| 1. |
- Hoel, Hakon, et al.
(author)
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Classical langevin dynamics derived from quantum mechanics
- 2020
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In: Discrete and continuous dynamical systems. Series B. - : AMER INST MATHEMATICAL SCIENCES-AIMS. - 1531-3492 .- 1553-524X. ; 25:10, s. 4001-4038
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Journal article (peer-reviewed)abstract
- The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.
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| 2. |
- Huang, Xin, et al.
(author)
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Canonical mean-field molecular dynamics derived from quantum mechanics
- 2022
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In: ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - : EDP Sciences. - 2822-7840 .- 2804-7214. ; 56:6, s. 2197-2238
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Journal article (peer-reviewed)abstract
- Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr(He-beta H)/Tr(e(-beta H)) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian (sic). It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M-1 + t epsilon(2)), for correlation time t where epsilon(2) is related to the variance of mean value approximation h. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.
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| 3. |
- Kammonen, Aku, 1984-
(author)
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Numerical algorithms for high dimensional integration with application to machine learning and molecular dynamics
- 2021
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Doctoral thesis (other academic/artistic)abstract
- This thesis contains results on high dimensional integration with two papers, paper I and paper II, presenting applications in machine learning and two papers, paper III and paper IV, presenting applications to molecular dynamics.In paper I we present algorithms based on a Metropolis test for training shallow neural networks with trigonometric activation functions. Numerical experiments are performed on both synthetic and real data. The trigonometric activation function gives access to the Fourier transform and its inverse transform. The algorithms gives equidistributed amplitudes.In paper II we derive smaller generalization error for deep residual neural networks compared to shallow ones. An algorithm that builds the residual neural network layer by layer based on an algorithm from paper I is presented both as a stand alone algorithm as well as a pre-step for a global optimizer like Stochastic gradient descent or Adam. Numerical test are performed with promising results.In paper III we make use of the semiclassical Weyl law to show that canonical quantum observables can be approximated by molecular dynamics with an error rate proportional to the electron-nuclei mass ratio. Numerical experiments are presented that confirms the expected theoretical result.In paper IV we consider canonical ensembles of molecular systems. We propose four numerical algorithms for efficient computation of the canonical ensemble molecular dynamics observables. The four algorithms can each be efficient in different situations. For example in low temperatures we can make use of the fact that the lowest electron energy levels contributes most to the observable. The work is an extension of the results in paper III.
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