of order $n>3$. The order is at least 3 because the next transition out of $L_3$ depends on whether we arrived from lump $L_1$ or $L_4$ two state-changes before. Since multiple jumps between $L_2 \leftrightarrow L_3$ may have occurred before, the order is in fact higher, but the influence of these back-and-forth jumps vanishes exponentially. (b) Entropy production rate estimated with $\dot{S}^{\rm est}_k$ as a function of $k$ from trajectories of different length. While undersampling effects occur for higher orders (red and blue lines do not agree for $k\geq 8$), the overshoot for lower orders is not an artifact of undersampling. This example explicitly shows that second-order estimators are not necessarily a lower bound. Notably, the numerical algorithm for determining all estimators up to order 10 completes within a few seconds (red line) and minutes (blue line) on a normal desktop computer; for details see SM.

Theorems & Definitions (13)

• Theorem 1
• proof : Proof of Theorem. \ref{['thm:estimator-gives-lower-bound']}
• Theorem 2: Robustness against overestimation
• Lemma 1
• proof : Proof of Lemma \ref{['lem:estimator-consistent-in-order']}
• proof : Proof of Theorem \ref{['thm:robustness-to-overestimation']}
• Lemma 2
• proof
• Corollary 1
• Corollary 2