Second quantization for classical nonlinear dynamics

TL;DR

This paper introduces a novel framework using quantum-inspired techniques to analyze classical nonlinear dynamical systems through infinite-dimensional rotation models, enabling data-driven approximations and applications to complex flows.

Contribution

It develops a new mathematical framework employing Fock spaces and spectral decompositions to model and approximate the evolution of observables in classical ergodic systems.

Findings

Constructed weighted Fock spaces with Banach algebra structure.

Decomposed spectra into infinite-dimensional tori.

Demonstrated data-driven approximation methods with kernel techniques.

Abstract

Using techniques from many-body quantum theory, we propose a framework for representing the evolution of observables of measure-preserving ergodic flows through infinite-dimensional rotation systems on tori. This approach is based on a class of weighted Fock spaces $F_w(\mathcal H_\tau)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_\tau$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_\tau)$ and show that their Banach algebra spectra, $\sigma(F_w(\mathcal H_\tau))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_\tau$ of the Koopman operator acting on $\mathcal H_\tau$ are then lifted to rotation systems on these tori akin to the topological models of ergodic systems with pure point spectra in the Halmos--von Neumann theorem. Our scheme also employs a procedure for representing observables of the original system by polynomial functions on finite-dimensional tori in $\sigma(F_w(\mathcal H_\tau))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_\tau$. This leads to models for the Koopman evolution of observables on $L^2$ built from tensor products of finite collections of approximate Koopman eigenfunctions. Numerically, the scheme is amenable to consistent data-driven implementation using kernel methods. We illustrate it with applications to Stepanoff flows on the 2-torus and the Lorenz 63 system. Connections with quantum computing are also discussed.