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arxiv: 2601.12135 · v1 · pith:T4PQYTODnew · submitted 2026-01-17 · ❄️ cond-mat.stat-mech · physics.class-ph

Stochastic dynamics from maximum entropy in action space

Fabricio de Souza Luiz , Jos\'e Carlos Bellizotti Souza , Lu\'isa Toledo Tude , Marcos C\'esar de Oliveira This is my paper

Pith reviewed 2026-05-25 07:06 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.class-ph
keywords stochastic dynamicsmaximum entropy principleaction spaceBrownian propagatorrelativistic stochastic processeslarge deviation theorydensity of statesChapman-Kolmogorov equation
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The pith

Maximizing entropy over joint action-endpoint distributions produces a Boltzmann-like distribution in action space.

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

The paper develops an information-theoretic approach to stochastic dynamics by treating the total action connecting spacetime points as the basic random variable. Maximizing Shannon entropy over the joint distribution of actions and endpoints, subject only to normalization and a fixed mean action, directly yields a Boltzmann-like probability distribution over actions. This recovers the standard Brownian propagator in the nonrelativistic limit and supplies a Lorentz-covariant extension to relativistic regimes where path-based Wiener constructions break covariance. The derivation uses large deviation theory to obtain an explicit Gaussian density of states centered on the minimal action and verifies that the resulting propagator obeys the Chapman-Kolmogorov equation.

Core claim

Maximizing Shannon entropy over a joint distribution of actions and endpoints, subject to normalization and a constraint on the mean action, produces a Boltzmann-like distribution in action space. This framework reproduces the standard Brownian propagator in the nonrelativistic limit and naturally extends to relativistic regimes, where the Wiener construction fails to preserve Lorentz covariance. The density of states is derived as Gaussian using large deviation theory, and the Markov property holds via the Chapman-Kolmogorov equation from additivity of the minimal action.

What carries the argument

The maximum entropy principle applied to the joint distribution over total actions and endpoints under normalization and mean-action constraints, yielding the Boltzmann-like action distribution.

If this is right

  • The resulting dynamics are governed by a competition between extremization of the action and entropic effects in the diffusive regime.
  • The propagator satisfies the Chapman-Kolmogorov equation, confirming the Markovian property.
  • The approach bypasses functional integration over paths and makes entropic degeneracy explicit through the action-space density of states.
  • The framework provides a transparent connection between the principle of least action and statistical inference.

Where Pith is reading between the lines

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

  • This construction could be extended to interacting systems by including interaction terms in the minimal action.
  • The effective action free energy interpretation might connect to variational principles in nonequilibrium thermodynamics.
  • Testing the relativistic extension could involve simulating particle diffusion at speeds close to light and comparing to covariant formulations.

Load-bearing premise

Large deviation theory independently supplies a Gaussian density of states centered on the minimal action that justifies the saddle-point approximation.

What would settle it

A direct calculation showing that the density of states for actions deviates from Gaussian form, or experimental violation of the derived propagator in relativistic stochastic processes.

Figures

Figures reproduced from arXiv: 2601.12135 by Fabricio de Souza Luiz, Jos\'e Carlos Bellizotti Souza, Lu\'isa Toledo Tude, Marcos C\'esar de Oliveira.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic comparison between the path-based and action-space formulations of stochastic dynamics. ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

We develop an information-theoretic formulation of stochastic dynamics in which the fundamental stochastic variable is the total action connecting spacetime points, rather than individual paths. By maximizing Shannon entropy over a joint distribution of actions and endpoints, subject to normalization and a constraint on the mean action, we obtain a Boltzmann-like distribution in action space. This framework reproduces the standard Brownian propagator in the nonrelativistic limit and naturally extends to relativistic regimes, where the Wiener construction fails to preserve Lorentz covariance. The approach bypasses functional integration over paths, makes the role of entropic degeneracy explicit through an action-space density of states, and provides a transparent connection between the principle of least action and statistical inference. We derive the density of states explicitly using large deviation theory, showing that it takes a Gaussian form centered at the minimal action, and rigorously justify the saddle-point approximation in the diffusive regime. The Markovian property of the resulting propagator is verified to hold via the Chapman--Kolmogorov equation, following from the additivity of the minimal action for free-particle dynamics. In the diffusive regime, the resulting dynamics are governed by a competition between extremization of the action and entropic effects, which can be interpreted in terms of an effective action free energy. Our results establish an unified, covariant, and information-based foundation for classical and relativistic stochastic processes.

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 / 0 minor

Summary. The manuscript presents an information-theoretic approach to stochastic dynamics by maximizing the Shannon entropy of a joint distribution over actions and endpoints, subject to a mean-action constraint. This yields a Boltzmann-like distribution in action space that is claimed to reproduce the standard Brownian propagator in the nonrelativistic limit, extend covariantly to relativistic regimes, and satisfy the Markov property via the Chapman-Kolmogorov equation. The paper asserts an explicit derivation of a Gaussian action-space density of states from large-deviation theory and a rigorous justification of the saddle-point approximation in the diffusive regime.

Significance. Should the central derivation of an independent Gaussian density of states hold without circularity in the mean-action constraint, the work would offer a novel, covariant foundation for stochastic processes that bypasses path integrals and explicitly incorporates entropic effects through an action free energy. This could unify classical and relativistic stochastic dynamics under a statistical-inference principle.

major comments (2)
  1. [large-deviation derivation of density of states] The section deriving the density of states via large-deviation theory asserts a Gaussian form centered on the minimal action that is independent of the target propagator, but provides no explicit demonstration that this form (and the numerical value of the mean-action constraint) is obtained externally rather than from the same minimal-action saddle; without this independence the construction risks circularity and the claimed reproduction of the Brownian kernel fails to follow.
  2. [saddle-point approximation justification] The abstract claims a rigorous justification of the saddle-point approximation in the diffusive regime together with error estimates or checks against known limits, yet the supplied text contains no such explicit derivations or verifications; this justification is load-bearing for both the nonrelativistic reproduction and the relativistic extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the derivations require greater explicitness. We agree that the manuscript would benefit from expanded sections addressing both concerns and will revise accordingly.

read point-by-point responses

  1. Referee: [large-deviation derivation of density of states] The section deriving the density of states via large-deviation theory asserts a Gaussian form centered on the minimal action that is independent of the target propagator, but provides no explicit demonstration that this form (and the numerical value of the mean-action constraint) is obtained externally rather than from the same minimal-action saddle; without this independence the construction risks circularity and the claimed reproduction of the Brownian kernel fails to follow.

    Authors: We acknowledge that the current text does not supply a sufficiently detailed, self-contained derivation demonstrating independence. In the revised manuscript we will expand the large-deviation section with an explicit step-by-step calculation: starting from the rate function for action fluctuations around the classical path, deriving the quadratic expansion that yields the Gaussian form and its variance from the Hessian of the action, and showing that both the centering and the variance are fixed before the mean-action constraint is imposed. This will remove any appearance of circularity and allow the subsequent reproduction of the Brownian kernel to follow directly. revision: yes

  2. Referee: [saddle-point approximation justification] The abstract claims a rigorous justification of the saddle-point approximation in the diffusive regime together with error estimates or checks against known limits, yet the supplied text contains no such explicit derivations or verifications; this justification is load-bearing for both the nonrelativistic reproduction and the relativistic extension.

    Authors: The referee is correct that the abstract asserts a rigorous justification with error estimates while the body of the paper does not contain the corresponding derivations or verifications. We will add a dedicated subsection that (i) derives the leading-order error term of the saddle-point approximation in the diffusive scaling limit, (ii) supplies explicit bounds on the remainder, and (iii) verifies the approximation against the exact non-relativistic Brownian propagator as well as consistency checks in the relativistic regime. These additions will make the claimed justification fully explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit large-deviation derivation of density of states supplies independent input

full rationale

The derivation maximizes entropy subject to normalization and a mean-action constraint to obtain the Boltzmann-like form, then separately invokes large-deviation theory to derive an explicit Gaussian density of states centered on the minimal action; this density is used to justify the saddle-point step. Because the density-of-states derivation is presented as an independent calculation (not obtained by fitting the target propagator or by self-reference), the mean-action constraint does not reduce to the output by construction. No self-citation chains, ansatz smuggling, or renaming of known results appear in the load-bearing steps. The reproduction of the Brownian propagator functions as a consistency check rather than a fitted prediction. The framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Central claim rests on the maximum-entropy principle applied to action distributions, the validity of large-deviation theory for the action density of states, and additivity of the classical minimal action for free particles.

free parameters (1)
  • mean action constraint value
    A numerical value for the mean action is imposed as a constraint; its origin is not derived from first principles within the abstract.
axioms (2)
  • domain assumption Additivity of the minimal action for free-particle dynamics
    Invoked to verify that the propagator satisfies the Chapman-Kolmogorov equation and is therefore Markovian.
  • domain assumption Large-deviation theory supplies a Gaussian density of states centered at the minimal action
    Used to justify the saddle-point approximation that yields the final propagator.

pith-pipeline@v0.9.0 · 5778 in / 1527 out tokens · 49230 ms · 2026-05-25T07:06:31.698524+00:00 · methodology

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    By maximizing Shannon entropy over a joint distribution of actions and endpoints, subject to normalization and a constraint on the mean action, we obtain a Boltzmann-like distribution in action space … We derive the density of states explicitly using large deviation theory, showing that it takes a Gaussian form centered at the minimal action

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
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The paper appears to rely on the theorem as machinery.
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The paper's claim conflicts with a theorem or certificate in the canon.
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Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · 3 internal anchors

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