1
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Domingo L, Borondo J, Borondo F. Using reservoir computing to construct scarred wave functions. Phys Rev E 2024; 109:044214. [PMID: 38755942 DOI: 10.1103/physreve.109.044214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2023] [Accepted: 01/22/2024] [Indexed: 05/18/2024]
Abstract
Scar theory is one of the fundamental pillars in the field of quantum chaos, and scarred functions are a superb tool to carry out studies in it. Several methods, usually semiclassical, have been described to cope with these two phenomena. In this paper, we present an alternative method, based on the novel machine learning algorithm known as reservoir computing, to calculate such scarred wave functions together with the associated eigenstates of the system. The resulting methodology achieves outstanding accuracy while reducing execution times by a factor of ten. As an illustration of the effectiveness of this method, we apply it to the widespread chaotic two-dimensional coupled quartic oscillator.
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Affiliation(s)
- L Domingo
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco - 28049 Madrid, Spain
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, 28035 Madrid, Spain
- Instituto de Ciencias Matemáticas (ICMAT), Campus de Cantoblanco, Nicolás Cabrera, 13-15, 28049 Madrid, Spain
| | - J Borondo
- Departamento de Telemática y Computación, Universidad Pontificia de Comillas, 28015 Madrid, Spain
| | - F Borondo
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco - 28049 Madrid, Spain
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2
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Domingo L, Borondo J, Borondo F. Adapting reservoir computing to solve the Schrödinger equation. CHAOS (WOODBURY, N.Y.) 2022; 32:063111. [PMID: 35778135 DOI: 10.1063/5.0087785] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2022] [Accepted: 05/17/2022] [Indexed: 06/15/2023]
Abstract
Reservoir computing is a machine learning algorithm that excels at predicting the evolution of time series, in particular, dynamical systems. Moreover, it has also shown superb performance at solving partial differential equations. In this work, we adapt this methodology to integrate the time-dependent Schrödinger equation, propagating an initial wavefunction in time. Since such wavefunctions are complex-valued high-dimensional arrays, the reservoir computing formalism needs to be extended to cope with complex-valued data. Furthermore, we propose a multi-step learning strategy that avoids overfitting the training data. We illustrate the performance of our adapted reservoir computing method by application to four standard problems in molecular vibrational dynamics.
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Affiliation(s)
- L Domingo
- Instituto de Ciencias Matemáticas (ICMAT), Campus de Cantoblanco UAM, Nicolás Cabrera, 13-15, 28049 Madrid, Spain
| | - J Borondo
- Departamento de Gestión Empresarial, Universidad Pontificia de Comillas ICADE, Alberto Aguilera 23, 28015 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Campus de Cantoblanco UAM, Nicolás Cabrera, 13-15, 28049 Madrid, Spain
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3
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Arranz FJ, Benito RM, Borondo F. Correspondence between classical and quantum resonances. Phys Rev E 2021; 103:062207. [PMID: 34271628 DOI: 10.1103/physreve.103.062207] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2020] [Accepted: 05/12/2021] [Indexed: 11/07/2022]
Abstract
Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li⇆Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to ℏ=0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit ℏ→0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.
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Affiliation(s)
- F J Arranz
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - R M Benito
- Grupo de Sistemas Complejos, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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4
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Montes J, Revuelta F, Borondo F. Bohr-Sommerfeld-like quantization in the theory of walking droplets. Phys Rev E 2021; 103:053110. [PMID: 34134206 DOI: 10.1103/physreve.103.053110] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2019] [Accepted: 04/29/2021] [Indexed: 11/07/2022]
Abstract
Recent experiments have shown that self-propelled millimetric walking droplets bouncing on a vibrating liquid surface exhibit phenomena, such as interference or tunneling, that so far were thought to be possible only in the microscopic realm. Here we present calculations showing that the surface wave satisfies, in the long-memory limit, a Bohr-Sommerfeld quantization-like relation. This strongly suggest the possibility of a novel fundamental type of quantization in these experiments, which can simultaneously explain their emulation of the quantum behavior and, more importantly, shed light into some of the interpretational difficulties of the standard quantum theory.
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Affiliation(s)
- J Montes
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
| | - F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avda. Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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5
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Revuelta F, Vergini E, Benito RM, Borondo F. Short-periodic-orbit method for excited chaotic eigenfunctions. Phys Rev E 2020; 102:042210. [PMID: 33212620 DOI: 10.1103/physreve.102.042210] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/17/2019] [Accepted: 09/18/2020] [Indexed: 11/07/2022]
Abstract
An alternative method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wave functions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wave functions is only of the order of the ratio t_{H}/t_{E}, with t_{H} the Heisenberg time and t_{E} the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - E Vergini
- Departamento de Física, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, Avenida Puerta de Hierro 2-4, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain.,Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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6
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Tekur SH, Kumar S, Santhanam MS. Exact distribution of spacing ratios for random and localized states in quantum chaotic systems. Phys Rev E 2018; 97:062212. [PMID: 30011473 DOI: 10.1103/physreve.97.062212] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2017] [Indexed: 06/08/2023]
Abstract
Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra are modeled through an appropriate ensemble described by random matrix theory. However, a small subset of states violates this principle and displays eigenstate localization, a counterintuitive feature known to arise due to purely quantum or semiclassical effects. In the spectrum of chaotic systems, the localized and random states interact with one another and modify the spectral statistics. In this work, a 3×3 random matrix model is used to obtain exact results for the ratio of spacing between a generic and localized state. We consider time-reversal-invariant as well as noninvariant scenarios. These results agree with the spectra computed from realistic physical systems that display localized eigenmodes.
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Affiliation(s)
- S Harshini Tekur
- Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India
| | - Santosh Kumar
- Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar Pradesh 201314, India
| | - M S Santhanam
- Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India
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7
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Párraga H, Arranz FJ, Benito RM, Borondo F. Above Saddle-Point Regions of Order in a Sea of Chaos in the Vibrational Dynamics of KCN. J Phys Chem A 2018. [DOI: 10.1021/acs.jpca.8b00113] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- H. Párraga
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. J. Arranz
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - R. M. Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. Borondo
- Instituto de Ciencias Matemáticas, Cantoblanco, 28049 Madrid, Spain
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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8
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Revuelta F, Vergini E, Benito RM, Borondo F. Semiclassical basis sets for the computation of molecular vibrational states. J Chem Phys 2017; 146:014107. [PMID: 28063420 DOI: 10.1063/1.4973376] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN molecule.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - E Vergini
- Departamento de Física, Comisión Nacional de Energía Atómica, Av. del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), 28049 Cantoblanco, Madrid, Spain
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9
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Revuelta F, Vergini E, Benito RM, Borondo F. Scar Functions, Barriers for Chemical Reactivity, and Vibrational Basis Sets. J Phys Chem A 2016; 120:4928-38. [DOI: 10.1021/acs.jpca.5b12541] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- F. Revuelta
- Grupo
de Sistemas Complejos, Escuela Técnica Superior de Ingeniería
Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
- Instituto
de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
| | - E. Vergini
- Departamento
de Física, Comisión Nacional de la Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R. M. Benito
- Grupo
de Sistemas Complejos, Escuela Técnica Superior de Ingeniería
Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. Borondo
- Instituto
de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
- Departamento
de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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