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arxiv: 2605.16247 · v1 · pith:WG7H7ULMnew · submitted 2026-05-15 · ❄️ cond-mat.stat-mech

Brownian motion: non-equilibrium states from equilibrium trajectories -- recovering hydrodynamic regimes from prepared displacement measurements

Jason Boynewicz , Michael C. Thumann , Giuseppe Procopio , Massimiliano Giona This is my paper

Pith reviewed 2026-05-19 18:31 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Brownian motionMarkovian dynamicsChapman-Kolmogorov equationhydrodynamic fluctuationsnon-equilibrium statesshort-time scalingfluid-particle interactionsthermal forces
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The pith

Any equilibrium Brownian trajectory decomposes into a superposition of non-equilibrium states via the Chapman-Kolmogorov equation.

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

The paper establishes that Markovian dynamics allow any single equilibrium path of a Brownian particle to be treated as an uncountable overlay of non-equilibrium trajectories. By extracting the second moments of position in trapped conditions, the approach isolates the effects of thermal-hydrodynamic forces and determines short-time displacement statistics directly from the force correlations. This recovers known hydrodynamic scalings such as t to the 5/2 while predicting a possible crossover to t to the 4 at even shorter times due to force regularity. A reader would care because the method extracts microscale fluid-particle details from ordinary equilibrium data without requiring specially prepared non-equilibrium experiments.

Core claim

Owing to the Chapman-Kolmogorov equation for Markovian dynamics, any equilibrium trajectory of a Brownian particle in a solvent fluid can be viewed as the superposition of an uncountable number of non-equilibrium states. This property permits the unraveling of fine details of fluid-particle interactions at microscales from the analysis of a single Brownian trajectory by considering the lower-order (second) moments of particle position in trapped conditions. In this way the acceleration due to thermal-hydrodynamic fluctuational forces is isolated and the short-time displacement statistics is completely determined by the correlation properties of the fluctuational thermal-hydrodynamic force.

What carries the argument

The Chapman-Kolmogorov equation for Markovian dynamics, which decomposes an equilibrium trajectory into a superposition of non-equilibrium states.

If this is right

  • Short-time displacement statistics are fixed solely by the correlation function of the thermal-hydrodynamic force.
  • Fluid inertial effects produce a t to the 5/2 scaling in mean-square displacement.
  • Correlated stochastic forcings can supersede this with a t to the 4 scaling at sufficiently short times.
  • Hydrodynamic regimes become recoverable from second-moment analysis of trapped-particle trajectories.

Where Pith is reading between the lines

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

  • The same decomposition could be applied to other Markovian processes to extract hidden force statistics from equilibrium records.
  • Precision experiments could test the crossover between t to the 5/2 and t to the 4 regimes to constrain the regularity of velocity paths.
  • The method offers a route to characterize solvent properties at microscales using only standard Brownian data.

Load-bearing premise

The particle dynamics must be strictly Markovian so that the Chapman-Kolmogorov equation applies directly to the equilibrium trajectory and permits its decomposition into non-equilibrium components.

What would settle it

High-resolution measurements of short-time particle displacements in a fluid that deviate from both the t to the 5/2 law and the predicted t to the 4 scaling once force correlations are accounted for.

Figures

Figures reproduced from arXiv: 2605.16247 by Giuseppe Procopio, Jason Boynewicz, Massimiliano Giona, Michael C. Thumann.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic representation of the subsampling proced [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Prototypical moment graphs associated with the Eins [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p039_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p043_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p045_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Portion of an orbit of a Brownian particle in a Sierpi [PITH_FULL_IMAGE:figures/full_fig_p046_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: shows the temporal evolution of mxx(t) starting from a Z-preparation, obtained from stochastic simulations of an ensemble of Np = 105 particles, in the case of a Stokesian dynamics (h(t) = δ(t), k(t) = 0), and for a Maxwell fluid (h(t) = λ e−λ t , k(t) = 0) character￾ized by a relaxation rate λ = 1. The Langevin equations have been integrated numerically via an Euler-Langevin algorithm with a step size hh… view at source ↗
read the original abstract

Owing to the Chapman-Kolmogorov equation for Markovian dynamics,any equilibrium trajectory of a Brownian particle in a solvent fluid can be viewed as the superposition of an uncountable number of non-equilibrium states. This property permits the unraveling of fine details of fluid-particle interactions at microscales defined by its non-equilibrium properties from the analysis of a single Brownian trajectory and to connect them to the hydrodynamics of the solvent fluid, simply considering the lower-order (second) moments of particle position in trapped conditions. In this way, the acceleration due to thermal-hydrodynamic fluctuational forces is isolated from the other factors and the short-time displacement statistics is completely determined by the correlation properties of the fluctuational thermal-hydrodynamic force. This approach not only confirms the $t^{5/2}$-law obtained by Boynewicz et al. (2026), related to fluid inertial effects, but indicates that this scaling may be superseded by a $t^4$-scaling at very short times once the correlated nature of the stochastic forcings is taken into account. The latter result is related to the regularity properties of particle velocity realizations.

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

1 major / 2 minor

Summary. The manuscript argues that the Chapman-Kolmogorov equation for Markovian dynamics permits any equilibrium Brownian trajectory to be viewed as a superposition of non-equilibrium states. This decomposition is used to extract microscale fluid-particle interaction details from the second moments of particle position under trapped conditions, isolating the effects of thermal-hydrodynamic fluctuational forces. The approach confirms the t^{5/2} scaling associated with fluid inertial effects and indicates that a t^4 scaling may supersede it at very short times once correlations in the stochastic forcing are accounted for, with the latter tied to the regularity properties of particle velocity realizations.

Significance. If the central decomposition and scaling results hold, the work would supply a practical route to probe hydrodynamic regimes directly from single equilibrium trajectories, avoiding the need for prepared non-equilibrium initial conditions. The explicit connection between force correlations and short-time displacement statistics yields falsifiable predictions that could be tested in optical-trap experiments. The manuscript builds on prior scaling results while extending them to incorporate correlated forcings.

major comments (1)
  1. Abstract: The central claim invokes the Chapman-Kolmogorov equation under the assumption of strictly Markovian dynamics to decompose the equilibrium trajectory into non-equilibrium components. Yet the t^4 scaling is introduced as arising from the correlated nature of the fluctuational forces, which (via a finite correlation time) generates a memory kernel in the generalized Langevin equation and renders the position process non-Markovian. This apparent inconsistency between the Markovian decomposition used for the overall framework and the non-Markovian short-time regime must be resolved explicitly, for instance by delineating the time scales on which each description applies or by showing how the superposition remains valid when correlations are present.
minor comments (2)
  1. Abstract: The phrase 'trapped conditions' is used without specifying the trap potential or how it modifies the second-moment analysis; a single clarifying sentence would improve accessibility.
  2. Abstract: The confirmation of the t^{5/2} law is referenced to Boynewicz et al. (2026) without a brief recap of the key steps or assumptions of that result; including such a summary would make the present manuscript more self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We have addressed the concern about the consistency between the Markovian framework and the non-Markovian short-time scaling by clarifying the applicable time scales in the revised manuscript.

read point-by-point responses

  1. Referee: Abstract: The central claim invokes the Chapman-Kolmogorov equation under the assumption of strictly Markovian dynamics to decompose the equilibrium trajectory into non-equilibrium components. Yet the t^4 scaling is introduced as arising from the correlated nature of the fluctuational forces, which (via a finite correlation time) generates a memory kernel in the generalized Langevin equation and renders the position process non-Markovian. This apparent inconsistency between the Markovian decomposition used for the overall framework and the non-Markovian short-time regime must be resolved explicitly, for instance by delineating the time scales on which each description applies or by showing how the superposition remains valid when correlations are present.

    Authors: We thank the referee for highlighting this potential inconsistency. Upon reflection, the Markovian assumption via the Chapman-Kolmogorov equation is valid for the equilibrium trajectory on time scales exceeding the correlation time of the thermal-hydrodynamic forces. The decomposition into non-equilibrium states is employed to interpret the displacement moments in this regime, confirming the t^{5/2} scaling due to fluid inertia. For shorter times, where force correlations lead to a memory kernel and non-Markovian behavior, we derive the t^4 scaling from the regularity of the velocity process. To address the referee's concern, we will explicitly delineate these regimes in the revised abstract and introduction: the superposition framework applies primarily to intermediate times, while the short-time analysis is based on the underlying stochastic differential equation with colored noise. We will also include a brief discussion showing that the decomposition remains useful as an approximation when the correlation time is small compared to observation times. This revision clarifies the scope without altering the core results. revision: yes

Circularity Check

1 steps flagged

Minor self-citation for scaling confirmation; central Markovian decomposition independent of inputs

specific steps
  1. self citation load bearing [Abstract]
    "This approach not only confirms the $t^{5/2}$-law obtained by Boynewicz et al. (2026), related to fluid inertial effects, but indicates that this scaling may be superseded by a $t^4$-scaling at very short times once the correlated nature of the stochastic forcings is taken into account."

    The confirmation step invokes a result from prior work sharing the lead author to support the scaling outcome of the new framework; while not the sole justification for the Chapman-Kolmogorov superposition itself, this creates a minor self-referential loop for the hydrodynamic regime claims without an independent external benchmark stated in the text.

full rationale

The paper's core derivation begins from the standard Chapman-Kolmogorov equation for Markovian dynamics, an external mathematical fact that directly enables the superposition of non-equilibrium states without reference to the paper's own results or fitted quantities. The t^{5/2} confirmation cites prior work by the lead author, but this functions as external validation of a specific scaling rather than load-bearing justification for the superposition or the new t^4 claim. The t^4 scaling is introduced as a consequence of force correlation properties within the same Markovian framework, without equations reducing by construction to the inputs or a self-citation chain that replaces independent derivation. No ansatz, renaming, or fitted-input-as-prediction patterns appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Markov property of the dynamics and on the prior t^{5/2} result; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Particle dynamics obey the Chapman-Kolmogorov equation because they are Markovian.
    Invoked in the first sentence to justify the superposition of non-equilibrium states.

pith-pipeline@v0.9.0 · 5744 in / 1452 out tokens · 46127 ms · 2026-05-19T18:31:00.118073+00:00 · methodology

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Reference graph

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