PPubMed7 Feb 2023Cited 4×

Symplectic Geometry Aspects of the Parametrically-Dependent Kardar–Parisi–Zhang Equation of Spin Glasses Theory, Its Integrability and Related Thermodynamic Stability †

Anatolij K. Prykarpatski, Petro Y. Pukach, Myroslava I. Vovk, Alexander Toikka, Dmitry Gromov

Abstract

'Alexander Toikka' 'Dmitry Gromov'] A thermodynamically unstable spin glass growth model described by means of the parametrically-dependent Kardar-Parisi-Zhang equation is analyzed within the symplectic geometry-based gradient-holonomic and optimal control motivated algorithms. The finitely-parametric functional extensions of the model are studied, and the existence of conservation laws and the related Hamiltonian structure is stated. A relationship of the Kardar-Parisi-Zhang equation to a so called dark type class of integrable dynamical systems, on functional manifolds with hidden symmetries, is stated.

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Using symplectic geometry, the authors prove a parametrically‑dependent KPZ model for spin‑glass growth is integrable, possessing conserved quantities and a Hamiltonian structure.

What’s new

i.Applied a symplectic‑geometry based gradient‑holonomic algorithm to the parametrically‑dependent KPZ equation.
ii.Demonstrated the existence of conservation laws and a Hamiltonian structure for the spin‑glass growth model.
iii.Established a link between the KPZ system and dark‑type integrable dynamical systems.
iv.Derived an infinite hierarchy of conserved quantities and explicit Hamiltonian functional.
v.Formulated an optimal‑control framework for the extended KPZ dynamics.

How they did it

The study employed a gradient‑holonomic algorithm motivated by symplectic geometry on functional manifolds, combined with optimal‑control theory, to analyze integrability. These techniques were used to construct conservation laws, Poisson structures, and Lax‑type representations for the parametrically‑extended KPZ system.

What they found

Proposition 1: the extended KPZ system possesses a single local conserved quantity γ₀=∫u vₓ dx, ensuring thermodynamic stability.
Proposition 2: an infinite hierarchy of conserved quantities γ₀, γ₁, γ₂,… is obtained, with explicit forms such as γ₁=−¼∫(u²vₓ²+4uₓvₓ)dx.
The Hamiltonian H=∫(uₓvₓ+u²vₓ²/4)dx generates the dynamics via a skew‑symmetric Poisson operator.

Caveats

✕The determining equations for the auxiliary density A are highly complex and not solved analytically in the paper.
✕The analysis remains theoretical; no numerical simulations or experimental validation are provided.
✕Results rely on specific functional extensions; broader generalizations are left for future work.

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Symplectic Geometry Aspects of the Parametrically-Dependent Kardar–Parisi–Zhang Equation of Spin Glasses Theory, Its Integrability and Related Thermodynamic Stability †

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