Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.

May 8, 201930 citations

Statistics and Nosé formalism for Ehrenfest dynamics

J L Alonso, A Castro, Jesús Clemente-Gallardo +3 more

arXiv:1104.2154Journal of Physics A Mathematical and TheoreticalSep 7, 201129 citations

Stochastic Quantum Chemistry for extended systems

George H. Booth

Bulletin of the American Physical SocietyMar 5, 20150 citations

Temperature-controlled open-quantum-system dynamics using a time-dependent variational method

Mantas Jakučionis, Darius Abramavičius

The Dirac-Frenkel variational method with Davydov ${\text{D}}_{2}$ trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite-size bath when in contact with the quantum system. Thermalization of the bath vibrational modes is realized via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical analysis of the exciton relaxation dynamics in a small molecular cluster reveals that thermalization additionally provides significant calculation speedup due to the reduced number of vibrational modes needed to obtain the convergence.The Dirac-Frenkel variational method with Davydov ${\text{D}}_{2}$ trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite-size bath when in contact with the quantum system. Thermalization of the bath vibrational modes is realized via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical analysis of the exciton relaxation dynamics in a small molecular cluster reveals that thermalization additionally provides significant calculation speedup due to the reduced number of vibrational modes needed to obtain the convergence.

arXiv:2103.02586Physical Review AMar 1, 20215 citations

Stochastic quantum dynamics beyond mean field

Denis Lacroix A, Sakir Ayik B

arXiv:1402.2393The European Physical Journal AJun 1, 2014110 citations

The Dirac-Frenkel variational method with Davydov

{\text{D}}_{2}

trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite-size bath when in contact with the quantum system. Thermalization of the bath vibrational modes is realized via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical analysis of the exciton relaxation dynamics in a small molecular cluster reveals that thermalization additionally provides significant calculation speedup due to the reduced number of vibrational modes needed to obtain the convergence.