Wigner-Moyal Equation Research Articles

Wave kinetics of drift-wave turbulence and zonal flows beyond the ray approximation.

Inhomogeneous drift-wave turbulence can be modeled as an effective plasma where drift waves act as quantumlike particles and the zonal-flow velocity serves as a collective field through which they interact. This effective plasma can be described by a Wigner-Moyal equation (WME), which generalizes the quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves the wavelength scale. Unlike waves governed by manifestly quantumlike equations, whose WMEs can be borrowed from quantum mechanics and are commonly known, drift waves have Hamiltonians very different from those of conventional quantum particles. This causes unusual phase-space dynamics that is typically not captured by the WKE. We demonstrate how to correctly model this dynamics with the WME instead. Specifically, we report full-wave phase-space simulations of the zonal-flow formation (zonostrophic instability), deterioration (tertiary instability), and the so-called predator-prey oscillations. We also show how the WME facilitates analysis of these phenomena, namely, (i) we show that full-wave effects critically affect the zonostrophic instability, particularly its nonlinear stage and saturation; (ii) we derive the tertiary-instability growth rate; and (iii) we demonstrate that, with full-wave effects retained, the predator-prey oscillations do not require zonal-flow collisional damping, contrary to previous studies. We also show how the famous Rayleigh-Kuo criterion, which has been missing in wave-kinetic theories of drift-wave turbulence, emerges from the WME.

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.

Effects of group velocity and multiplasmon resonances on the modulation of Langmuir waves in a degenerate plasma.

We study the nonlinear wave modulation of Langmuir waves (LWs) in a fully degenerate plasma. Using the Wigner-Moyal equation coupled to the Poisson equation and the multiple scale expansion technique, a modified nonlocal nonlinear Schrödinger (NLS) equation is derived which governs the evolution of LW envelopes in degenerate plasmas. The nonlocal nonlinearity in the NLS equation appears due to the group velocity and multiplasmon resonances, i.e., resonances induced by the simultaneous particle absorption of multiple wave quanta. We focus on the regime where the resonant velocity of electrons is larger than the Fermi velocity and thereby the linear Landau damping is forbidden. As a result, the nonlinear wave-particle resonances due to the group velocity and multiplasmon processes are the dominant mechanisms for wave-particle interaction. It is found that in contrast to classical or semiclassical plasmas, the group velocity resonance does not necessarily give rise the wave damping in the strong quantum regime where ℏk∼mv_{F} with ℏ denoting the reduced Planck's constant, m the electron mass, and v_{F} the Fermi velocity; however, the three-plasmon process plays a dominant role in the nonlinear Landau damping of wave envelopes. In this regime, the decay rate of the wave amplitude is also found to be higher compared to that in the modest quantum regime where the multiplasmon effects are forbidden.

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.

Nonlinear Landau damping of wave envelopes in a quantum plasma

The nonlinear theory of Landau damping of electrostatic wave envelopes (WEs) is revisited in a quantum electron-positron pair plasma. Starting from a Wigner-Moyal equation coupled to the Poisson equation and applying the multiple scale technique, we derive a nonlinear Schrödinger (NLS) equation which governs the evolution of electrostatic WEs. It is shown that the coefficients of the NLS equation, including the nonlocal nonlinear term, which appears due to the resonant particles having a group velocity of the WEs, are significantly modified by the particle dispersion. The effects of the quantum parameter H (the ratio of the plasmon energy to the thermal energy densities), associated with the particle dispersion, are examined on the Landau damping rate of carrier waves, as well as on the modulational instability of WEs. It is found that the Landau damping rate and the decay rate of the solitary wave amplitude are greatly reduced compared to their classical values (H = 0).

Nonlinear evolution of a single coherent mode in a turbulent plasma

We consider the excitation, nonlinear evolution and saturation of a single coherent mode in a turbulent plasma background. We adopt a generic wave-kinetic description of plasma turbulence, based on a Wigner–Moyal equation, which stays valid well beyond the geometric optics approximation. We illustrate our model by considering the case of a geodesic acoustic mode in a tokamak plasma, in the presence of a broad spectrum of drift wave turbulence.

Wigner function and the probability representation of quantum states

The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.

Two-frequency radiative transfer: Maxwell equations in random dielectrics

The paper addresses the space-frequency correlations of electromagnetic waves in general random, bianisotropic media whose constitutive tensors are complex Hermitian matrices. The two-frequency Wigner distribution (2f-WD) for polarized waves is introduced to describe the space-frequency correlations, and the closed form Wigner-Moyal equation is derived from the Maxwell equations. Two-frequency radiative transfer (2f-RT) equations are then derived from the Wigner-Moyal equation by using the multiple-scale expansion. For the simplest isotropic medium, the result coincides with Chandrasekhar's transfer equation. In birefringent media, the 2f-RT equations take the scalar form due to the absence of depolarization. A number of birefringent media such as chiral, uniaxial, and gyrotropic media are examined. For the unpolarized wave in an isotropic medium the 2f-RT equations reduces to the 2f-RT equation previously derived in part I of this research [J. Opt. Soc. Am. A24, 2248 (2007)]. A similar Fokker-Planck-type equation is derived from the scalar 2f-RT equation for the birefringent media.

Mutual coherence of polarized light in disordered media: two-frequency method extended

The paper addresses the two-point correlations of electromagnetic waves in general random, bi-anisotropic media whose constitutive tensors are complex Hermitian, positive- or negative-definite matrices. A simplified version of the two-frequency Wigner distribution (2f-WD) for polarized waves is introduced and the closed form Wigner–Moyal equation is derived from the Maxwell equations. In the weak-disorder regime with an arbitrarily varying background the two-frequency radiative transfer (2f-RT) equations for the associated 2 × 2 coherence matrices are derived from the Wigner–Moyal equation by using the multiple-scale expansion. In birefringent media, the coherence matrix becomes a scalar and the 2f-RT equations take the scalar form due to the absence of depolarization. A paraxial approximation is developed for spatially anisotropic media. Examples of isotropic, chiral, uniaxial and gyrotropic media are discussed.

Influence of nonlinear dispersion on modulation instability of coherent and partially coherent ultrashort pulses in metamaterials

The combination of dispersive magnetic permeability with nonlinear polarization leads to a series of nonlinear dispersion terms in the propagation equations for ultrashort pulses in metamaterials. Here we present an investigation of modulation instability (MI) of both coherent and partially coherent ultrashort pulses in metamaterials to identify the role of nonlinear dispersion in pulse propagation. The Wigner–Moyal equation for partially coherent ultrashort pulses and the nonlinear dispersion relation for MI in metamaterials are derived. Combining the standard MI theory with the unique properties of the metamaterial, the influence of the controllable first-order nonlinear dispersion, namely self-steepening, and the second-order nonlinear dispersion on both coherent and partially coherent MI, in both negative-index and positive-index regions of the metamaterial for all physically possible cases is analyzed in detail. For the first time to our knowledge, we demonstrate that the role of the second-order nonlinear dispersion in MI is equivalent to that of group-velocity dispersion (GVD) to some extent, and thus due to the role of the second-order nonlinear dispersion, MI may appear in the otherwise impossible cases, such as in the normal GVD regime.

Local and non-local force analysis for Wigner function barrier scattering

The Derivative Propagation Method (DPM) can be used to classify the components of the total force acting upon a trajectory into local and non-local components, without the use of finite differences or function fittings. In this study, the DPM was used to generate and analyze classical local and quantum non-local forces for evolving trajectories given by a phase space Wigner distribution function governed by the Wigner–Moyal equation for Eckart barrier scattering. Solutions were verified with finite difference methods.

A kinetic approach to Bose-Einstein condensates: Self-phase modulation and Bogoliubov oscillations

A kinetic approach to the Bose-Einstein condensates (BECs) is proposed, based on the Wigner-Moyal equation (WME). In the quasi-classical limit, the WME reduces to the particle number conservation equation. Two examples of application are: i) a self-phase modulation of a BE condensate beam where we show that part of the beam is decelerated and eventually stops as a result of the gradient of the effective self-potential; ii) the derivation of a kinetic dispersion relation for sound waves in the BECs, including a collisionless Landau damping.

White-Noise and Geometrical OpticsLimits of Wigner–Moyal Equation for Beam Waves in Turbulent Media II: Two-Frequency Formulation

We introduce two-frequency Wigner distribution in the setting of parabolic approximation to study the scaling limits of the wave propagation in a turbulent medium at two different frequencies. We show that the two-frequency Wigner distribution satisfies a closed-form equation (the two-frequency Wigner–Moyal equation). In the white-noise limit we show the convergence of weak solutions of the two-frequency Wigner–Moyal equation to a Markovian model and thus prove rigorously the Markovian approximation with power-spectral densities widely used in the physics literature. We also prove the convergence of the simultaneous geometrical optics limit whose mean field equation has a simple, universal form and is exactly solvable.

The classical and commutative limits of noncommutative quantum mechanics: a superstar * Wigner-Moyal equation

We are interested in the similarities and differences between the quantum-classical (Q-C) and the noncommutative-commutative (NC-Com) correspondences. As one useful platform to address this issue we derive the superstar Wigner-Moyal equation for noncommutative quantum mechanics (NCQM). A superstar *-product combines the usual phase space * star and the noncommutative * star-product. Having dealt with subtleties of ordering present in this problem we show that the Weyl correspondence of the NC Hamiltonian has the same form as the original Hamiltonian, but with a non-commutativity parameter theta-dependent, momentum-dependent shift in the coordinates. Using it to examine the classical and the commutative limits, we find that there exist qualitative differences between these two limits. Specifically, if <FONT FACE=Symbol>q ¹</FONT> 0 there is no classical limit. Classical limit exists only if <FONT FACE=Symbol>q ®</FONT> 0 at least as fast as h ->0, but this limit does not yield Newtonian mechanics, unless the limit of theta/h vanishes as <FONT FACE=Symbol>q ®</FONT> 0. For another angle towards this issue we formulate the NC version of the continuity equation both from an explicit expansion in orders of theta and from a Noether's theorem conserved current argument. We also examine the Ehrenfest theorem in the NCQM context.

Self-Averaging Scaling Limits for Random Parabolic Waves

We consider several types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the standard radiative transfer limit, the geometrical-optics limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by one of the 6 different types of transport (Boltzmann or Fokker-Planck) equations depending on the specific scaling involved. We discuss the connection to the statistical stability of time-reversal procedure and the decoherence effect in quantum mechanics.

White-Noise and Geometrical Optics Limits of Wigner-Moyal Equation for Wave Beams in Turbulent Media

Starting with the Wigner distribution formulation for beam wave propagation in Hölder continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the Itô type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its n-point correlation functions are governed by closed form equations.

Influence of higher-order dispersion on modulational instability and pulse broadening of partially incoherent light.

The Wigner-Moyal equation for the Wigner distribution of a partially incoherent optical wave field propagating in dispersive and nonlinear media has been generalized to include the effects of both arbitrary order of dispersion and arbitrary nonlinearity. The theory predicts partial incoherence to enhance the pulse broadening during linear wave pulse propagation. Furthermore, an application of the theory to the modulational instability of constant amplitude waves shows how higher-order dispersion affects the instability growth rate.

Nonlinear Dynamics of Partially Incoherent Optical Waves Based on the Wigner Transform Method

A novel statistical approach based on the Wigner transform method is proposed for the description of partially incoherent optical wave dynamics in nonlinear media. The Wigner–Moyal equation is derived for the Wigner distribution of the optical wave field governed by the nonlinear Schrödinger equation with an arbitrary nonlinearity. An application to incoherent light propagation in dispersive Kerr media shows that random phase fluctuations of a plane wave solution lead to a linear Landau-like damping effect, which can stabilize the nonlinear modulational instability. A similar effect is shown to occur in the case of the two-stream instability of two partially incoherent optical waves interacting with each other through cross-phase modulation initiated by the nonlinearity. In the limit of the geometrical optics approximation, it is shown that 1D and 2D self-trapped, stationary and incoherent wave pulse structures may exist for a wide class of nonlinear media. Furthermore, time-dependent self-similar 1D and 2D structures have been found in the case of the Kerr nonlinearity.

Analog of the wigner-moyal equation for the electromagnetic field

The evolution equation for the Wigner distribution of the classical electromagnetic field is derived for a nonstationary and inhomogeneous optical medium, which is formally similar to the Wigner-Moyal equation for a quantum system. The geometric optics approximation is discussed in detail, and the conservation equation for the number of photons is justified. The influence of dispersion is also considered.