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arxiv: 2603.01278 · v3 · pith:SITQBIBDnew · submitted 2026-03-01 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Linearization Principle: The Geometric Origin of Nonlinear Fokker-Planck Equations

Hiroki Suyari This is my paper

Pith reviewed 2026-05-25 06:35 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords anomalous diffusionnonlinear Fokker-PlanckTsallis statisticsq-GaussianLinearization PrincipleH-theoremq-deformed geometrypower-law distributions
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The pith

Nonlinear Fokker-Planck equations emerge from linear drift via the Linearization Principle in q-deformed geometry.

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

The paper shows how to derive nonlinear Fokker-Planck equations geometrically by applying the Linearization Principle at the dynamical stage. This is done by taking the generalized chemical potential as the dynamical ansatz, which keeps the drift term linear in the probability density. A sympathetic reader cares because this approach gives a consistent foundation for anomalous diffusion and power-law distributions in complex systems without using ad-hoc nonlinear forces. The framework reveals that in q-deformed geometry the dynamic index q is dual to the thermodynamic index 2-q, so that the stationary state is a q-Gaussian that minimizes the free energy defined by the entropy of index 2-q. The paper also proves the H-theorem for the equation and shows its use for the harmonic oscillator and free particle.

Core claim

By introducing the Linearization Principle directly at the dynamical stage through the generalized chemical potential as the natural dynamical ansatz, the nonlinear Fokker-Planck equation is derived such that the drift term is linear. In the q-deformed geometry corresponding to Tsallis statistics, there is a duality between the dynamic index q and the thermodynamic index 2-q. The stationary state is a q-Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index 2-q. The standard form of the Einstein relation is preserved.

What carries the argument

The Linearization Principle, which uses the generalized chemical potential to construct the dynamical equation while keeping the drift linear.

If this is right

  • The equation satisfies the H-theorem.
  • It describes the dynamics of the harmonic oscillator and the free particle.
  • Anomalous diffusion is captured without relying on phenomenological nonlinear drift forces.
  • The stationary distribution minimizes the free energy with the 2-q entropy.

Where Pith is reading between the lines

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

  • This geometric approach might apply to other forms of deformed statistics beyond Tsallis.
  • Experimental systems with power-law tails could be checked for the predicted duality between diffusion and entropy indices.
  • The linear drift preservation could simplify numerical simulations of nonlinear diffusion processes.

Load-bearing premise

The generalized chemical potential can be introduced directly as the dynamical ansatz without additional constraints.

What would settle it

If measurements in a system exhibiting anomalous diffusion show a stationary distribution that is not a q-Gaussian or if the drift term deviates from linearity while following the derived dynamics, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2603.01278 by Hiroki Suyari.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of stationary distributions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by introducing the Linearization Principle directly at the dynamical stage. By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the $q$-deformed geometry, corresponding to Tsallis statistics, exhibits a fundamental duality between the dynamic index $q$ and the thermodynamic index $2-q$: the stationary state is a $q$-Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index $2-q$. We prove the $H$-theorem for the derived equation and demonstrate its application to the harmonic oscillator and the free particle. This framework describes anomalous diffusion without relying on ad-hoc constraints or phenomenological nonlinear drift forces.

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

1 major / 0 minor

Summary. The manuscript claims a geometric derivation of nonlinear Fokker-Planck equations via a Linearization Principle imposed at the dynamical stage. By identifying the generalized chemical potential as the natural ansatz, the drift remains linear in density while the FP equation is nonlinear; for q-deformed geometry (Tsallis), this yields a duality between dynamic index q and thermodynamic index 2-q, with the stationary state a q-Gaussian minimizing a 2-q entropy. The paper states that the H-theorem is proved and demonstrates applications to the harmonic oscillator and free particle.

Significance. If the derivation is rigorous and the ansatz is forced by the geometry rather than chosen to enforce linearity, the framework would supply a non-phenomenological geometric origin for anomalous diffusion and Tsallis statistics, unifying dynamics with thermodynamics through the claimed q-duality. The H-theorem and explicit applications would strengthen its utility for modeling complex systems.

major comments (1)
  1. [Linearization Principle derivation] The Linearization Principle section: the identification of the generalized chemical potential as the natural dynamical ansatz is load-bearing for the q-duality claim, yet the manuscript supplies no explicit geometric axioms or intermediate identities from the q-deformed metric that force this choice without auxiliary conditions on the chemical potential or metric; if the ansatz is instead selected to produce linear drift, the duality and H-theorem follow by construction rather than as independent geometric predictions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comment. We address it point by point below.

read point-by-point responses

  1. Referee: [Linearization Principle derivation] The Linearization Principle section: the identification of the generalized chemical potential as the natural dynamical ansatz is load-bearing for the q-duality claim, yet the manuscript supplies no explicit geometric axioms or intermediate identities from the q-deformed metric that force this choice without auxiliary conditions on the chemical potential or metric; if the ansatz is instead selected to produce linear drift, the duality and H-theorem follow by construction rather than as independent geometric predictions.

    Authors: The Linearization Principle is introduced as the geometric axiom requiring that the dynamical drift remain linear in the probability density, thereby preserving the Einstein relation inside the q-deformed geometry. Within this geometry the generalized chemical potential is the variable conjugate to the density via the q-metric, as already defined in the thermodynamic sector of Tsallis statistics. We agree that the manuscript would benefit from an explicit derivation of the intermediate identities that connect the q-metric tensor to this ansatz; we will add a short subsection containing those steps so that the choice is seen to follow directly from the geometry rather than from an auxiliary linearity condition. revision: yes

Circularity Check

1 steps flagged

Linearization Principle introduced via chemical-potential ansatz that enforces linear drift, making q-duality a constructed consequence

specific steps
  1. self definitional [Abstract]
    "By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density, preserving the standard form of the Einstein relation. Within this framework, we show that the q-deformed geometry, corresponding to Tsallis statistics, exhibits a fundamental duality between the dynamic index q and the thermodynamic index 2-q: the stationary state is a q-Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index 2-q."

    The Linearization Principle is introduced precisely by adopting the ansatz that enforces linearity of the drift; the duality and H-theorem are then derived inside the framework built from that choice. The claimed geometric necessity therefore reduces to the definitional step rather than following from independent geometric axioms.

full rationale

The paper's derivation begins by defining the Linearization Principle through the explicit choice of generalized chemical potential as dynamical ansatz to keep drift linear in density. The q-duality (dynamic q paired with thermodynamic 2-q) and associated stationary q-Gaussian minimizing 2-q entropy are then shown inside the resulting framework. This reduces the claimed geometric origin to a direct consequence of the initial ansatz rather than an independent derivation from prior q-geometry axioms, matching the self-definitional pattern. No self-citations, fitted predictions, or uniqueness theorems are invoked in the provided text to support the ansatz, so the circularity is localized to this definitional step.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the newly introduced Linearization Principle and the identification of the generalized chemical potential as the dynamical ansatz. These are presented without independent evidence outside the paper. The q-deformed geometry and generalized entropy of index 2-q are taken from Tsallis statistics.

free parameters (1)
  • q
    Deformation parameter of the q-geometry that controls both dynamics and the stationary distribution; its value is not derived from first principles in the abstract.
axioms (1)
  • ad hoc to paper The generalized chemical potential is the natural dynamical ansatz for the system.
    This identification is stated as the key step that allows the Linearization Principle to be applied at the dynamical stage.
invented entities (1)
  • Linearization Principle no independent evidence
    purpose: To derive the nonlinear Fokker-Planck equation geometrically while keeping the drift term linear.
    New principle introduced directly in the paper; no independent evidence or prior reference is given in the abstract.

pith-pipeline@v0.9.0 · 5702 in / 1473 out tokens · 25663 ms · 2026-05-25T06:35:18.457749+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 echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the q-deformed geometry... exhibits a fundamental duality between the dynamic index q and the thermodynamic index 2-q: the stationary state is a q-Gaussian distribution that minimizes a free energy functional defined by a generalized entropy of index 2-q

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

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density

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

13 extracted references · 13 canonical work pages

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    Suyari, Exact density profiles of 1D quantum fluids in the thomas-fermi limit: Geometric hierarchy to the tonks-girardeau gas (2026), arXiv:2603.01273

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    work page arXiv 2026

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    Tsallis and D

    C. Tsallis and D. J. Bukman, Tsallis’ thermostatistics and the porous medium equation, Physical Review E54, R2197 (1996)

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    A. Ohara and T. Wada, Information geometry ofq- gaussian densities and behaviors of solutions to related diffusion equations, Journal of Physics A: Mathematical and Theoretical43, 035002 (2010)

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