Second quantization for classical nonlinear dynamics
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916649245868032 |
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| author | Giannakis, Dimitrios Jebelli, Mohammad Javad Latifi Montgomerry, Michael Pfeffer, Philipp Schumacher, Jörg Slawinska, Joanna |
| author_facet | Giannakis, Dimitrios Jebelli, Mohammad Javad Latifi Montgomerry, Michael Pfeffer, Philipp Schumacher, Jörg Slawinska, Joanna |
| contents | 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 $F_w(\mathcal H_τ)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_τ$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_τ)$ and show that their Banach algebra spectra, $σ(F_w(\mathcal H_τ))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_τ$ of the Koopman operator acting on $\mathcal H_τ$ 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 $σ(F_w(\mathcal H_τ))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_τ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07419 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second quantization for classical nonlinear dynamics Giannakis, Dimitrios Jebelli, Mohammad Javad Latifi Montgomerry, Michael Pfeffer, Philipp Schumacher, Jörg Slawinska, Joanna Dynamical Systems Fluid Dynamics Quantum Physics 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 $F_w(\mathcal H_τ)$ generated by a 1-parameter family of reproducing kernel Hilbert spaces $\mathcal H_τ$, and endowed with commutative Banach algebra structure under the symmetric tensor product using a subconvolutive weight $w$. We describe the construction of the spaces $F_w(\mathcal H_τ)$ and show that their Banach algebra spectra, $σ(F_w(\mathcal H_τ))$, decompose into a family of tori of potentially infinite dimension. Spectrally consistent unitary approximations $U^t_τ$ of the Koopman operator acting on $\mathcal H_τ$ 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 $σ(F_w(\mathcal H_τ))$ of arbitrarily large degree, with coefficients determined from pointwise products of eigenfunctions of $U^t_τ$. 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. |
| title | Second quantization for classical nonlinear dynamics |
| topic | Dynamical Systems Fluid Dynamics Quantum Physics |
| url | https://arxiv.org/abs/2501.07419 |