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Evaluating non-equilibrium trajectories via mean back relaxation: Dependence on length and time scales

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Pith's one-line read For Gaussian trajectories, the variance of mean back relaxation is set exactly by the mean square displacement, and the statistical error is minimized at l ≈ sqrt(MSD(τ)); cell data deviate, marking non-Gaussianity.

desk verdict The Gaussian VBR formula is a genuine, useful analytical result, but the cell-level conclusions need a stationarity control and public data before they convince. read the letter →

arxiv 2507.05912 v2 pith:UO74Z53N submitted 2025-07-08 cond-mat.stat-mech cond-mat.softphysics.bio-ph

Gabriel Knotz , Till M. Muenker , Timo Betz , Matthias Krüger This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softphysics.bio-ph
keywords meanbackrelaxationvarianceofeffectiveenergyGaussianprocessnon-Gaussianitymicrorheologydetailedbalanceactivematter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Mean back relaxation (MBR) compares a particle's future displacement with its past displacement and has been proposed as a probe of broken detailed balance. This paper asks how the two time parameters (conditioning time $\tau$ and observation time $t$) and one length cutoff $l$ control MBR and its statistical error. The central result is an exact formula for the variance of back relaxation (VBR) valid for any stationary Gaussian process, expressed purely through the mean squared displacement (MSD). The formula shows VBR has a unique minimum in $l$, located at $l \approx 0.925\sqrt{\Delta x^2(\tau)}$, so the rule of thumb $l \approx \sqrt{\Delta x^2(\tau)}$ yields the smallest error bar in MBR measurements. Comparing VBR from probe particles in A549 cells with the Gaussian prediction reveals a clear excess, serving as a non-Gaussianity marker for intracellular motion.

What carries the argument

The central object is the variance of back relaxation (VBR), the mean-square fluctuation of the ratio $\delta x(t)/\delta x(-\tau)$ weighted by the cutoff $\vartheta_l$. For Gaussian processes the paper derives the closed-form identity $$\mathrm{VBR} = \mathrm{MBR}^2\left[(\$\alpha$-1)g(\eta)+h(\eta)\right],$$ where $g$ and $h$ are explicit functions of $\eta = l/\sqrt{2\Delta x^2(\tau)}$ built from error functions, and $\alpha$ is a dimensionless ratio of MSDs. This identity does three pieces of work: it proves VBR diverges at small and large $l$, it locates the minimum of VBR in $l$ at $\eta_{\min}\le 0.654$ (hence $l\approx 0.925\sqrt{\Delta x^2(\tau)}$), and it supplies a data-only null model—since all inputs are the measured MSD and MBR—against which cell data can be tested for Gaussianity.

What would settle it

Compute VBR at fixed τ and t from long simulated trajectories of a stationary Gaussian process (e.g., Brownian motion in a harmonic trap) and check that the l-dependence follows the closed-form formula with its minimum at η_min ≤ 0.654; any systematic deviation would falsify the 'any Gaussian process' claim. For the non-Gaussianity conclusion, repeat the VBR comparison on an equilibrium viscoelastic sample with the same MSD as the cells; if the excess over the Gaussian curve also appears there, the marker is not specific to active non-Gaussian intracellular motion.

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Extended reading notes

Core claim

The authors establish that for any stationary Gaussian process the variance of the mean back relaxation is exactly $$\mathrm{VBR}(\tau,t,l) = \mathrm{MBR}(\tau,t)^2\left[(\$\alpha$(\tau,t)-1)\,g(\eta)+h(\eta)\right],$$ with $\eta = l/\sqrt{2\Delta x^2(\tau)}$, $\alpha = \frac{1}{\mathrm{MBR}^2}\frac{\Delta x^2(t)}{\Delta x^2(\tau)}$, and $g,h$ given in terms of complementary error functions. MBR itself is independent of $l$ for Gaussian processes, but VBR depends on $l$ and diverges for both $l\to 0$ and $l\to\infty$; its single minimum lies at $\eta_{\min}\le 0.654$, i.e., at $l\approx 0.925\sqrt{\Delta x^2(\tau)}$. The authors further show that the 'Random Horse and Cart' model, a linear Gaussian model with a nonreciprocally driven trap and a memory-adding bath particle, reproduces the characteristic non-monotonic dependence of MBR on $t$ and its dependence on $\tau$ found in living cells. They extend the phenomenological linear relation between effective energy amplitude $E_0$ and the long-time MBR value to a larger range of $\tau$ values. Finally, they compare VBR of the cell data with the Gaussian prediction and find that the cell VBR lies above it, demonstrating that the intracellular process is non-Gaussian.

Load-bearing premise

The cell trajectories are treated as a stationary, time-translation-invariant process with a well-defined mean, although the paper concedes that no mean position can be claimed for probe particles in cells; if slow drift or active remodeling occurs on the experimental window, the Gaussian reference and the MBR–E0 correlation are contaminated.

Editorial extensions

If this is right

  • For any stationary Gaussian process, the VBR formula gives the statistical error of MBR as a function of the length cutoff, so experiments should set l ≈ √Δx²(τ) to minimize the error bar.
  • Because MBR is l-independent but VBR is not, the length parameter can be tuned to reduce noise without biasing the MBR value, for Gaussian data.
  • The cell-data VBR lying above the Gaussian prediction provides a practical non-Gaussianity test that only requires passive trajectory data and the MSD.
  • The E0–MBR linear relation holds for a wider range of conditioning times τ (up to 0.01 s in the cell data), making effective-energy estimates accessible to setups with lower temporal resolution.
  • The Random Horse and Cart model with a bath particle reproduces the non-monotonic MBR(t) curves seen in cells, indicating that memory is the source of this shape.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Eq. (40) is purely a function of MSD and MBR, it can be evaluated in any experiment that already measures trajectories; the same formula could be used to retroactively re-analyze existing passive microrheology data sets and report error bars on previously published MBR values.
  • The VBR excess could be turned into a quantitative non-Gaussianity index, e.g., the ratio of measured VBR to the Gaussian prediction at a fixed η; this index might correlate with the activity parameter Dq or with the effective energy amplitude E0 across cell types.
  • The η_min ≤ 0.654 bound is derived for Gaussian processes; for genuinely non-Gaussian cell dynamics, the optimal l for MBR evaluation may differ, but the Gaussian rule of thumb remains a safe starting point because the VBR curve is flat around the minimum.
  • A testable extension would be to measure VBR in reconstituted active networks with controlled activity; if the VBR excess scales with ATP-driven activity, VBR becomes a simple optical-microscopy readout of nonequilibrium activity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. This paper analyzes the mean back relaxation (MBR) observable for stochastic trajectories, focusing on its dependence on the conditioning time τ, observation time t, and length cutoff l. The authors introduce a modified "Random Horse and Cart" (RHC) model with memory and show it reproduces qualitative features of MBR in cell data, including non-monotonic time dependence and a τ-dependent long-time plateau. They extend the phenomenological relation between the effective energy amplitude E0 and the long-time MBR to finite τ, for both cells and the model. The central new result is the variance of back relaxation (VBR): for stationary Gaussian processes, the authors derive a closed-form expression, Eq. (40), linking VBR to the MSD and a dimensionless length η, and identify the optimal length l ≈ 0.925√(Δx²(τ)) that minimizes VBR. They verify this formula in simulations of the RHC model and then compare cell data to the Gaussian prediction, finding an excess in VBR that they interpret as evidence of non-Gaussian dynamics.

Significance. The Gaussian VBR result is a substantive and useful contribution: it gives a parameter-free (up to the MSD) expression for the statistical error of MBR, provides a concrete rule for choosing the length cutoff, and yields a potential non-Gaussianity diagnostic. The derivation in Appendix A1 is internally consistent and is corroborated by the RHC simulations in Fig. 10b, which strengthens confidence in the algebra. The RHC model, though simple, captures several non-trivial features of the cell MBR phenomenology, and the extension of the E0–MBR linear relation to larger τ is of practical value for lower-resolution experiments. The main caveat is that the two cell-data conclusions—non-Gaussianity and the optimal-l rule—assume stationary increments, and this assumption is not tested in the manuscript.

major comments (2)
  1. [Sec. V D, Fig. 10(a)] The conclusion that cell trajectories are non-Gaussian rests on comparing VBR computed from the cell data with the prediction of Eq. (40), which is derived in Appendix A1 under the explicit assumption of a stationary Gaussian process. For the prediction to be a valid reference, the cell increments must be time-translation invariant so that the empirical MSD equals the ensemble MSD and the three-time distribution W3 in Eq. (3) is stationary. The manuscript performs no stationarity control: there is no detrending, no sub-window comparison of MBR or VBR, and no check of increment time-reversal symmetry. Since Sec. II concedes that a mean position ⟨x⟩ cannot be claimed to exist for probe particles in cells, slow drift, cytoskeletal remodeling, or active transport over the roughly one-second windows could break increment stationarity and generate an excess VBR even for a Gaussian process. This would make the non-Gaussianity claim a false positive. Please add explicit stationarity checks (e.g., comparing MBR/VBR from disjoint sub-windows, detrending, or testing time-reversal symmetry of increments) or substantially weaken the claim to report a deviation from the stationary Gaussian reference rather than a demonstration of non-Gaussianity.
  2. [Sec. V D, Discussion] The practical recommendation l ≈ √(Δx²(τ)) is derived from the Gaussian minimum of VBR, η_min ≤ 0.654 (Eq. (44) and Fig. 9). The transfer of this rule to cell data is justified by the qualitative flatness of the cell VBR curve around the minimum, but the location of the minimum depends on α(τ,t), which is itself computed from the same empirical MSD. If the cell process has non-stationary increments, the empirical MSD is not the stationary MSD, so both the reference VBR curve and the estimated optimal l are shifted. The rule of thumb is therefore only as reliable as the stationarity assumption. A stationarity control, or at least an explicit discussion of the sensitivity of η_min to drift, is needed before recommending l ≈ √(Δx²(τ)) for cell experiments.
minor comments (7)
  1. [Abstract and Sec. V B] The phrase "we determine its absolute minimum as a function of the length and time parameters" overstates the result: the minimum is computed with respect to η (or l) for fixed τ and t, while the dependence on t and τ is analyzed asymptotically but not jointly minimized. Please rephrase to avoid ambiguity.
  2. [Sec. V B, after Eq. (45)] The sentence "the statement hols true for τ and t exchanged" contains a typo; it should read "holds true."
  3. [Eq. (20)] The displayed formula for MBR(τ → ∞, t → ∞) is garbled in the text; please rewrite the expression with clear notation for the prefactor and the τ dependence.
  4. [Fig. 10(a)] The red data points for cell VBR are shown without error bars or an uncertainty estimate; adding them, even as approximate bootstrap intervals, would strengthen the comparison with the Gaussian prediction.
  5. [Sec. IV C] The sentence "limiting ourselves to τ ≤ 0.01 s; In Fig. 2(a), we see that, in this regime, MBR decreases with τ" uses a semicolon and capitalization awkwardly; also clarify that this restriction is chosen to avoid the MBR minimum.
  6. [Fig. 4 caption] The caption contains "fir τ → 0" which should be "for τ → 0."
  7. [Appendix A2] The word "ration" in the sentence about the critical activity should be "ratio."

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central VBR formula is derived from Gaussian conditioning and validated against simulation; self-citations are applied results rather than fitted predictions.

full rationale

The paper's central new result, Eq. (40) for VBR of a Gaussian process, is derived in Appendix A1 from the Gaussian conditional distribution p(b|d) in Eq. (A2) and is not assumed as an input. It is then checked against numerical simulations of the RHC model in Fig. 10(b), which provides independent support. The Gaussian MBR relation in Eq. (5) is cited from Ref. 34, but it is a parameter-free mathematical identity for stationary Gaussian processes and is used as a tool rather than as a fitted or target-dependent input; even the VBR derivation leaves MBR as an explicit quantity and does not require Eq. (5) for its validity. The E0-MBR relation in cell data, Eq. (38), is presented as an empirical fit with reported R2 values and is not called a prediction, so no fitted input is being relabeled as a prediction. The non-Gaussianity test in Fig. 10(a) constructs a Gaussian null prediction from the empirical MSD of the same trajectories, but this is a standard goodness-of-fit construction: the MSD is a different observable from VBR, and no VBR value is used to tune the prediction, so the comparison is not circular by construction. The stationarity caveat for cell data is explicitly acknowledged by the authors in Section II, and while non-stationary drift could confound the Gaussian reference, that is a correctness risk rather than a circular reduction. I therefore find no circular step that reduces a predicted result to its own input, and the derivation is self-contained apart from routinely applying prior mathematical identities.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles, fields, forces, or conserved quantities are postulated. The diffusing trap q (the horse) and the bath particle x2 are mechanical degrees of freedom of a linear Langevin model, used to generate phenomenology, not claims about physical components of the cell. The free parameters are model ratios chosen for qualitative agreement, plus fit parameters (E0, λ(τ)) extracted from cell data. The axioms are standard Gaussian process tools, the model's own structure, and the unproven stationarity premise for cells.

free parameters (5)
  • RHC model dimensionless ratios k1/k2, γ1/γ2, Dq/D1 = k1/k2 = 1.0, γ1/γ2 = 0.2, Dq/D1 = 0.5 (Fig. 1b)
    Chosen by hand to reproduce the non-monotonic MBR(t) shapes and timescales observed in A549 cell data; the qualitative agreement is the evidence, not a fit to cell data.
  • Effective energy amplitude E0 = per cell type, from EEff = E0(f0/f)^ν + kBT with f0 = 1 Hz, ν ≈ 1 (Sec. IV C)
    E0 is extracted by fitting the measured EEff(f) spectra; it is the y-coordinate in the linear E0-MBR relation.
  • Slope λ(τ) of E0 versus (MBR - 1/2) = one value per τ (Fig. 7a, Supplementary Table I)
    The linear relation E0 = λ(τ)(MBR - 1/2) is fit per τ across roughly seven cell types; λ(τ) is then compared to the model's predicted slope.
  • Length cutoff l for cell MBR = l = 0.002 µm (Figs. 1a, 2a)
    Fixed at roughly the experimental tracking resolution (about 2 nm); the paper does not investigate MBR's l-dependence in cells.
  • Effective energy exponent ν = ν ≈ 1
    Fixed at about 1 for all cell types (Eq. 37), based on prior work (Ref. 33); the fit then determines E0.
assumptions (5)
  • domain assumption Gaussian MSD-MBR relation, Eq. (5): MBR = 1/2 (1 - [Δx²(t+τ) - Δx²(t)]/Δx²(τ))
    Bridges the RHC model's MSD to its MBR; published by the same authors (Ref. 34) and used here rather than re-derived. Exact for the linear Gaussian RHC model, so its use inside the model is safe.
  • domain assumption Stationarity and time-translation invariance of cell trajectories
    The experimental MBR, VBR, and E0-MBR analyses assume time-translation-invariant statistics; Sec. II admits a mean position ⟨x⟩ cannot be claimed to exist in cells, leaving this premise unproven for the central experimental object.
  • domain assumption Overdamped Langevin description of the probe with white noise and Einstein relations for x1 and x2 (Eq. 6)
    Structural assumption of the RHC model used for all analytic MSD, MBR, and EEff results in Secs. III and IV.
  • domain assumption Gaussianity of the joint (b, d) process
    Premise of the VBR derivation in Appendix A1; the paper then tests and rejects it against cell data, so it functions as a null model rather than an asserted fact about cells.
  • domain assumption Effective energy definition and power-law fit form EEff = E0(ω0/ω)^ν + kBT with ν ≈ 1
    EEff = ωC/(2χ'') is the standard FDT-violation measure (Eq. 24); the specific ω^-ν ansatz with ν ≈ 1 comes from Ref. 33 and is used to assign E0 per cell type.

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Pith. "Pith review of Evaluating non-equilibrium trajectories via mean back relaxation: Dependence on length and time scales." pith.science (2026). https://pith.science/paper/UO74Z53N

@misc{pith2026250705912,
  author       = {Pith},
  title        = {Pith review of: Evaluating non-equilibrium trajectories via mean back relaxation: Dependence on length and time scales},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UO74Z53N}},
  note         = {Machine review of arXiv:2507.05912}
}
read the original abstract

The mean back relaxation (MBR) relates the value of a stochastic process at three different time points. It has been shown to detect broken detailed balance under certain conditions. For experiments of probe particles in living and passivated cells, MBR was found to be related to the so called effective energy, which quantifies the violation of the fluctuation dissipation theorem. In this manuscript, we discuss the dependence on the length and time parameters that enter MBR, both for cells as well as for a model system, finding qualitative agreement between the two. For the cell data, we extend the phenomenological relation between MBR and effective energy to a larger range of time parameters compared to previous work, allowing to test it in systems with limited resolution. We analyze the variance of back relaxation (VBR) in dependence of the mentioned parameters, relevant for the statistical error in MBR evaluation. For Gaussian systems, the variance is found analytically in terms of the mean squared displacement, and we determine its absolute minimum as a function of the length and time parameters. Comparing VBR from cell data to a Gaussian prediction demonstrates a non-Gaussian process.

Figures

Figures reproduced from arXiv: 2507.05912 by the authors.

Figure 1
Figure 1. (a) MBR obtained from experimental data on A549 cells as a function of time [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. (a) Experiments: MBR(t, τ, l) at t =1 s and l = 0.002 µm for A549 cells as a function of time τ . For increasing τ , MBR decreases, gets negative and decays to 0 from below. The minimum is approximately reached at τ ≈ 0.04 s. In the main panel, the x-axis is in log scale, and in the inset, we see the same data but in linear scale. (b) RHC Model: The long time limit, t → ∞, of MBR as a function of τ for different act… view at source ↗
Figure 3
Figure 3. Sketch of the ”Random Horse and Cart” (RHC) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Regimes of qualitatively different shapes of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Illustration of MBR as resulting from MSD of RHC, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 4
Figure 4. Figure 4: In the following, we focus on parameters for which MBR shows a maximum as a function of t, as this is the behavior observed in the cell data in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: (a) Amplitude E0 of effective energy against the long time limit of MBR for multiple τ between 5 ×10−5 s and 0.01 s, for different cell types as labeled. For each τ value the values from different cells fall on a straight line, which is fitted and also shown. For incre…
Figure 7
Figure 7. Figure 7: (a) Slope λ(t) of the linear relation between E0 and long time limit of MBR, extracted from [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Variance of back relaxation (VBR) versus the length [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: Position of minimum of VBR, ηmin, as a function of α. Solid blue line shows the numerically exact solution, and the dotted grey and red lines show the zeroth and first order terms of Eq. (44), respectively. Inset shows VBR, evaluated at ηmin, as a function of α − 1, di…
Figure 10
Figure 10. Figure 10: (a) VBR from A549 cells (red dots) compared to the Gaussian prediction using the particle’s MSD via Eq. (40) (grey [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.