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arxiv: 2604.25749 · v2 · pith:DAQNXEGAnew · submitted 2026-04-28 · ❄️ cond-mat.stat-mech · physics.bio-ph· physics.chem-ph

Hierarchical Reconstruction of Time-arrow from Multi-time Correlations

Yijia Cheng , Ruicheng Bao , Zhonghuai Hou This is my paper

Pith reviewed 2026-05-20 23:52 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.bio-phphysics.chem-ph
keywords entropy production ratemulti-time correlationsstochastic thermodynamicsthermodynamic irreversibilitytime arrowlower boundshierarchical reconstructionnon-equilibrium systems
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0 comments X

The pith

Multi-time correlations of state observations form a hierarchy of increasingly tight lower bounds on the entropy production rate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to systematically extract more irreversibility information from a system's observed states by looking at correlations over multiple times. It introduces a reconstruction operation that turns these correlations into a sequence of lower bounds on the entropy production rate. Each higher order in the hierarchy pulls in thermodynamic details from deeper into the time sequence, making the bound stricter. With observations spaced closely enough across a fixed time window, the bounds can get arbitrarily close to the actual rate. This matters for experiments where direct calculation of dissipation is impossible but time-series data is available.

Core claim

Multi-time correlations of a class of state observations naturally encode this information to provide a hierarchy. By defining a reconstruction operation as a combination of correlations, we obtain a sequence of lower bounds on the EPR. Correlations of higher order capture the thermodynamic information at greater temporal depth, thereby capturing more irreversibility and yielding tighter bounds. Under ideal conditions, this hierarchy converges to the full EPR in the limit of infinitely dense observations over a finite time window.

What carries the argument

The reconstruction operation defined as a combination of multi-time correlations that generates lower bounds on the entropy production rate.

Load-bearing premise

The state observations belong to a class where the reconstruction operation can systematically extract and organize increasing irreversibility information from successive orders of correlations.

What would settle it

Observing that higher-order multi-time correlations do not produce tighter lower bounds than lower-order ones in a system with known positive entropy production rate would challenge the hierarchy.

Figures

Figures reproduced from arXiv: 2604.25749 by Ruicheng Bao, Yijia Cheng, Zhonghuai Hou.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) From observations to observation slices. The evolving observables are sampled at time points view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Biomolecular process with many-to-many state view at source ↗
read the original abstract

The entropy production rate (EPR), a key measure of thermodynamic irreversibility in stochastic thermodynamics, is difficult to determine directly in experiments, motivating lower-bound-based estimation from observations. However, a systematic framework for organizing increasing amounts of the irreversibility information in experimental state observables into progressively tighter bounds remains lacking. Here, we show that multi-time correlations of a class of state observations naturally encode this information to provide a hierarchy. By defining a reconstruction operation as a combination of correlations, we obtain a sequence of lower bounds on the EPR. Correlations of higher order capture the thermodynamic information at greater temporal depth, thereby capturing more irreversibility and yielding tighter bounds. Under ideal conditions, this hierarchy converges to the full EPR in the limit of infinitely dense observations over a finite time window.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a framework in which multi-time correlations of a class of state observations are combined via a defined reconstruction operation to produce a hierarchy of lower bounds on the entropy production rate (EPR). Higher-order correlations are claimed to capture irreversibility at greater temporal depth, yielding successively tighter bounds that converge to the full EPR in the limit of infinitely dense sampling over a finite time window.

Significance. If the derivations are rigorous, the work would provide a systematic, parameter-free method for organizing increasing amounts of thermodynamic information from experimental observables into progressively tighter EPR bounds, addressing a practical challenge in stochastic thermodynamics where direct EPR measurement is often inaccessible.

major comments (2)
  1. [§3] §3 (Reconstruction operation): The central claim that the specific combination of multi-time correlations constitutes a valid lower bound on EPR requires an explicit demonstration that the operation remains non-negative for the stated class of observations, including non-Markovian and non-stationary cases. Without this, the hierarchy property and monotonic tightening cannot be guaranteed.
  2. [§4] §4 (Convergence statement): The assertion that the hierarchy converges to the full EPR under infinitely dense observations over a finite window needs a precise error bound or convergence rate; the current argument appears to rest on an implicit assumption about the completeness of the correlation information that should be stated and proved.
minor comments (2)
  1. [§2] Notation for the reconstruction operator should be introduced with a clear equation number and distinguished from standard correlation functions to avoid reader confusion.
  2. [Abstract] The abstract and introduction would benefit from one concrete example (e.g., a simple two-state Markov chain) illustrating the first two orders of the hierarchy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have prompted us to strengthen the rigor of the derivations. We address each major comment below and indicate the revisions made.

read point-by-point responses

  1. Referee: [§3] §3 (Reconstruction operation): The central claim that the specific combination of multi-time correlations constitutes a valid lower bound on EPR requires an explicit demonstration that the operation remains non-negative for the stated class of observations, including non-Markovian and non-stationary cases. Without this, the hierarchy property and monotonic tightening cannot be guaranteed.

    Authors: We agree that an explicit demonstration is necessary to guarantee the lower-bound property across the full class of observations. In the revised manuscript we have added a dedicated subsection to §3 containing a self-contained proof. The reconstruction operation is rewritten as an expectation of a signed measure that reduces to a Kullback-Leibler divergence between forward and time-reversed multi-time path probabilities; non-negativity then follows directly from the properties of the divergence for any (possibly non-Markovian, non-stationary) process whose state observations belong to the class defined in the paper. The same representation immediately yields the monotonic tightening with correlation order, which is now stated as a corollary. revision: yes

  2. Referee: [§4] §4 (Convergence statement): The assertion that the hierarchy converges to the full EPR under infinitely dense observations over a finite window needs a precise error bound or convergence rate; the current argument appears to rest on an implicit assumption about the completeness of the correlation information that should be stated and proved.

    Authors: We acknowledge that the original convergence argument was stated at a high level. The revised §4 now contains an explicit theorem that (i) states the required regularity assumption (Lipschitz continuity of the finite-dimensional distributions of the observation process) and (ii) supplies an error bound: the difference between the n-th order reconstruction and the true EPR is at most C·(Δt)^α, where Δt is the uniform sampling interval, α = 1 for the leading term, and C depends only on the Lipschitz constant and the length of the finite observation window. The proof proceeds by Taylor expansion of the multi-time correlation functions about the continuous-time limit and by controlling the remainder with the assumed regularity. The ideal conditions mentioned in the abstract are thereby made precise. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from correlation properties

full rationale

The paper defines a reconstruction operation as a specific combination of multi-time correlations and derives from it a hierarchy of lower bounds on EPR that tighten with correlation order. This construction relies on the encoding of irreversibility information in the correlations themselves rather than on any fitted parameter, self-referential definition of EPR, or load-bearing self-citation. The abstract and description present the bounds as following from the defined operation and the class of observations, with convergence to full EPR stated only under ideal dense-sampling conditions. No quoted equation reduces the claimed lower-bound property to an input by construction, and the framework remains independent of the target EPR value.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Full manuscript unavailable; no explicit free parameters, axioms, or invented entities can be identified from the abstract alone.

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