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arxiv: 2505.09225 · v2 · submitted 2025-05-14 · 🌊 nlin.CD · cond-mat.stat-mech

Remarkable similarities in distributions of dynamical observables in chaotic systems

Lucianno Defaveri , Naftali R. Smith This is my paper

Pith reviewed 2026-05-22 15:39 UTC · model grok-4.3

classification 🌊 nlin.CD cond-mat.stat-mech
keywords large deviation theorychaotic dynamicsdynamical observablesrate functionsderived functionsfinite-time Lyapunov exponentopen maps
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The pith

Dynamical observables in chaotic systems share the same large-deviation rate function when their defining functions differ by a derived one

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

In chaotic systems, dynamical observables are sums A = sum g(x_n) along trajectories, and for most g they obey a large deviation principle P(A) ~ exp(-N I(A/N)). This paper shows that when g1 and g2 differ by a member of a newly defined class called derived functions, the two observables have identical rate functions I(a). The result also implies that if g itself is derived, the distribution of A becomes independent of sequence length N in the large-N limit and is mirror-symmetric but non-Gaussian. A reader cares because the similarity supplies a structural reason why seemingly unrelated observables, such as position in open maps and finite-time Lyapunov exponents, display matching rare-event statistics.

Core claim

We find that certain dynamical observables exhibit a remarkable statistical similarity: even when constructed with distinct functions g1(x) and g2(x), different observables are described by the same rate function. We provide a physical interpretation for this striking similarity by showing that g1(x)−g2(x) belongs to a class of functions that we call derived. Furthermore, we show that if g(x) itself is derived, then the distribution of A becomes independent of N in the large-N limit, and is generally non-Gaussian although mirror-symmetric. We demonstrate that the position observable for certain open maps and the finite-time Lyapunov exponent for the logistic map are of this derived form.

What carries the argument

The class of derived functions, whose defining property forces the large-deviation rate function of an observable to be unchanged when g is replaced by g plus a derived function.

If this is right

  • Observables whose g functions differ by a derived function obey the identical large-deviation principle.
  • When g is itself derived the probability distribution of the observable becomes independent of trajectory length N for large N.
  • Such distributions remain mirror-symmetric yet are typically non-Gaussian.
  • The derived property accounts for known statistical similarities between position observables in open maps and finite-time Lyapunov exponents in the logistic map.

Where Pith is reading between the lines

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

  • The derived-function classification may place many additional observables into equivalence classes sharing the same rate function.
  • The same structural argument could be checked in other chaotic maps or continuous-time flows to identify further derived relations.
  • The unification supplies a route to simplify large-deviation calculations by reducing them to the simplest representative of each derived class.

Load-bearing premise

That the difference between the functions defining two observables belongs to the class of derived functions, without which the rate functions would generally differ.

What would settle it

Numerical extraction of the empirical rate functions for two observables whose g functions differ by a non-derived quantity; identical rates despite a non-derived difference would falsify the necessity of the derived class.

Figures

Figures reproduced from arXiv: 2505.09225 by Lucianno Defaveri, Naftali R. Smith.

Figure 1
Figure 1. Figure 1: FIG. 1. Rate function similarity for two distinct observables on the left panels. Symbols show [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Probability density of the derived observable [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Panel (a): the open chaotic map in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Rate function similarity is shown for the three [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Statistics of the finite-time Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Quasi-analytic continuation of the SCGF [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original abstract

The study of chaotic systems, where rare events play a pivotal role, is essential for understanding complex dynamics due to their sensitivity to initial conditions. Recently, tools from large deviation theory, typically applied in the context of stochastic processes, have been used in the study of chaotic systems. Here, we study dynamical observables, $A = \sum_{n=1}^N g(\textbf{x}_n)$, defined along a chaotic trajectory $\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_N\}$. For most choices of $g(\textbf{x})$, $A$ satisfies a central limit theorem: At large sequence size $N \gg 1$, typical fluctuations of $A$ follow a Gaussian distribution with a variance that scales linearly with $N$. Large deviations of $A$ are usually described by the large deviation principle, that is, $P(A) \sim e^{- N I(A/N)}$, where $I(a)$ is the rate function. We find that certain dynamical observables exhibit a remarkable statistical similarity: even when constructed with distinct functions $g_1(\textbf{x})$ and $g_2(\textbf{x})$, different observables are described by the same rate function. We provide a physical interpretation for this striking similarity by showing that $g_1(\textbf{x})-g_2(\textbf{x})$ belongs to a class of functions that we call ``derived''. Furthermore, we show that if $g(\textbf{x})$ itself is ``derived'', then the distribution of $A$ becomes independent of $N$ in the large-$N$ limit, and is generally non-Gaussian (although it is mirror-symmetric). We demonstrate that the position observable for certain open maps, used to model random walks and the finite-time Lyapunov exponent (FTLE) for the logistic map are of this derived form, thus providing a simple explanation for some existing results.

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 studies dynamical observables A = sum_{n=1}^N g(x_n) along trajectories of chaotic maps. It claims that distinct observables built from g1 and g2 share the identical large-deviation rate function I(a) whenever g1(x) − g2(x) belongs to a newly introduced class of 'derived' functions. When g itself is derived, the distribution of A becomes N-independent for large N and is non-Gaussian yet mirror-symmetric. Concrete examples are the position observable on certain open maps (modeling random walks) and the finite-time Lyapunov exponent on the logistic map; these are asserted to be derived, thereby explaining prior observations of non-Gaussian or N-independent statistics.

Significance. If the definition and closure properties of the derived class are made rigorous, the result supplies a unifying physical mechanism for the observed coincidence of rate functions across different observables and accounts for the N-independent non-Gaussian limits reported in the literature for open maps and the logistic map. The work thereby strengthens the link between large-deviation theory and deterministic chaos without introducing free parameters or ad-hoc fitting.

major comments (2)
  1. [§3] §3 (definition of derived functions): the manuscript introduces the class of 'derived' functions to establish both the equality of rate functions I(a) for g1 and g2 and the N-independent limit when g is derived, yet supplies neither an explicit functional equation (e.g., g = f ∘ T − f or a cocycle condition) nor a proof that the class is closed under addition and composition with the observables needed for the large-deviation principle. Without these, the central equivalence does not follow from standard LDT results for deterministic systems.
  2. [§4.2] §4.2 (logistic-map FTLE example): membership of the finite-time Lyapunov exponent in the derived class is asserted rather than verified by direct substitution into the map's transfer operator or by exhibiting the requisite f such that g = f ∘ T − f. Because this example is used to illustrate the N-independent non-Gaussian claim, an explicit check is required to confirm that the asserted rate-function identity is not circular.
minor comments (2)
  1. [Abstract] The abstract and introduction use the phrase 'derived' before its formal definition appears; a forward reference or brief parenthetical characterization would improve readability.
  2. [§2] Notation for the observable A and the scaled variable a = A/N is introduced inconsistently across sections; a single consolidated definition would reduce ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We have revised the manuscript to address the concerns regarding the rigor of the derived functions definition and the verification of the example. Below we respond point by point.

read point-by-point responses

  1. Referee: [§3] §3 (definition of derived functions): the manuscript introduces the class of 'derived' functions to establish both the equality of rate functions I(a) for g1 and g2 and the N-independent limit when g is derived, yet supplies neither an explicit functional equation (e.g., g = f ∘ T − f or a cocycle condition) nor a proof that the class is closed under addition and composition with the observables needed for the large-deviation principle. Without these, the central equivalence does not follow from standard LDT results for deterministic systems.

    Authors: We appreciate the referee's observation that the definition of 'derived' functions requires more rigor. In the revised manuscript, we now provide an explicit functional equation: a function g is derived if there exists a function f such that g(x) = f(T(x)) - f(x), where T is the dynamical map. We also prove that this class is closed under addition and under composition with observables in the context of the large deviation principle. This allows the equality of rate functions to follow directly from standard results in large deviation theory for deterministic dynamical systems. revision: yes

  2. Referee: [§4.2] §4.2 (logistic-map FTLE example): membership of the finite-time Lyapunov exponent in the derived class is asserted rather than verified by direct substitution into the map's transfer operator or by exhibiting the requisite f such that g = f ∘ T − f. Because this example is used to illustrate the N-independent non-Gaussian claim, an explicit check is required to confirm that the asserted rate-function identity is not circular.

    Authors: We agree that an explicit verification for the finite-time Lyapunov exponent (FTLE) is essential to avoid any appearance of circularity. In the revised version, we explicitly construct the function f for the logistic map such that the FTLE observable g satisfies g = f ∘ T - f. We verify this by direct substitution and confirm the resulting N-independent distribution. This strengthens the illustration of the non-Gaussian, mirror-symmetric limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new 'derived' class defined and applied independently

full rationale

The paper introduces the 'derived' function class explicitly to interpret why rate functions coincide for observables differing by such a function, and separately shows that when g itself is derived the large-N distribution becomes N-independent and non-Gaussian. No equation or step in the abstract or described derivation reduces the claimed equivalence to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The examples (position observable on open maps, FTLE on logistic map) are asserted to belong to the class after the definition is given, rather than the membership being verified only by assuming the rate-function identity. The derivation chain therefore remains self-contained against external large-deviation results and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the newly introduced concept of 'derived' functions together with the standard assumption that the large deviation principle applies to sums along chaotic trajectories.

axioms (1)
  • domain assumption The large deviation principle applies to the dynamical observable A in chaotic systems.
    This is invoked as the framework for describing P(A) ~ exp(-N I(A/N)).
invented entities (1)
  • derived functions no independent evidence
    purpose: Class of functions g such that adding them to another observable does not change the rate function and makes the distribution N-independent.
    Newly postulated class introduced to account for the observed statistical similarities.

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