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Applications of the positive definite wigner distribution function

A. D. Jannussis, A. Leodaris, X. Patargias +4 more

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoJul 1, 19825 citations

The Wigner Distribution Function: Analytical Evaluation

Amalia Torre

Elsevier eBooksJan 1, 20051 citation

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions

Jian Liu, William Hughes Miller

We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum canonical distribution-equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.

The Journal of Chemical PhysicsMar 8, 201138 citations

Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation?

Matteo Villani, Xavier Oriols

Abstract To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.Abstract To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.

arXiv:2201.07574Journal of Computational ElectronicsNov 23, 20215 citations

Properties of Wigner distribution functions applied to quantum mechanics

A V Gorbachev

Вестник Российского университета дружбы народов. Серия: Математика, информатика, физикаJan 1, 20120 citations

WIGNER PHASE-SPACE DISTRIBUTION FUNCTIONS

World Scientific lecture notes in physicsMar 1, 19910 citations

Dynamics of the wigner distribution function: conservation of positivity in time

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