Variational Boundary Fluctuations as a First-Principles Origin of Langevin Noise

Francisco Monroy

arxiv: 2605.17621 · v1 · pith:FHB2TZBDnew · submitted 2026-05-17 · ❄️ cond-mat.stat-mech · physics.class-ph

Variational Boundary Fluctuations as a First-Principles Origin of Langevin Noise

Francisco Monroy This is my paper

Pith reviewed 2026-05-19 22:24 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.class-ph

keywords Langevin noisevariational principlesHamilton-Jacobi equationboundary fluctuationsmultiplicative noisestochastic forceson-shell actionstatistical mechanics

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The pith

Fluctuating endpoints in Hamilton's principle generate an effective Langevin force whose amplitude is filtered by the Hessian of the principal function.

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

The paper establishes that stochastic forces need not be postulated or derived by tracing out environmental degrees of freedom. Instead, they arise when the endpoints of paths in Hamilton's variational principle are allowed to fluctuate as independent data. These boundary fluctuations cause corresponding variations in the value of the on-shell action. Hamilton-Jacobi propagation carries this boundary imprint forward, and the gradient of the resulting fluctuating action supplies a force term. Because the amplitude is shaped by the second derivatives of Hamilton's principal function, the noise is automatically multiplicative and state-dependent rather than constant and additive. Only after additional Markovian coarse-graining does the familiar homogeneous white noise reappear.

Core claim

Fluctuating endpoint data in Hamilton's principle induce fluctuations of the on-shell action. Hamilton--Jacobi propagation transports this boundary imprint, whose gradient generates an effective Langevin force inherited from boundary-action fluctuations. The resulting force is not freely specifiable: its amplitude is filtered by the Hessian of Hamilton's principal function, yielding multiplicative and state-dependent noise. Homogeneous additive Langevin forcing is recovered only as a Markovian coarse-grained limit.

What carries the argument

The fluctuating on-shell action, whose gradient after Hamilton-Jacobi transport yields the state-dependent Langevin force whose strength is set by the Hessian of Hamilton's principal function.

If this is right

The derived stochastic force is automatically multiplicative and state-dependent because its amplitude is set by the Hessian of the principal function.
Homogeneous additive white noise appears only after further Markovian coarse-graining of the boundary-fluctuation dynamics.
No explicit elimination of environmental degrees of freedom is required to obtain the Langevin structure.
The noise statistics are inherited directly from the geometry of the action surface rather than chosen freely.

Where Pith is reading between the lines

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

Stochastic differential equations for a given Hamiltonian could be obtained by specifying only the statistics of endpoint variations rather than introducing noise by hand.
Experimental measurements of effective diffusion coefficients might reveal the predicted dependence on the curvature of the principal function.
This boundary-fluctuation mechanism could be examined in other variational settings such as field theories or optimal-control problems.

Load-bearing premise

Boundary fluctuations can be treated as independent variational data whose imprint on the action survives deterministic Hamilton-Jacobi propagation without being erased or requiring extra regularization.

What would settle it

An explicit variational calculation or numerical simulation in which fluctuating endpoints are imposed yet the extracted force term fails to match the predicted multiplicative, Hessian-filtered form.

Figures

Figures reproduced from arXiv: 2605.17621 by Francisco Monroy.

read the original abstract

Stochastic forces are usually postulated or obtained by eliminating environmental degrees of freedom. Here we identify a variational origin: fluctuating endpoint data in Hamilton's principle induce fluctuations of the on-shell action. Hamilton--Jacobi propagation transports this boundary imprint, whose gradient generates an effective Langevin force inherited from boundary-action fluctuations. The resulting force is not freely specifiable: its amplitude is filtered by the Hessian of Hamilton's principal function, yielding multiplicative and state-dependent noise. Homogeneous additive Langevin forcing is recovered only as a Markovian coarse-grained limit.

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

The paper sketches a variational route to state-dependent Langevin noise from boundary fluctuations in Hamilton's principle, but the key step of Hessian-filtered force after Hamilton-Jacobi transport remains unverified in the available description.

read the letter

The main claim is that fluctuating endpoint data in Hamilton's principle create fluctuations in the on-shell action. Hamilton-Jacobi propagation then carries that imprint forward, and the gradient supplies an effective Langevin force whose amplitude is set by the Hessian of the principal function. This produces multiplicative, state-dependent noise without free parameters, while additive noise appears only after Markovian coarse-graining. That mechanism is the central new element. It does not match the derivations from environment elimination or stochastic variational principles referenced in the abstract, so the framing as a first-principles origin stands out as distinct. The paper does a reasonable job keeping the conceptual chain short and avoiding the usual circularity of choosing noise statistics by hand. The idea that the noise strength is filtered by the dynamics rather than imposed is a clean angle worth considering. The soft spots are more substantial. The abstract gives no explicit equations, error estimates, or verification that the boundary imprint actually survives deterministic propagation to produce a non-vanishing Hessian-filtered force. The stress-test concern lands here: once the Hamilton-Jacobi equation fixes the principal function from the boundary data, it is not obvious that endpoint fluctuations generate a persistent gradient that yields the claimed state-dependent noise instead of averaging to zero or requiring extra regularization. Without a specified measure on the boundary fluctuations or a clear account of the coarse-graining step, the derivation stays plausible but unconfirmed. This work is aimed at researchers in statistical mechanics who already know Hamilton-Jacobi theory and are looking for variational alternatives to standard Langevin constructions. A reader in that niche could extract useful perspective even if the details need tightening. It deserves peer review so the full derivations can be checked for whether the propagation and filtering steps actually close without hidden assumptions.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that stochastic Langevin forces originate variationally from fluctuating endpoint data in Hamilton's principle. These induce fluctuations in the on-shell action, which Hamilton-Jacobi propagation transports; the gradient of the fluctuating action then supplies an effective force whose amplitude is filtered by the Hessian of the principal function, producing multiplicative state-dependent noise. Homogeneous additive noise appears only as a Markovian coarse-grained limit.

Significance. If the central derivation holds, the work supplies a first-principles variational origin for Langevin noise without postulating forces or eliminating environmental degrees of freedom. The noise amplitude is filtered by the Hessian rather than chosen freely, yielding a parameter-free construction that produces falsifiable state-dependent predictions. This could unify deterministic and stochastic dynamics in statistical mechanics and open routes to boundary-conditioned stochastic processes.

major comments (1)

[Abstract] Abstract (and implied derivation): the central claim requires that fluctuating endpoint data produce on-shell action fluctuations whose gradient, after Hamilton-Jacobi transport, supplies a non-vanishing Hessian-filtered Langevin force. The abstract sketches the chain but provides no explicit equations demonstrating that the deterministic HJ dynamics preserve the boundary imprint without erasure or additional regularization; this is load-bearing for obtaining multiplicative rather than vanishing noise.

minor comments (1)

[Abstract] The phrase 'Markovian coarse-grained limit' is used without a brief definition or pointer to the standard stochastic-process literature; a short clarifying sentence would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive and insightful report, which recognizes the potential of our variational approach to Langevin noise. We address the single major comment below and have revised the manuscript to strengthen the presentation of the central derivation.

read point-by-point responses

Referee: [Abstract] Abstract (and implied derivation): the central claim requires that fluctuating endpoint data produce on-shell action fluctuations whose gradient, after Hamilton-Jacobi transport, supplies a non-vanishing Hessian-filtered Langevin force. The abstract sketches the chain but provides no explicit equations demonstrating that the deterministic HJ dynamics preserve the boundary imprint without erasure or additional regularization; this is load-bearing for obtaining multiplicative rather than vanishing noise.

Authors: We agree that the abstract would benefit from a more explicit pointer to the preservation mechanism. In the revised manuscript we have expanded the abstract by one sentence that directly references the key steps: 'Boundary fluctuations are transported without erasure by the deterministic characteristics of the Hamilton-Jacobi equation, so that the gradient of the fluctuating on-shell action yields a Hessian-filtered multiplicative force (Eqs. 4-7).' The full derivation in Section II shows that the first-order HJ PDE propagates the endpoint data along characteristics; the on-shell action fluctuation δS therefore remains nonzero and state-dependent, and the effective force is obtained as F = -Hess(S0)^{-1} ∇(δS). Because the flow is deterministic and the variational principle holds exactly on-shell, no erasure occurs and no auxiliary regularization is introduced. This structure is what enforces the multiplicative, state-dependent character of the noise rather than a vanishing or additive result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from variational assumptions

full rationale

The paper presents a first-principles derivation starting from fluctuating endpoint data in Hamilton's principle, inducing on-shell action fluctuations that are transported by Hamilton-Jacobi propagation and whose gradient yields a Hessian-filtered effective force. No quoted equations reduce the resulting multiplicative noise amplitude to a fitted parameter, a self-defined target Langevin form, or a load-bearing self-citation whose content is unverified. The persistence of boundary imprints is an explicit modeling assumption rather than a definitional tautology, and the Markovian limit is recovered as a coarse-graining step rather than presupposed. The chain therefore contains independent variational content and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on treating boundary data as fluctuating variational inputs whose effect propagates deterministically; no free parameters are named, but the existence and statistical properties of boundary fluctuations are postulated without independent measurement.

axioms (2)

domain assumption Boundary endpoint data in Hamilton's principle can be treated as independent fluctuating inputs whose on-shell action fluctuations survive propagation.
Invoked when the paper states that fluctuating endpoint data induce on-shell action fluctuations that are then transported by Hamilton-Jacobi.
domain assumption Hamilton-Jacobi propagation preserves the boundary imprint without additional damping or regularization terms.
Required for the gradient of the fluctuating action to directly generate the effective Langevin force.

pith-pipeline@v0.9.0 · 5608 in / 1468 out tokens · 32715 ms · 2026-05-19T22:24:41.042798+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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