Transverse electrical conductivity of a quantum collisional plasma in the Mermin approach

We derive formulas for the transverse electrical conductivity and the permittivity in a quantum collisional plasma using the kinetic equation for the density matrix in the relaxation approximation in the momentum space. We show that the derived formula becomes the classical formula when the Planck constant tends to zero and that when the electron collision rate tends to zero (i.e., the plasma becomes collisionless), the derived formulas become the previously obtained Lindhard formulas. We also show that when the wave number tends to zero, the quantum conductivity becomes classical. We compare the obtained conductivity with the conductivity obtained by Lindhard and with the classical conductivity     

Generation of a longitudinal current by a transverse electromagnetic field in collisional degenerate plasma

From the Vlasov–Boltzmann kinetic equation for a collisional degenerate plasma, the electron distribution function is constructed in the quadratic approximation in the electric field strength. A formula for calculating the electric current is derived. It is shown that nonlinearity leads to the rise of a longitudinal electric current directed along the wave vector. The longitudinal current is orthogonal to the known transverse classical current obtained in the linear analysis. When the collision frequency tends to zero, all results obtained for a collisional plasma pass into the corresponding results for a collisionless plasma. The case of small wavenumbers is considered. It is shown that, when the collision frequency tends to zero, the expression for the current passes into the corresponding expression for the current in a collisionless plasma. Graphic analysis of the real and imaginary parts of the current density is performed. The dependence of the electromagnetic field oscillation frequency and electron–plasma-particle collision frequency on the wavenumber is studied.     

Longitudinal electric current generated by a transverse electromagnetic field in a collisional plasma

We consider a collisional plasma with an arbitrary degree of degeneration of the electron gas. The plasma is located in an external electromagnetic field. We calculate the electric current generated in the plasma by the electromagnetic field. We show that the electric current has two nonzero components. One component is a transverse current, obtained by a linear analysis. The second component is a longitudinal current directed along the wave vector and orthogonal to the transverse current. We consider the case of small wave numbers. As the collision rate tends to zero, all the derived formulas pass into formulas for a collisionless plasma. We perform a graphical investigation of the dimensionless current density depending on the wave number, the oscillation frequency of the electromagnetic field, and the rate of electron collisions with plasma particles.     

Longitudinal permittivity of a quantum degenerate collisional plasma

We find the permittivity of a degenerate electron gas for a collisional plasma. We use the Wigner-Vlasov-Boltzmann kinetic equation with the collision integral in the relaxation form in the coordinate space. We study the Kohn permittivity singularities and reveal their spreading in the collisionless plasma.     

Decay in time for a one‐dimensional two‐component plasma

The motion of a collisionless plasma is described by the Vlasov–Poisson (VP) system, or in the presence of large velocities, the relativistic VP system. Both systems are considered in one space and one momentum dimension, with two species of oppositely charged particles. A new identity is derived for both systems and is used to study the behavior of solutions for large times. Copyright © 2008 John Wiley & Sons, Ltd.     

Solitons of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions: Hirota bilinear method

We use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.     

Asymptotic properties of the solution of a delay system

We consider a differential-difference system with constant delay of special form. We show that the derivative of the solution of the system tends to zero by the exponential law. This fact is used to construct a numerical solution for large times. We give an example of numerical simulation for a two-dimensional system.     

Burgers turbulence initialized by a regenerative impulse

In this article we look at a one-dimensional infinitesimal particle system governed by the completely inelastic collision rule. Considering uniformly spread mass, we feed the system with initial velocities, so that when time evolves the corresponding velocity field fulfils the inviscid Burgers equation. More precisely, we suppose here that the initial velocities are zero, except for particles located on a stationary regenerative set for which the velocity is some given constant number. We give results of a large deviation type. First, we estimate the probability that a typical particle is located at time 1 at distance at least D from its initial position, when D tends to infinity. Its behaviour is related to the left tail of the gap measure of the regenerative set. We also show the same asymptotics for the tail of the shock interval length distribution. Second, we analyse the event that a given particle stands still at time T as T tends to infinity. The data to which we relate its behaviour are the right tail of the gap measure of the regenerative set. We conclude with some results on the shock structure.     

Estimation of a change-point in the mean function of functional data

The paper develops a comprehensive asymptotic theory for the estimation of a change-point in the mean function of functional observations. We consider both the case of a constant change size, and the case of a change whose size approaches zero, as the sample size tends to infinity. We show how the limit distribution of a suitably defined change-point estimator depends on the size and location of the change. The theoretical insights are confirmed by a simulation study which illustrates the behavior of the estimator in finite samples.     

Relaxation limit and initial layer to hydrodynamic models for semiconductors

We study a relaxation limit of a solution to the initial-boundary value problem for a hydrodynamic model to a drift-diffusion model over a one-dimensional bounded domain. It is shown that the solution for the hydrodynamic model converges to that for the drift-diffusion model globally in time as a physical parameter, called a relaxation time, tends to zero. It is also shown that the solutions to the both models converge to the corresponding stationary solutions as time tends to infinity, respectively. Here, the initial data of electron density for the hydrodynamic model can be taken arbitrarily large in the suitable Sobolev space provided that the relaxation time is sufficiently small because the drift-diffusion model is a coupled system of a uniformly parabolic equation and the Poisson equation. Since the initial data for the hydrodynamic model is not necessarily in “momentum equilibrium”, an initial layer should occur. However, it is shown that the layer decays exponentially fast as a time variable tends to infinity and/or the relaxation time tends to zero. These results are proven by the decay estimates of solutions, which are derived through energy methods.     

Projective line over the finite quotient ring GF(2)[x]/〈x 3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram

In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres “magic” square and the Mermin pentagram. The former is a 3×3 array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2) ⊗ GF(2). Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column’s observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over GF(2)[x]/〈x³ ™ x〉. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over GF(2), their three “Jacobson” counterparts, and the four points whose both coordinates are zero divisors. The other con.guration features the neighborhood of the point (1, 0) (or, equivalently, that of (0, 1)). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 219–227, May, 2007.     

Quasi-exchange Formulas for Wavelet Transforms of Distributions

We investigate the wavelet transforms of tempered distributions in a way that closely links their Fourier transforms and wavelet transforms. Two exchange formulas of the convolution and the multiplication of wavelet transforms of tempered distributions are established. We call these formulas the quasi-exchange formulas for wavelet transforms of distributions, because of the resemblance between these formulas and the well-known exchange formula for Fourier transforms.     

Optimal time-decay rates of the Boltzmann equation

We consider the optimal time-convergence rates of the global solution to the Cauchy problem for the Boltzmann equation in R3.We show that the global solution tends to the global Maxwellian at the optimal time-decay rate(1+t)-3/4,where the macroscopic density,momentum and energy decay at the optimal rate(1+t)-3/4 and the microscopic part decays at the optimal rate(1+t)-5/4.We also show that the solution tends to the Maxwellian at the optimal time-decay rate(1+t).5/4 in the case of the macroscopic part of the initial data is zero,where the macroscopic density,momentum and energy decay at the optimal rate(1+t)-5/4 and the microscopic part decays at the optimal rate(1+t)-7/4.These convergence rates are shown to be optimal for the Boltzmann equation.     

Interactions of nonlinear ion-acoustic waves in a collisionless plasma

In the present work, based on a one-dimensional model, the interaction of two solitary waves propagating in opposite directions in a collisionless plasma is investigated by use of the extended Poincaré–Lighthill–Kuo (PLK) method. It is shown that bi-directional solitary waves are propagated and the head-on collision of these two solitons occur. The phase shifts and the trajectories of these two solitons after the collision are obtained.     

Dynamics of a spatially distributed logistic equation with small diffusion and small delay

For the spatially distributed Hutchinson equation with transport and small diffusion constant, we show that the loss of stability of the equilibrium can occur even for asymptotically small values of the delay coefficient. Here infinitely many roots of the characteristic quasipolynomial tend to the imaginary axis as the small parameter, the diffusion constant, tends to zero. Thus, the critical (in the problem on the equilibrium stability) case of infinite dimension is realized. We construct special quasinormal forms, namely, nonlinear parabolic systems and families of degenerate parabolic systems whose nonlocal dynamics describes the behavior of solutions of the original equation in a small neighborhood of the equilibrium. These quasinormal forms can have a rather complicated dynamics; moreover, the onset and disappearance of steady-state modes as the small parameter tends to zero is a typical phenomenon. Therefore, the local dynamics of the Hutchinson equation with and without transport are very distinct.     

Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space

In this paper, we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas. This system takes the form of the bipolar Euler-Poisson model with electric field and frictional damping added to the momentum equations. In the framework of the Besov space theory, we establish the global existence of smooth solutions for Cauchy problems when the initial data are sufficiently close to the constant equilibrium. Next, based on the special structure of the nonlinear system, we also show the uniform estimate of solutions with respect to the relaxation time by the high- and low-frequency decomposition methods. Finally we discuss the relaxation-time limit by compact arguments. That is, it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift-diffusion model, as the relaxation time tends to zero.     

Polarization of the Electron–Positron Vacuum by a Strong Magnetic Field in the Theory with a Fundamental Mass

In the framework of the theory with a fundamental mass in the one-loop approximation, we evaluate the exact Lagrange function of the strong constant magnetic field, replacing the Heisenberg–Euler Lagrangian in the traditional QED. We establish that the derived generalization of the Lagrange function is real for arbitrary values of the magnetic field. In the weak field, the evaluated Lagrangian coincides with the known Heisenberg–Euler formula. In extremely strong fields, the field dependence of the Lagrangian completely disappears; in this range, the Lagrangian tends to the limit value determined by the ratio of the fundamental and lepton masses.     

On global symmetric solutions to the relativistic Vlasov–Poisson equation in three space dimensions

A collisionless plasma is modelled by the Vlasov–Maxwell system. In the presence of very large velocities, relativistic corrections are meaningful. When magnetic effects are ignored this formally becomes the relativistic Vlasov–Poisson equation. The initial datum for the phase space density ƒ₀(x, v) is assumed to be sufficiently smooth, non‐negative and cylindrically symmetric. If the (two‐dimensional) angular momentum is bounded away from zero on the support of ƒ₀(x, v), it is shown that a smooth solution to the Cauchy problem exists for all times. Copyright © 2001 John Wiley & Sons, Ltd.     

Stabilization of linear systems by rotation

We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.     

Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system

We investigate the zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system.We justify this singular limit rigorously in the framework of smooth solutions and obtain the nonisentropic compressible magnetohydrodynamic equations as the dielectric constant tends to zero.