Moyal Equation Research Articles

The Wigner-Vlasov formalism for time-dependent quantum oscillator

This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space

Is the Moyal equation for the Wigner function a quantum analogue of the Liouville equation?

The Moyal equation describes the evolution of the Wigner function of a quantum system in the phase space. The right-hand side of the equation contains an infinite series with coefficients proportional to powers of the Planck constant. There is an interpretation of the Moyal equation as a quantum analogue of the classical Liouville equation. Indeed, if one uses the notion of the classical passage to the limit as the Planck constant tends to zero, then formally the right-hand side of the Moyal equation tends to zero. As a result, the Moyal equation becomes the classical Liouville equation for the distribution function. In this paper, we show that the right side of the Moyal equation does not explicitly depend on the Planck constant, and all terms of the series can make a significant contribution. The transition between the classical and quantum descriptions is related not to the Planck constant, but to the spatial scale. For a model quantum system with a potential in the form of a «quadratic funnel», an exact 3D solution of the Schrödinger equation is found and the corresponding Wigner function is constructed in the paper. Using trajectory analysis in the phase space, based on the representation of the right-hand side of the Moyal equation, it is shown that on the spatial microscale there is an infinite number of «trajectories» of the particle motion (thereby the concept of a trajectory is indefinite), and when passing to the macroscale, all «trajectories» concentrate around the classical trajectory.

Wigner function of the 4-th rank

The Lorentz-Abraham-Dirac differential motion equation is of the third order since it contains the second order acceleration v→¨, which stands for the radiation friction force. To achieve a rigorous and comprehensive assessment of the dynamics of such radiation-friction systems it is necessary to expand the phase space {r→,v→} and to introduce higher kinematic values, such as v→˙ and v→¨.Our study is dedicated to the problem of building of an expanded phase space {r→,v→,v→˙,v→¨}, within which the fourth Vlasov equation for the distribution function f(r→,v→,v→˙,v→¨,t) is considered. Since the second Vlasov equation for the distribution function f(r→,v→,t) in a particular case is shown to transform into the Moyal equation for the Wigner function W(r→,p→,t), the authors have elaborated and present a way to expand the Wigner function on the phase space {r→,v→,v→˙,v→¨} and also propose a new Moyal equation, which this function satisfies.

Data-driven reconstruction of stochastic dynamical equations based on statistical moments

Stochastic processes are encountered in many contexts, ranging from generation sizes of bacterial colonies and service times in a queueing system to displacements of Brownian particles and frequency fluctuations in an electrical power grid. If such processes are Markov, then their probability distribution is governed by the Kramers–Moyal (KM) equation, a partial differential equation that involves an infinite number of coefficients, which depend on the state variable. The KM coefficients must be evaluated based on measured time series for a data-driven reconstruction of the governing equations for the stochastic dynamics. We present an accurate method of computing the KM coefficients, which relies on computing the coefficients’ conditional moments based on the statistical moments of the time series. The method’s advantages over state-of-the-art approaches are demonstrated by investigating prototypical one-dimensional stochastic processes with well-known properties.

Stable stadial and interstadial states of the last glacial's climate identified in a combined stable water isotope and dust record from Greenland

Abstract. During the last glacial interval, the Northern Hemisphere climate was punctuated by a series of abrupt changes between two characteristic climate regimes. The existence of stadial (cold) and interstadial (milder) periods is typically attributed to a hypothesised bistability in the glacial North Atlantic climate system, allowing for rapid transitions from the stadial to the interstadial state – the so-called Dansgaard–Oeschger (DO) events – and more gradual yet still fairly abrupt reverse shifts. The physical mechanisms driving these regime transitions remain debated. DO events are characterised by substantial warming over Greenland and a reorganisation of the Northern Hemisphere atmospheric circulation, which are evident from concomitant shifts in the δ18O ratios and dust concentration records from Greenland ice cores. Treating the combined δ18O and dust record obtained by the North Greenland Ice Core Project (NGRIP) as a realisation of a two-dimensional, time-homogeneous, and Markovian stochastic process, we present a reconstruction of its underlying deterministic drift based on the leading-order terms of the Kramers–Moyal equation. The analysis reveals two basins of attraction in the two-dimensional state space that can be identified with the stadial and interstadial regimes. The drift term of the dust exhibits a double-fold bifurcation structure, while – in contrast to prevailing assumptions – the δ18O component of the drift is clearly mono-stable. This suggests that the last glacial's Greenland temperatures should not be regarded as an intrinsically bistable climate variable. Instead, the two-regime nature of the δ18O record is apparently inherited from a coupling to another bistable climate process. In contrast, the bistability evidenced in the dust drift points to the presence of two stable circulation regimes of the last glacial's Northern Hemisphere atmosphere.

Moyal equation—Wigner distribution functions for anharmonic oscillators

We investigate the structure of Wigner distribution functions of energy eigenstates of quartic and sextic anharmonic oscillators. The corresponding Moyal equations are shown to be solvable, revealing new properties of Schrödinger eigenfunctions of these oscillators. In particular, they provide alternative approaches to their determination and corresponding eigenenergies, away from the conventional differential equation method, which usually relies on the not so widely known bi- and tri-confluent Heun functions.

Phase-space studies of backscattering diffraction of defective Schr\xf6dinger cat states

The coherent superposition of two well separated Gaussian wavepackets, with defects caused by their imperfect preparation, is considered within the phase-space approach based on the Wigner distribution function. This generic state is called the defective Schrödinger cat state due to this imperfection which significantly modifies the interference term. Propagation of this state in the phase space is described by the Moyal equation which is solved for the case of a dispersive medium with a Gaussian barrier in the above-barrier reflection regime. Formally, this regime constitutes conditions for backscattering diffraction phenomena. Dynamical quantumness and the degree of localization in the phase space of the considered state as a function of its imperfection are the subject of the performed analysis. The obtained results allow concluding that backscattering communication based on the defective Schrödinger cat states appears to be feasible with existing experimental capabilities.

On the generalization of moyal equation for an arbitrary linear quantization

For an arbitrary linear combination of quantizations, the kernel of the inverse operator is constructed. An equation for the evolution of the Wigner function for an arbitrary linear quantization is derived and it is shown that only for Weyl quantization this equation does not contain a source of quasi-probability. Stationary solutions for the Wigner function of a harmonic oscillator are constructed, depending on the characteristic function of the quantization rule. In the general case of Hermitian linear quantization these solutions are real but not positive. We found the representation of Weyl quantization in the form of the limit of a sequence of linear Hermitian quantizations, such that for each element of this sequence the stationary solution of the Moyal equation is positive.

Generalized Evolution Equation of Wigner Function for an Arbitrary Linear Quantization

The arbitrary linear quantization is considered for definition of Wigner function. The evolution equation of this function is derived with the use of inversion of linear quantization kernel. In general case the Moyal equation depends on the characteristic function of quantization. It is shown that only for Weyl quantization this equation does not consist the source of quasi-probability. The stationary solutions of Moyal equation are constructed for an arbitrary linear quantization of a harmonic oscillator. The case of anharmonic oscillator is presented as a practical example of quantum system model in the so-called post-newtonian approximation. This model exhibits the intersection effect between quantization dependences of Hamiltonian and Wigner function.

Extended Wigner function for the harmonic oscillator in the phase space

New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.

Wigner function of a quantum system with polynomial potential

The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.

Conditions for Quantum and Classical Tomogram-Like Functions to Describe System States and to Retain Normalizations During Time Evolution

It is shown that dynamical equations for quantum tomograms retain the normalizations of their solutions during time evolution only if the solutions satisfy a set of special conditions. These conditions are found explicitly. On the contrary, it is also illustrated that the classical Liouville equation, Moyal equation for Wigner function, and evolution equation for Husimi function retain normalizations of any initially normalized and quickly decaying at infinity functions on the phase space. Necessary and sufficient conditions for optical and symplectic tomogram-like functions to be tomograms of physical states are also discussed.

The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

Abstract Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator.

Semiclassical Moyal dynamics.

We present a semiclassical Moyal dynamics (SMD) method based on the phase space formulation of quantum dynamics. In contrast to the quantized Hamiltonian dynamics approach by Prezhdo et al. using the Heisenberg equation of motion, SMD adopts the Moyal equation of motion to derive the time evolution of expectation values and uses an auxiliary phase space distribution technique to systematically terminate the coupled equations of motion. Thereby, tedious derivation of commutators is essentially not required, and semiclassical dynamics of arbitrary orders can be realized. The appealing simplicity, flexibility, and reliability of SMD are demonstrated in three representative model systems with strong quantum effects.

Nonlinear wave damping due to multi-plasmon resonances

For short wavelengths, it is well known that the linearized Wigner–Moyal equation predicts wave damping due to wave-particle interaction, where the resonant velocity shifted from the phase velocity by a velocity . Here is the reduced Planck constant, k is the wavenumber and m is the electron mass. Going beyond linear theory, we find additional resonances with velocity shifts , giving rise to a new wave-damping mechanism that we term multi-plasmon damping, as it can be seen as the simultaneous absorption (or emission) of multiple plasmon quanta. Naturally this wave damping is not present in classical plasmas. For a temperature well below the Fermi temperature, if the linear (n = 1) resonant velocity is outside the Fermi sphere, the number of linearly resonant particles is exponentially small, while the multi-plasmon resonances can be located in the bulk of the distribution. We derive sets of evolution equations for the case of two-plasmon and three-plasmon resonances for Langmuir waves in the simplest case of a fully degenerate plasma. By solving these equations numerically for a range of wave-numbers we find the corresponding damping rates, and we compare them to results from linear theory to estimate the applicability. Finally, we discuss the effects due to a finite temperature.

On the Stochastic Origin of Quantum Mechanics

The quantum Liouville equation, which describes the phase space dynamics of a quantum system of fermions, is analyzed from stochastic point of view as a particular example of the Kramers–Moyal expansion. Quantum mechanics is extended to relativistic domain by generalizing the Wigner–Moyal equation. Thus, an expression is derived for the relativistic mass in the Wigner quantum phase space presentation. The diffusion with an imaginary diffusion coefficient is discussed. An imaginary stochastic process is proposed as the origin of quantum mechanics.

Quantum kinetic theories in degenerate plasmas

In this review we give an overview of the recent work on quantum kinetic theories of plasmas. We focus, in particular, on the case where the electrons are fully degenerate. For such systems, perturbation methods using the distribution function can be problematic. Instead we present a model that considers the dynamics of the Fermi surface. The advantage of this model is that, even though the value of the distribution function can be greatly perturbed outside the equilibrium Fermi surface, deformation of the Fermi surface is small up to very large amplitudes. Next, we investigate the short-scale dynamics for which the Wigner–Moyal equation replaces the Vlasov equation. In particular, we study wave–particle interaction, and deduce that new types of wave damping can occur due to the simultaneous absorption (or emission) of multiple wave quanta. Finally, we consider exchange effects within a quantum kinetic formalism to find a model that is more accurate than those using exchange potentials from density functional theory. We deduce the exchange corrections to the dispersion relations for Langmuir and ion-acoustic waves. In comparison to results based on exchange potentials deduced from density functional theory we find that the latter models are reasonably accurate for Langmuir waves, but rather inaccurate for ion acoustic waves.

Quantum mechanics emerging from stochastic dynamics of virtual particles

It is shown how quantum mechanics emerges from the stochastic dynamics of force carriers. It is demonstrated that the Moyal equation corresponds to dynamic correlations between the real particle momentum and the virtual particle position, which are not present in classical mechanics. This new concept throws light on the physical meaning of quantum theory, showing that the Planck constant square is a second-second position-momentum cross-cumulant.

Initial-value semiclassical propagators for the Wigner phase space representation: Formulation based on the interpretation of the Moyal equation as a Schrödinger equation.

We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.

Exact cumulant Kramers–Moyal-like expansion

We derive an exact equation, a Cumulant Kramers–Moyal Equation (CKME), quite similar to the Kramers–Moyal Equation (KME), for the probability distribution of a Markovian dynamical system. It can be applied to any well behaved (converging cumulants) continuous time systems, such as Langevin equations or other models. An interesting but significant difference with respect to the KME is that their jump-moments are proportional to cumulants of the dynamical variables, but not proportional to central moments, as is the case for the KME. In fact, they still obey a weaker version of Pawula’s theorem, namely Marcinkiewicz’s theorem. We compare the results derived from the equations herein with the ones obtained by computing via Gaussian and biased, and unbiased, Poisson Langevin dynamics and a Poisson non-Langevin model. We obtain the exact CKME time-evolution equation for the systems, and in several cases, those are distinct from the Fokker–Planck equation or the KME.