arxiv: 2604.06162 · v1 · submitted 2026-04-07 · ❄️ cond-mat.stat-mech

Recognition: 2 theorem links

· Lean Theorem

Mutual Linearity in and out of Stationarity for Markov Jump Processes: A Trajectory-Based Approach

Jiming Zheng , Zhiyue Lu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:20 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords Markov jump processesmutual linearitytrajectory linear responsenonequilibrium responsenon-stationary dynamicsstate observablescounting observables

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The pith

Trajectory-based derivation extends mutual linearity to non-stationary relaxation in Markov jump processes.

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

The paper develops a derivation of mutual linearity that does not require the system to be in a stationary state. Using linear response expressed along individual stochastic trajectories, it shows that changing the rate of one transition makes two different observables linearly dependent even while the process is still relaxing. The result covers both observables that depend on the current state and those that count transitions. A reader would care because the relation now applies during the time-dependent approach to equilibrium rather than only after long times have passed.

Core claim

By applying trajectory-level linear response theory to a Markov jump process whose single-edge transition rate is perturbed, the authors derive that the resulting changes in any two observables remain linearly dependent throughout the non-stationary relaxation phase, for both state observables and counting observables.

What carries the argument

trajectory-level linear response theory, which expresses the first-order change in an observable expectation as an average over perturbed trajectories

If this is right

Mutual linearity applies directly to non-stationary relaxation for state observables.
The same linear dependence holds for counting observables during relaxation.
The trajectory formulation identifies the path-level origin of the mutual linearity relation.
The method opens a route to the same relation in diffusion processes and open quantum systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests checking whether similar linear relations appear in experimental time-series data from chemical reaction networks that have not yet reached steady state.
It may link to other nonequilibrium response identities that are currently stated only for stationary conditions.
Simple two- or three-state jump models could be simulated to quantify how quickly the linear dependence emerges after a rate perturbation.

Load-bearing premise

The derivation assumes trajectory-level linear response theory remains valid when applied to the non-stationary relaxation dynamics of Markov jump processes.

What would settle it

A numerical simulation of a small Markov jump network in which the linear dependence between two observables is measured during the transient after a single-edge rate perturbation and found to deviate systematically from the predicted relation while still holding in the long-time limit.

Figures

Figures reproduced from arXiv: 2604.06162 by Jiming Zheng, Zhiyue Lu.

read the original abstract

Nonequilibrium response theory is a fundamental framework for understanding how physical systems respond to perturbations. Recently, a mutual linearity has been discovered for Markov jump processes using linear algebra analysis. This mutual linearity states that two observables are linearly dependent on each other in the long-time limit when the transition rate of a single edge is altered. It has also been extended to non-stationary cases for current observables. In this work, we provide a trajectory-based derivation of mutual linearity utilizing the trajectory-level linear response theory. The trajectory approach allows us to generalize the mutual linearity to non-stationary relaxation dynamics for state observables and counting observables. Our results shed light on the fundamental response properties far from equilibrium and the trajectory-level origin of mutual linearity. Our trajectory-based approach makes it possible to generalize the mutual linearity to a broader class of systems, including diffusion processes and open quantum systems.

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.

Desk Editor's Note private letter to a colleague

Trajectory derivation extends mutual linearity to non-stationary state observables in MJPs, but the non-stationary claim likely needs explicit conditions on initial distributions and perturbation timing to hold exactly.

read the letter

The main point is that Zheng and Lu derive mutual linearity for Markov jump processes via trajectory-level linear response and use that to generalize it to non-stationary relaxation, including for state observables rather than just currents. This is the concrete advance over the earlier linear-algebra treatment they cite. The trajectory route is a genuine alternative and they correctly note it could carry over to diffusions or open quantum systems if the steps are solid. That part is useful for people who prefer physical pictures over matrix manipulations. The derivations themselves are not visible in the abstract, but the approach avoids obvious circularity with the prior work. The soft spot is the non-stationary extension. Standard trajectory response formulas are usually stated for stationary or long-time regimes, and a sudden rate change acting on an arbitrary initial distribution can produce transient corrections or dependence on the starting measure. The abstract gives no indication that the paper spells out the exact regime—whether the perturbation starts at t=0, whether the initial probability vector is restricted, or how transients are handled—so the claimed generality for relaxation dynamics may be narrower than stated or may require additional assumptions. No other red flags appear in the abstract: the citation pattern is straightforward, and there are no free parameters or invented entities. This is for readers already working on nonequilibrium response in stochastic systems who want a trajectory-based handle on mutual linearity. Someone in that niche would get value from the method even if they have to supply the missing regime conditions themselves. It deserves peer review because the idea is specific and the alternative derivation is distinct, though referees will need to see the full steps and any checks on the non-stationary case.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a trajectory-based derivation of mutual linearity for Markov jump processes, extending a prior linear-algebra result to non-stationary relaxation dynamics. Using trajectory-level linear response theory, it claims to establish that two observables remain linearly dependent when a single edge rate is perturbed, now holding for both state observables and counting observables during relaxation from arbitrary initial conditions. The approach is positioned as enabling future generalizations to diffusion processes and open quantum systems.

Significance. If the central derivation is rigorous and the non-stationary extension holds without hidden restrictions, the work supplies a trajectory-level origin for mutual linearity and broadens its scope beyond the long-time stationary limit. This could unify response properties across nonequilibrium Markovian systems and provide a template for similar relations in continuous-state or quantum settings.

major comments (2)

[Abstract and trajectory-response derivation] The non-stationary extension for state observables (Abstract and the section deriving the trajectory response) invokes trajectory-level linear response without stating the precise initial probability vector or the instant at which the rate perturbation is applied. Standard trajectory response formulas acquire transient corrections when the system starts far from stationarity; the manuscript must specify whether the perturbation occurs at t=0 from an arbitrary p(0) or after equilibration, and whether any additional averaging or long-time limit is taken to recover exact mutual linearity.
[Extension to counting observables] The claim that the same mutual-linearity relation holds for counting observables in the non-stationary regime (the paragraph extending prior work) should be accompanied by an explicit check that the integrated current response does not pick up initial-condition-dependent terms that would violate linearity. If the derivation relies on the same trajectory formula used for state observables, the paper must demonstrate why the counting case is free of the transient issues that could affect state observables.

minor comments (2)

[Notation] Notation for the perturbed rate matrix and the trajectory weight should be introduced once and used consistently; currently the abstract and main text alternate between different symbols for the same object.
[Introduction] The manuscript would benefit from a short table or diagram contrasting the stationary linear-algebra result with the new trajectory result, highlighting which assumptions are relaxed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. The comments correctly identify places where the original manuscript was insufficiently explicit about initial conditions and the handling of transients for counting observables. We have revised the manuscript to address both points directly, adding the requested specifications and an explicit verification that no initial-condition-dependent terms violate the linearity for counting observables.

read point-by-point responses

Referee: [Abstract and trajectory-response derivation] The non-stationary extension for state observables (Abstract and the section deriving the trajectory response) invokes trajectory-level linear response without stating the precise initial probability vector or the instant at which the rate perturbation is applied. Standard trajectory response formulas acquire transient corrections when the system starts far from stationarity; the manuscript must specify whether the perturbation occurs at t=0 from an arbitrary p(0) or after equilibration, and whether any additional averaging or long-time limit is taken to recover exact mutual linearity.

Authors: We agree that the original text did not state these details with sufficient precision. In the revised manuscript we now explicitly specify that the single-edge rate perturbation is applied at t=0 to a system prepared in an arbitrary initial distribution p(0). The trajectory-level linear response is used at finite times without any long-time limit or additional ensemble averaging. The mutual linearity follows directly from the structure of the first-order response for a single perturbed edge; the transient corrections that appear in the individual response functions are identical (up to the observable) for the two observables under consideration and therefore cancel in the linear dependence relation. A new clarifying paragraph has been inserted in the derivation section. revision: yes
Referee: [Extension to counting observables] The claim that the same mutual-linearity relation holds for counting observables in the non-stationary regime (the paragraph extending prior work) should be accompanied by an explicit check that the integrated current response does not pick up initial-condition-dependent terms that would violate linearity. If the derivation relies on the same trajectory formula used for state observables, the paper must demonstrate why the counting case is free of the transient issues that could affect state observables.

Authors: We thank the referee for this observation. The integrated counting observable is constructed from the same trajectory response formula, but the initial-distribution contributions appear only in the unperturbed part of the current and therefore drop out of the first-order response. Consequently they cannot break the linear relation between the two counting observables. To make this transparent we have added an explicit appendix calculation that isolates the initial-condition term and shows it cancels identically in the mutual-linearity identity. This demonstration is independent of the particular form of p(0) and confirms that no additional transient corrections arise for the counting case. revision: yes

Circularity Check

0 steps flagged

Independent trajectory derivation; prior linear-algebra result cited but not load-bearing

full rationale

The paper cites a prior linear-algebra discovery of mutual linearity but then supplies a distinct trajectory-based derivation that invokes trajectory-level linear response theory to extend the relation to non-stationary relaxation for both state and counting observables. No equation or claim reduces by construction to the cited result or to a fitted parameter; the derivation chain stands on its own trajectory formalism. The single self-citation is therefore minor and non-load-bearing, consistent with normal scholarly practice when an independent route is presented.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard assumptions of Markov jump processes and trajectory linear response theory.

axioms (1)

domain assumption Markov jump processes obey standard stochastic trajectory dynamics under rate changes
Invoked implicitly for the trajectory-level linear response framework.

pith-pipeline@v0.9.0 · 5446 in / 1050 out tokens · 54253 ms · 2026-05-10T18:20:23.231416+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

d⟨Q(τ)⟩/dr_ij = ∫ dt (1/r_ij) ⟨Q(τ) dε_ij(t)⟩; response function R_rij(τ,t) = p_j(t) ∑_l c_l ∫ ds [P(l,s|i)-P(l,s|j)]
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

lim τ→∞ d⟨Q_m(τ)⟩/dr_ij = π_j τ ∑_l c_l^{(m)} (Z_li - Z_lj) + O(1); ratio χ independent of r_ij

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Graph theoretic derivation of mutual linearity for transient probabilities and hitting time distributions in Markov networks
cond-mat.stat-mech 2026-04 unverdicted novelty 7.0

Graph theory yields explicit combinatorial formulas showing mutual linearity for transient occupation probabilities and hitting time distributions in Markov networks.
Mutual Linearity in Nonequilibrium Langevin Dynamics
cond-mat.stat-mech 2026-05 unverdicted novelty 6.0

Local perturbations in nonequilibrium Langevin dynamics induce linear relations between stationary densities and currents at different positions due to an underlying one-dimensional response structure.

Reference graph

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