Accuracy balancing for the finite-difference-based solution of the discrete Wigner transport equation

Kim, Kyoung-Youm; Kim, Saehwa; Tang, Ting-Wei

doi:10.1007/s10825-016-0944-9

Cited by 9 publications

(11 citation statements)

References 41 publications

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“…A calculation analogous to Eq. (12) gives Hence, the extension of Eq. ( 13) to N particles in three dimensions is given by with ∑ i T i given by Eq.…”

Section: Grw Theorymentioning

confidence: 98%

“…We can now combine Eqs. ( 8), (9), and (12) to the GRW master equation for Wigner functions, given by To simplify Eq. ( 13), we make two approximations.…”

Section: Grw Theorymentioning

confidence: 99%

“…(20) (which is what we are interested in) due to the mathematical relation of Langevin and Fokker-Planck equations. 12 Third, we have made the approximation of moving from a master to a Fokker-Planck equation [e.g., from Eq. ( 13) to Eq.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Wigner functions provide a description of quantum systems in phase space. Since their initial development [1], they have found a significant number of applications [2][3][4][5] in fields such as atomic physics [6], quantum optics [7][8][9], visualization of quantum effects [10,11], computational electronics [12][13][14], and solid-state theory [15][16][17][18][19]. Besides these practical aspects, they are also of interest for fundamental questions in quantum mechanics such as the theory of quantum chaos [20].…”

Section: Introductionmentioning

confidence: 99%

“… Drossel [132] takes, in a sense, the opposite standpoint as she argues for the existence of limitations to quantum dynamics based on the success of classical molecular dynamics simulations 12. This line of thought is similar to Wallace's[29] own argument, who explains the success of classical statistical mechanics (even in the case of probably inaccurate microdynamics) based on the fact that it arises as a limiting case of quantum many-body dynamics.…”

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confidence: 99%

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Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

2021

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Wigner functions, allowing for a reformulation of quantum mechanics in phase space, are of central importance for the study of the quantum-classical transition. A full understanding of the quantum-classical transition, however, also requires an explanation for the absence of macroscopic superpositions to solve the quantum measurement problem. Stochastic reformulations of quantum mechanics based on spontaneous collapses of the wavefunction are a popular approach to this issue. In this article, we derive the dynamic equations for the four most important spontaneous collapse models—Ghirardi–Rimini–Weber (GRW) theory, continuous spontaneous localization (CSL) model, Diósi-Penrose model, and dissipative GRW model—in the Wigner framework. The resulting master equations are approximated by Fokker–Planck equations. Moreover, we use the phase-space form of GRW theory to test, via molecular dynamics simulations, David Albert’s suggestion that the stochasticity induced by spontaneous collapses is responsible for the emergence of thermodynamic irreversibility. The simulations show that, for initial conditions leading to anti-thermodynamic behavior in the classical case, GRW-type perturbations do not lead to thermodynamic behavior. Consequently, the GRW-based equilibration mechanism proposed by Albert is not observed.

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“…A calculation analogous to Eq. (12) gives Hence, the extension of Eq. ( 13) to N particles in three dimensions is given by with ∑ i T i given by Eq.…”

Section: Grw Theorymentioning

confidence: 98%

“…We can now combine Eqs. ( 8), (9), and (12) to the GRW master equation for Wigner functions, given by To simplify Eq. ( 13), we make two approximations.…”

Section: Grw Theorymentioning

confidence: 99%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

2021

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Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

Iwamoto

Tanimura

2021

J Comput Electron

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Discretizing distribution function in a phase space for an efficient quantum dynamics simulation is non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we find that a twodimensional (2D) periodically invariant system-bath (PISB) model with two heat baths is an ideal platform not only for a periodical system but also for a system confined by a potential. We then derive the numerically "exact" hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. The stability of the present scheme is demonstrated in a high-temperature Markovian case by numerically integrating the discrete QFPE with by a coarse mesh for a 2D free rotor and harmonic potential systems for an initial condition that involves singularity.

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Recent advances in Wigner function approaches

Weinbub

Ferry

2018

Applied Physics Reviews

229

174

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The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.

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Accuracy balancing for the finite-difference-based solution of the discrete Wigner transport equation

Cited by 9 publications

References 41 publications

Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility

Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

Recent advances in Wigner function approaches

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