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arxiv: 2503.01537 · v2 · pith:SOINNV3Cnew · submitted 2025-03-03 · 🧮 math.PR

Monge-Amp\`ere gravitating fluids. Least action principles and particle systems

Christian L\'eonard , Roya Mohayaee This is my paper

Pith reviewed 2026-05-23 01:36 UTC · model grok-4.3

classification 🧮 math.PR
keywords Monge-Ampère gravitationlarge deviationsoptimal transportparticle systemsfluidsOtto-Wasserstein manifoldquantum force fieldconditional Gibbs principle
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The pith

A particle system with splitting produces the Monge-Ampère gravitation fluid action plus a thermal term that a quantum force field on the Otto-Wasserstein manifold can cancel.

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

The paper extends Monge-Ampère gravitation theory from particles to fluids. It introduces a particle system subject to a splitting mechanism whose large deviation rate function equals the MAG action functional plus an extra term from thermal fluctuations, obtained through a conditional Gibbs principle. To recover the exact MAG functional, the authors propose adding a quantum force field on the Otto-Wasserstein manifold of fluids to balance the fluctuations. This builds on earlier double large deviation principles for Brownian particles while offering a more physically interpretable microscopic model. The explicit microscopic particle system realizing the quantum force via the conditional Gibbs principle is presented as an open problem.

Core claim

A system of particles equipped with a splitting mechanism yields, through its conditional large deviation principle, an action functional on fluid space equal to the Monge-Ampère gravitation functional plus an extra term associated with thermal fluctuations; introducing a quantum force field on the Otto-Wasserstein manifold cancels the extra term and recovers the pure MAG action.

What carries the argument

The conditional Gibbs principle generated by the splitting mechanism, which produces the fluid action functional as MAG's plus the thermal term before the quantum force is applied.

Load-bearing premise

A microscopic particle system with a splitting mechanism exists whose conditional Gibbs principle produces exactly the extra thermal term that the proposed quantum force field can cancel.

What would settle it

Construction of an explicit particle system with splitting whose large deviation rate function on fluid space matches the MAG action exactly after addition of the quantum force, or demonstration that no such splitting mechanism produces a cancellable thermal term.

Figures

Figures reproduced from arXiv: 2503.01537 by Christian L\'eonard, Roya Mohayaee.

Figure 1
Figure 1. Figure 1: Cosmic microwave background One aim of EUR is to give estimates of the field of very early fluctuations from this uniformity µ ′ 0 (x) := lim t→0+ µt(x) − L −d t , a crucial information to test the cosmic inflation theory and provide details about the initial quantum fluctuations. An effective theory in computational cosmology. As an illustration of the good performance of MAG in cosmology, we provide at … view at source ↗
Figure 2
Figure 2. Figure 2: Typical structures of the actual Universe Courtesy of Bruno Lévy MAG works well with EUR. Let us discuss a little bit about MAG being a good ap￾proximation of the Newtonian gravity in the special setting of the EUR problem. The symmetric operator Hess φ admits an orthogonal basis of eigenvectors. Let K := {x ∈ R 3 ; Hess φ(x) = 0} denotes its kernel, and K⊥ be K’s orthogonal subspace in R 3 . Sup￾pose that… view at source ↗
Figure 3
Figure 3. Figure 3: Optimal transport vs. N-body simulation, [18]. MAG’s approximation of SNS is effective in the regime of short time and weak gravity. From now on, following Brenier and his co-authors [9, 1, 2] we drop the relativistic dynamics (1.7) and go back to the usual Newton equation (1.1). With this choice the underlying physics is less realistic, but the mathematics are clearer and easier to handle. This simplifica… view at source ↗
Figure 4
Figure 4. Figure 4: The force field depends on D MAG’s force field. Replacing MAG’s equation (1.13) by (2.7), the right hand side of Newton’s equation (1.1) is −∇φ(y) = y − ∇θ(y) = y − ←− T (y), y ∈ −→T (D), a.e. (2.8) This is the explicit connection of MAG with quadratic optimal transport. It holds both on T d and R d , since on T d one chooses D = T d . Remarks 2.9. (i) Since the force field −∇φ depends on the choice of the… view at source ↗
read the original abstract

The Monge-Amp\`ere gravitation theory (MAG) was introduced by Brenier in 2011 to obtain an approximate solution of the early Universe reconstruction problem. It is a modification of Newtonian gravitation which is based on quadratic optimal transport. Later, Brenier in 2016, then Ambrosio, Baradat and Brenier in 2020 discovered a double large deviation principle for Brownian particles whose rate function is precisely MAG's action functional. In the present article, following Brenier we first recap MAG's theory. Then, we slightly extend it from particles to fluid. This allows us to revisit the Ambrosio-Baradat-Brenier particle system. We propose another particle system which is easier to interpret in physics and whose large deviation rate function is half the way to MAG's action functional for fluids. While the setting of the Schr\"odinger problem is a system of noninteracting particles, our particle system is subject to some splitting mechanism which regulates the thermal fluctuations. This gives rise to some conditional Gibbs principle that leaves us with an action functional on the fluid space which is MAG's action functional plus an extra term associated with thermal fluctuations. In order to recover MAG's action functional, we have to remove this extra term. To do so, we propose to add some quantum force field on the Otto-Wasserstein manifold of fluids to balance the thermal fluctuations. A microscopic description of a system of particles leading to a conditional Gibbs principle whose action functional generates such a quantum force remains a challenging open problem.

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 paper recaps the Monge-Ampère gravitation (MAG) theory of Brenier, extends it from particles to fluids, revisits the Ambrosio-Baradat-Brenier Brownian particle system, and introduces a new splitting particle system. The large-deviation rate function of the new system is claimed to equal the MAG fluid action plus an extra thermal-fluctuation term arising from a conditional Gibbs principle on the Otto-Wasserstein manifold. The manuscript proposes that this extra term can be cancelled by a quantum force field defined on the same manifold, while explicitly stating that a microscopic particle realization of the required conditional Gibbs principle remains an open problem.

Significance. If the open construction of the quantum force field and the associated splitting mechanism can be carried out, the work would supply a physically motivated large-deviation derivation of the MAG fluid action, clarifying the role of thermal fluctuations and linking optimal transport, large deviations, and modified gravity. The explicit isolation of the extra thermal term constitutes a concrete intermediate result that future constructions could target.

major comments (2)
  1. [Abstract] Abstract and final paragraph: the central claim that the proposed splitting particle system yields a rate function that is 'half the way' to the MAG fluid action rests on the existence of a splitting mechanism whose conditional Gibbs principle produces precisely the extra thermal term. No definition of the splitting rule, no statement of the conditional Gibbs principle, and no derivation or verification of the resulting large-deviation rate function are supplied.
  2. [Abstract] Abstract and final paragraph: the proposal to cancel the extra thermal term by adding a quantum force field on the Otto-Wasserstein manifold is introduced without any equation defining the force field, any expression for its action on the manifold, or any argument showing that it exactly offsets the thermal term. The recovery of the pure MAG action is therefore presented as dependent on an unresolved existence question rather than a derived result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to clarify the status of the proposed constructions. We agree that the splitting mechanism, conditional Gibbs principle, and quantum force field are presented at a conceptual level without explicit definitions or derivations, as the manuscript already states that a microscopic realization remains an open problem. We will revise the abstract and final paragraph to emphasize the conjectural nature of these proposals.

read point-by-point responses

  1. Referee: [Abstract] Abstract and final paragraph: the central claim that the proposed splitting particle system yields a rate function that is 'half the way' to the MAG fluid action rests on the existence of a splitting mechanism whose conditional Gibbs principle produces precisely the extra thermal term. No definition of the splitting rule, no statement of the conditional Gibbs principle, and no derivation or verification of the resulting large-deviation rate function are supplied.

    Authors: We agree that no explicit definition or derivation of the splitting rule and conditional Gibbs principle is provided. The manuscript frames the splitting particle system as a proposed construction whose large-deviation rate function would equal the MAG fluid action plus an extra thermal term, conditional on the existence of such a mechanism (explicitly noted as an open problem in the final sentence of the abstract). The 'half the way' phrasing is therefore intended to describe a direction for future work rather than a completed result. In revision we will insert clarifying language in the abstract and conclusion to state that the rate-function claim is conjectural pending construction of the splitting rule. revision: partial

  2. Referee: [Abstract] Abstract and final paragraph: the proposal to cancel the extra thermal term by adding a quantum force field on the Otto-Wasserstein manifold is introduced without any equation defining the force field, any expression for its action on the manifold, or any argument showing that it exactly offsets the thermal term. The recovery of the pure MAG action is therefore presented as dependent on an unresolved existence question rather than a derived result.

    Authors: This observation is accurate. The quantum force field is introduced only as a conceptual device to cancel the extra thermal-fluctuation term on the Otto-Wasserstein manifold; no equations, action functional, or cancellation argument are supplied because the construction is left open. The manuscript already concludes by stating that a microscopic particle system realizing the required conditional Gibbs principle remains a challenging open problem. We will revise the abstract and final paragraph to make explicit that recovery of the pure MAG action is conditional on resolving this existence question. revision: partial

Circularity Check

0 steps flagged

No significant circularity; paper explicitly flags open problem rather than claiming closed derivation

full rationale

The paper recaps existing MAG theory from Brenier (2011, 2016) and Ambrosio-Baradat-Brenier (2020), extends the setting from particles to fluids, and constructs a new splitting particle system whose large-deviation rate function equals the target MAG fluid action plus one extra thermal term obtained from the conditional Gibbs principle. It then states that exact recovery requires an additional quantum force field on the Otto-Wasserstein manifold whose microscopic realization is an acknowledged open problem (abstract and final paragraph). No equation reduces a derived quantity to a fitted parameter by construction, no load-bearing premise rests on self-citation by the present authors, and the central contribution is presented as partial progress with an explicit gap rather than a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The construction rests on standard large-deviation theory for interacting particle systems and on the Otto-Wasserstein geometry already developed in the cited MAG papers; the only new postulated object is the quantum force field.

axioms (2)
  • domain assumption Large-deviation principles exist for the proposed splitting particle system and yield a rate function on path space.
    Invoked when the authors state that the new system 'whose large deviation rate function is half the way to MAG's action functional'.
  • domain assumption The Otto-Wasserstein manifold carries a geometry on which a force field can be added to cancel thermal fluctuations.
    Used in the final proposal to recover the exact MAG action.
invented entities (1)
  • quantum force field on the Otto-Wasserstein manifold no independent evidence
    purpose: To exactly cancel the extra thermal-fluctuation term arising from the splitting mechanism so that the fluid action matches MAG.
    Introduced in the last sentence of the abstract; no microscopic realization or independent evidence is provided.

pith-pipeline@v0.9.0 · 5810 in / 1537 out tokens · 27908 ms · 2026-05-23T01:36:48.908974+00:00 · methodology

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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.

    We propose another particle system ... whose large deviation rate function is half the way to MAG's action functional for fluids... To do so, we propose to add some quantum force field on the Otto-Wasserstein manifold of fluids to balance the thermal fluctuations. A microscopic description ... remains a challenging open problem.

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the action functional ... + ϵ² ∫ I(ps|rϵs) κs ds ... subtracting the Fisher information term

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.

Reference graph

Works this paper leans on

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