Stochastic and deterministic molecular dynamics derived from the time-independent Schr\"odinger equation

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Szepessy, Anders
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Ehrenfest, Born-Oppenheimer, Langevin and Smoluchowski dynamics are shown to be accurate approximations of time-independent Schr\"odinger observables for a molecular system avoiding caustics, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schr\"odinger equation and stability of the corresponding Hamilton-Jacobi equation, bypasses the usual separation of nuclei and electron wave functions, includes crossing electron eigenvalues, and gives a different perspective on the Born-Oppenheimer approximation, Schr\"odinger Hamiltonian systems, stochastic electron equilibrium states and numerical simulation in molecular dynamics modeling.
Comment: 36 pages: improved convergence rates for Ehrenfest observables with crossing eigenvalues and corrected estimates for wave functions approximating electron eigenstates; stability proved for crossing electron eigenvalues, temperature dependent drift correction included
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Mathematical Physics, 81Q20, 82C10, 82C31
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