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arxiv: 2511.12574 · v3 · submitted 2025-11-16 · ❄️ cond-mat.stat-mech

Symmetry-based nonlinear fluctuating hydrodynamics in one dimension

Yuki Minami , Hiroyoshi Nakano , Keiji Saito This is my paper

Pith reviewed 2026-05-17 22:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords nonlinear fluctuating hydrodynamicsKPZ scalingone-dimensional systemsdynamical renormalization groupsymmetry-based formulationthermalizationuniversal transport
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0 comments X

The pith

Symmetry and conservation laws determine the nonlinear fluctuating hydrodynamics equations in one dimension, producing a KPZ fixed point with dynamical exponent 3/2 for sound and heat modes.

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

The paper develops a formulation of nonlinear fluctuating hydrodynamics for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. It derives the hydrodynamic equations solely from symmetry and conservation principles in a manner fully consistent with thermalization and independent of microscopic details. Dynamic renormalization group analysis identifies a KPZ-type fixed point where both sound and heat modes share the dynamical exponent z=3/2. Numerical simulations of the resulting equations confirm the exponent and show that the scaling functions for both modes approach the universal Prahofer-Spohn form.

Core claim

A symmetry-based formulation yields the nonlinear fluctuating hydrodynamics equations for one-dimensional systems directly from symmetry and conservation principles, ensuring consistency with thermalization and producing a KPZ-type fixed point with dynamical exponent z=3/2 for both sound and heat modes, with simulations confirming closeness to the Prahofer-Spohn scaling function.

What carries the argument

Symmetry-based derivation of the nonlinear fluctuating hydrodynamics equations, analyzed via dynamic renormalization group to locate the KPZ fixed point.

If this is right

  • The dynamical exponent z=3/2 governs scaling for both sound and heat modes.
  • Fluctuations of both modes approach the universal Prahofer-Spohn scaling function.
  • Transport and fluctuation phenomena become universal across systems with different microscopic details.
  • The description remains consistent with thermalization in the nonequilibrium steady state.

Where Pith is reading between the lines

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

  • The same symmetry arguments could organize fluctuating hydrodynamics in other one-dimensional systems obeying similar conservation laws.
  • Extensions to weakly perturbed higher-dimensional cases might reveal when the z=3/2 scaling persists or crosses over.
  • Direct comparison of measured correlation functions in lattice models against the derived equations could test the independence from interaction specifics.

Load-bearing premise

The hydrodynamic equations for generic homogeneous nearest-neighbor interactions can be derived solely from symmetry and conservation principles while remaining fully consistent with thermalization and independent of microscopic details.

What would settle it

Observation of a dynamical exponent other than 3/2 for the sound or heat modes in numerical simulations or physical experiments on one-dimensional systems with nearest-neighbor interactions.

Figures

Figures reproduced from arXiv: 2511.12574 by Hiroyoshi Nakano, Keiji Saito, Yuki Minami.

Figure 1
Figure 1. Figure 1: FIG. 1. Numerical results for the dynamical scaling of the space–time correlation [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Self-energy diagrams. The left panel shows the Σ [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Diagrams of vertex corrections [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical results for the time correlation function [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We present a symmetry-based formulation of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. We derive the hydrodynamic equations solely from symmetry and conservation principles, ensuring full consistency with thermalization. Using the dynamic renormalization group, we identify a KPZ-type fixed point, characterized by the dynamical exponent $z=3/2$ for both the sound and heat modes. Extensive numerical simulations of the derived NFH equations confirm this exponent and further reveal that both modes are close to the universal KPZ scaling function, the Prahofer-Spohn function. These findings establish a unified, symmetry-based framework for understanding universal transport and fluctuation phenomena in one-dimensional nonequili brium systems, independent of microscopic details.

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 presents a symmetry-based formulation of nonlinear fluctuating hydrodynamics (NFH) for one-dimensional many-particle systems with generic homogeneous nearest-neighbor interactions. The hydrodynamic equations are derived solely from symmetry and conservation principles while ensuring consistency with thermalization. Dynamic renormalization group analysis identifies a KPZ-type fixed point with dynamical exponent z=3/2 for both sound and heat modes. Numerical simulations of the derived NFH equations are used to confirm this exponent and to show that both modes approach the universal KPZ scaling function (Prahofer-Spohn function).

Significance. If the central claims hold, the work supplies a unified, parameter-free framework for universal transport and fluctuations in one-dimensional nonequilibrium systems that is independent of microscopic details. It would extend KPZ universality to both sound and heat modes within a symmetry-derived hydrodynamic description and strengthen the link between conservation laws and dynamical scaling in low-dimensional systems.

major comments (2)
  1. [Derivation of the NFH equations] The central derivation asserts that the complete set of NFH equations, including stochastic noise terms and their correlators, follows from symmetry and conservation alone. However, consistency with thermalization requires that noise amplitudes satisfy the fluctuation-dissipation theorem and match equilibrium susceptibilities (compressibility, specific heat). The manuscript does not explicitly demonstrate how these amplitudes are fixed without additional thermodynamic input, which is the load-bearing step for the claimed independence from microscopic details.
  2. [Dynamic renormalization group analysis] The DRG analysis identifies the KPZ fixed point with z=3/2 for the heat mode on the basis of the nonlinearities and noise strengths obtained in the symmetry derivation. If the noise correlators are not rigorously constrained by symmetry alone, the assignment of the universality class for the heat mode rests on an assumption that requires explicit verification, for example by showing that the noise matrix is determined solely by the same symmetry constraints used for the deterministic terms.
minor comments (2)
  1. [Abstract] The abstract contains a typographical error: 'nonequili brium' should read 'nonequilibrium'.
  2. [Numerical simulations] The description of the numerical simulations would benefit from additional detail on integration scheme, system sizes, boundary conditions, and the precise procedure used to extract and compare scaling functions with the Prahofer-Spohn distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below and have made revisions to the manuscript to clarify the points raised.

read point-by-point responses

  1. Referee: [Derivation of the NFH equations] The central derivation asserts that the complete set of NFH equations, including stochastic noise terms and their correlators, follows from symmetry and conservation alone. However, consistency with thermalization requires that noise amplitudes satisfy the fluctuation-dissipation theorem and match equilibrium susceptibilities (compressibility, specific heat). The manuscript does not explicitly demonstrate how these amplitudes are fixed without additional thermodynamic input, which is the load-bearing step for the claimed independence from microscopic details.

    Authors: We appreciate the referee pointing out the need for explicit demonstration. The symmetry and conservation laws determine the form of both the deterministic currents and the noise terms. To ensure consistency with thermalization, the noise strengths are fixed by the fluctuation-dissipation theorem, which relates them to the equilibrium susceptibilities. In the revised manuscript, we have added a detailed explanation in Section II showing how these amplitudes are determined solely from the thermodynamic relations implied by the conservation laws and symmetries, without reference to specific microscopic interactions. The susceptibilities themselves are part of the hydrodynamic description and do not depend on microscopic details beyond the symmetry class. This addresses the concern while preserving the independence from microscopic details for the universal scaling. revision: yes

  2. Referee: [Dynamic renormalization group analysis] The DRG analysis identifies the KPZ fixed point with z=3/2 for the heat mode on the basis of the nonlinearities and noise strengths obtained in the symmetry derivation. If the noise correlators are not rigorously constrained by symmetry alone, the assignment of the universality class for the heat mode rests on an assumption that requires explicit verification, for example by showing that the noise matrix is determined solely by the same symmetry constraints used for the deterministic terms.

    Authors: We agree that explicit verification is important. The noise matrix is determined by the same symmetry constraints as the deterministic terms, augmented by the fluctuation-dissipation relation required for thermalization. In the revised version, we have included an explicit calculation demonstrating that the noise correlators follow directly from these symmetry-based constraints and the equilibrium conditions. This confirms that no additional assumptions are needed beyond those used for the deterministic part, thereby rigorously supporting the KPZ fixed point for the heat mode as well. revision: yes

Circularity Check

0 steps flagged

Symmetry-conservation derivation of NFH equations stands independent of fitted parameters or self-citation chains

full rationale

The paper derives the NFH equations from symmetry and conservation laws for generic nearest-neighbor interactions, then applies dynamic renormalization group analysis to recover the known KPZ fixed point (z=3/2) for sound and heat modes. No load-bearing step reduces by construction to a fitted input, self-citation, or ansatz smuggled from prior work; the noise terms are stated to ensure thermalization consistency without explicit parameter fitting to the target exponents. Numerical simulations of the derived equations provide external confirmation of the scaling. The derivation chain remains self-contained against external benchmarks, with the KPZ identification serving as an application rather than a redefinition of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on symmetry and conservation principles as the sole inputs; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • ad hoc to paper Hydrodynamic equations for 1D systems with generic homogeneous nearest-neighbor interactions can be derived solely from symmetry and conservation principles.
    This is the foundational methodological claim of the paper.
  • domain assumption The resulting equations remain fully consistent with thermalization.
    Invoked to ensure physical validity of the hydrodynamic description.

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    We thus obtain H2 =   H2 11 0 (a 3/a0)H2 11 0H 2 22 0 (a3/a0)H2 11 0 (a 2 3/a2 0)H2 11   ,(S39) H3 =   0 (1/a 0)H2 11 0 (1/a0)H2 11 0 (a 3/a2 0)H2 11 0 (a 3/a2 0)H2 11 0   .(S40) B. Special case with the space-inversion symmetry Here, we consider the special case where the Hamiltonian has the space-inversion symmetry,i→N−i. This case is special, s...

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    Nonlinear term [G α βγ(ϕβϕγ)i+1/2]: This term is approximated by Gα βγ(ϕβϕγ)i+1/2 =G α βγ ϕβ i ϕγ i+1 +ϕ β i+1ϕγ i 12 + ϕβ i ϕγ i +ϕ β i+1ϕγ i+1 6 ! .(S168) Importantly, in the absence of noise, this formulation strictly conserves the quantity P α=+,−,0(ϕα)2 [54]. The semi-discrete equations Eq. (S164) form a system of stochastic ordinary differential equ...

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