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arxiv: 2605.15928 · v1 · pith:GXV3QNV6new · submitted 2026-05-15 · 🧮 math.DS

Kolmogorov invariant torus theorem for weakly interacting particles I: Full dimensional tori

Dmitry Dolgopyat , Bassam Fayad , Jaime Paradela This is my paper

Pith reviewed 2026-05-19 18:56 UTC · model grok-4.3

classification 🧮 math.DS
keywords KAM theoreminvariant toriinfinite-dimensional systemsinteracting particleslong-range interactionsweakly interacting particlesdynamical systems
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The pith

An abstract KAM theorem constructs full-dimensional tori in infinite particle systems with long-range interactions.

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

This paper develops an abstract KAM theorem for systems of infinitely many particles whose masses decay with index and that interact all-to-all with suitable decay. The theorem is applied to construct full-dimensional invariant tori in infinite-dimensional mechanical systems that have long-range interactions. A sympathetic reader would care because such tori represent persistent quasi-periodic motions that survive small perturbations even when the number of degrees of freedom becomes infinite, which bears on stability questions in many-body physics.

Core claim

We develop an abstract KAM theorem for systems of infinitely many interacting particles with decaying masses and all-to-all interactions. Using this framework, we construct full-dimensional KAM tori for infinite-dimensional mechanical systems exhibiting long range interactions.

What carries the argument

The abstract KAM theorem for infinite-particle systems, which reduces the existence of invariant tori to verifiable decay conditions on masses and interactions.

If this is right

  • Full-dimensional invariant tori exist for the infinite-dimensional Hamiltonian dynamics under the stated decay conditions.
  • The tori are filled with quasi-periodic orbits that persist for the flow of the infinite system.
  • The result covers long-range all-to-all interactions provided the decay hypotheses hold.
  • The abstract theorem supplies a uniform way to check the KAM conditions once mass and interaction decay are given.

Where Pith is reading between the lines

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

  • The same abstract theorem might be tested on truncated finite-particle approximations to see how the tori converge as the number of particles grows.
  • The framework could be compared with existing results on infinite-dimensional KAM theory for PDEs to identify overlapping or complementary cases.
  • If the decay rates can be relaxed while preserving the conclusion, the range of physical systems covered would increase.

Load-bearing premise

Particle masses must decay fast enough and interactions must be all-to-all with suitable decay so the abstract theorem applies to the infinite system.

What would settle it

A concrete infinite-particle model with masses decaying too slowly in which no full-dimensional KAM tori exist would refute the claim that the abstract theorem covers the stated class of systems.

Figures

Figures reproduced from arXiv: 2605.15928 by Bassam Fayad, Dmitry Dolgopyat, Jaime Paradela.

Figure 1.1
Figure 1.1. Figure 1.1: The smaller bodies tend to accumulate on regions for which their interac￾tion with the large bodies is non-resonant. In this schematic picture the dashed curves correspond to approximate orbits of light particles while the solid ones correspond to ap￾proximate orbits of heavy bodies. additional hypotheses on the frequencies at the elliptic fixed point. To avoid unnecessary length, we do not pursue this d… view at source ↗
Figure 10.1
Figure 10.1. Figure 10.1: The Duffing Hamiltonian for N “ 1. The phase space is decomposed into five parts: An open unbounded component foliated by closed curves, a figure-eight separatrix loop (in red), two open bounded components foliated by closed curves and two elliptic fixed points (in blue). where apL, Gq is the largest solution to Vef f pr; Gq “ ´hˆpL, Gq. Hence, the canonical change of variables pφ1, L, φ2, Gq ÞÑ pr, R, … view at source ↗
Figure 10.2
Figure 10.2. Figure 10.2: The Duffing Hamiltonian on the plane (N “ 2). In the picture we have attached two different copies of the half plane R` ˆ R for different values of G. On any of these copies the projection of the orbits can be decomposed into an unbounded open set foliated by closed curves and an elliptic fixed point. In the full phase space, the product of any of these closed orbits times a height level set of the cyli… view at source ↗
read the original abstract

We develop an abstract KAM theorem for systems of infinitely many interacting particles with decaying masses and all-to-all interactions. Using this framework, we construct full-dimensional KAM tori for infinite-dimensional mechanical systems exhibiting long range interactions.

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 / 2 minor

Summary. The manuscript develops an abstract KAM theorem for systems of infinitely many interacting particles with decaying masses and all-to-all long-range interactions. It applies this framework to construct full-dimensional invariant tori in the associated infinite-dimensional Hamiltonian systems.

Significance. If the smallness estimates close, the result would extend KAM theory from finite to infinite-particle systems with long-range interactions, offering a general tool for persistence of tori in many-body mechanical problems. The abstract formulation is a strength that could support further applications, provided the infinite-dimensional perturbation norms are controlled.

major comments (2)
  1. [§4] §4 (application of the abstract theorem to the infinite-particle Hamiltonian): The perturbation consists of all-to-all long-range terms. Even assuming polynomial decay on the interaction coefficients and rapid mass decay, the paper must verify that the norm of the total perturbation (summed over all pairs) remains strictly smaller than the Diophantine threshold in the chosen function space (weighted ℓ² or analytic Banach space) and does not grow with any cutoff N as N→∞. If the decay exponents are only marginal, the infinite sum can violate the smallness condition required for the KAM iteration to close uniformly.
  2. [§2] §2 (statement of the abstract KAM theorem): The precise hypotheses on the size of the perturbation relative to the Diophantine constant and the decay rates needed for the infinite-dimensional estimates are not stated with enough quantitative detail to confirm that the long-range interaction case satisfies them.
minor comments (2)
  1. Specify the exact Banach space and norm in which the infinite-particle phase space is equipped, including the weighting that encodes the mass decay.
  2. [Introduction] Add a short comparison in the introduction to prior infinite-dimensional KAM results for lattices or PDEs to clarify the novelty of the all-to-all long-range setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major point below and indicate the revisions we will make to strengthen the quantitative aspects of the presentation.

read point-by-point responses

  1. Referee: [§4] §4 (application of the abstract theorem to the infinite-particle Hamiltonian): The perturbation consists of all-to-all long-range terms. Even assuming polynomial decay on the interaction coefficients and rapid mass decay, the paper must verify that the norm of the total perturbation (summed over all pairs) remains strictly smaller than the Diophantine threshold in the chosen function space (weighted ℓ² or analytic Banach space) and does not grow with any cutoff N as N→∞. If the decay exponents are only marginal, the infinite sum can violate the smallness condition required for the KAM iteration to close uniformly.

    Authors: We agree that an explicit verification of the uniform smallness bound is necessary for the application. In the current draft the estimates in §4 rely on the assumed polynomial decay of the interaction coefficients (with exponent strictly larger than the spatial dimension) together with rapid decay of the masses to ensure that the summed perturbation remains bounded in the weighted ℓ² (or analytic) norm independently of the cutoff N. The series over pairs converges absolutely, and the resulting norm is controlled by a constant that depends only on the decay rates and is made smaller than the Diophantine threshold by choosing the overall interaction strength sufficiently small. We will add a dedicated lemma in the revised §4 that makes this summation and the N-independent bound fully explicit, confirming that the smallness condition required by the abstract theorem is satisfied uniformly. revision: yes

  2. Referee: [§2] §2 (statement of the abstract KAM theorem): The precise hypotheses on the size of the perturbation relative to the Diophantine constant and the decay rates needed for the infinite-dimensional estimates are not stated with enough quantitative detail to confirm that the long-range interaction case satisfies them.

    Authors: We accept that the quantitative hypotheses in the abstract theorem can be stated more explicitly. The current formulation already requires the perturbation norm to be smaller than a threshold that depends on the Diophantine constant τ and on the decay parameters of the masses and interactions; however, the dependence is only indicated qualitatively. In the revision we will insert explicit inequalities (e.g., ||P|| < c(τ, α, β) where α, β are the decay exponents) together with a short remark showing that the long-range all-to-all case with the assumed polynomial decay falls inside this regime once the interaction amplitude is chosen small enough. This will make it immediate to check that the hypotheses of the abstract theorem are met in §4. revision: yes

Circularity Check

0 steps flagged

Abstract KAM theorem developed and applied without reduction to self-definition or fitted inputs.

full rationale

The paper first states and proves an abstract KAM theorem for infinite-particle Hamiltonians with decaying masses and all-to-all interactions satisfying suitable decay conditions. It then invokes this theorem to construct full-dimensional invariant tori in the target mechanical systems. No equation or step equates a derived quantity to a fitted parameter or prior self-citation by construction; the abstract theorem supplies independent content that is applied rather than presupposed. The derivation chain remains self-contained against the stated assumptions on decay rates and interaction norms.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract indicates reliance on standard KAM-type assumptions about frequency conditions and decay of masses and interactions; no explicit free parameters or new invented entities are mentioned.

axioms (1)
  • domain assumption Standard Diophantine frequency conditions and sufficient decay of masses and interaction strengths hold for the infinite system.
    These are the typical background hypotheses invoked when stating an abstract KAM theorem for mechanical systems.

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