One-dimensional Kinetic Equation Research Articles

Remarks on the $H$ Theorem for a non involutive Boltzmann like kinetic model

In this paper, we consider a one-dimensional kinetic equation of Boltzmann type in which the binary collision process is described by the linear transformation v^* = pv + qw , w^* = qv + pw , where (v, w) are the pre-collisional velocities and (v^*, w^*) the post-collisional ones and p ≥ q > 0 are two positive parameters. This kind of model has been extensively studied by Pareschi and Toscani (in J. Stat. Phys. , 124(2–4):747–779, 2006) with respect to the asymptotic behavior of the solutions in a Fourier metric. In the conservative case p^2 + q^2 = 1 , even if the transformation has Jacobian J ≠ 1 and so it is not involutive, we remark that the H Theorem holds true. As a consequence we prove exponential convergence in L^1 of the solution to the stationary state, which is the Maxwellian.

Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation’s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ν α, then the limit is a scale mixture of ν α. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.

Towards physically realizable and hyperbolic moment closures for kinetic theory

The numerical prediction of continuum and non-equilibrium flows by using fully hyperbolic and realizable mathematical descriptions that follow from moment closures of gas kinetic theory is reviewed. A brief review is first given of some of the current capabilities and limitations of moment closures for predicting a range of continuum and non-equilibrium flow. Next, an extended but hyperbolic Gaussian closure for diatomic gases that does not account for heat-transfer effects, as well as a regularized version of this closure that incorporates anisotropic thermal-diffusion effects via the inclusion of higher order terms having an elliptic nature, are both reviewed and applied to a number of canonical flow problems. The numerical results for the Gaussian closures clearly demonstrate the capabilities and potential of moment closures and purely hyperbolic treatments. Following these reviews, a somewhat novel hierarchy of physically realizable and hyperbolic moment closures is considered and described. This alternative hierarchy is based on modifications to the more common maximum-entropy hierarchies, so as to ensure the validity and integrability of the approximate distribution function for all values of the velocity moments that are physically realizable. The predictive capabilities of this new closure hierarchy are then explored by considering numerical solutions of the closures for a one-dimensional kinetic equation with a relaxation-time collision operator and by comparing the closure solutions to discrete numerical solutions of this simplified kinetic equation. The study concludes with a brief summary of the findings and a discussion of the potential of the physically realizable and hyperbolic moment closures for application to fully three-dimensional physics.

A One-Dimensional Energy Diffusion Approach to Multidimensional Dynamical Processes in the Condensed Phase

We propose a generalized one-dimensional energy diffusion approach for describing the dynamics of multidimensional dynamical processes in the condensed phase. On the basis of a formalism originally due to Zwanzig, we obtain a one-dimensional kinetic equation for a properly selected relevant dynamical quantity and derive new analytical results for the dynamics of a multidimensional electron-transfer process, nonequilibrium solvation, and diffusive escape from a potential well. The calculated results for electron-transfer reactions in solvent-separated and contact ion pair systems are found to be in good agreement with the experimental results. We are able to explain the rate of the electron-transfer reaction using much smaller and reasonable values of the solvent reorganization energy in contrast to earlier works that had to use a much larger value. The proposed theory is not only conceptually simpler than the conventional approaches but is also free from many of their limitations. More importantly, it provides a single theoretical framework for describing a wide class of dynamical phenomena.

Kinetic modeling of a sheath layer in a magnetized collisionless plasma

A sheath layer in a magnetized collisionless plasma is analyzed by the one-dimensional kinetic equation. The plasma is bounded by an absorbing wall and a plasma source with a shifted Maxwellian velocity distribution function that is characterized by a temperature, a drifting velocity parallel to the magnetic field, and a cutoff velocity. The magnetic field is assumed to be strong enough so that the ion Larmor radius is comparable to the Debye length. In order to include the polarization effect of ions due to a strong electric field, equations describing the potential profile are derived from the gyrokinetic Vlasov equation in a frame moving with the E×B drift. A new algorithm for evaluating the loss of particles in gyration at the wall is introduced. The condition of the stable sheath formation for a magnetized plasma is discussed. The dependence of the electric field at the wall on the angle and the strength of the magnetic field is examined and compared with the results of full-kinetic particle-in-cell simulation. The effect of the polarization and the loss of gyration particles on the wall electric field are also discussed.

Limiting transformation of the equilibrium spectrum of electrons and photons in strong electric fields of thunderclouds

The upper limit of transformation of the equilibrium electron-photon spectrum (E > 10 MeV) is obtained for the case of strong fields by solving the one-dimensional kinetic equation. Comparison with the experimental data from the Carpet array at the Baksan Neutrino Observatory (Institute for Nuclear Research, Russian Academy of Sciences) shows that variations in the intensity of the electron-photon component are observed during thunderstorms, which cannot be explained only by transformation of the spectrum in strong thunderstorm fields. One has to hypothesize some process of particle production in the energy range below ∼30 MeV.

Nonlinear propagation of broadband intense electromagnetic waves in an electron-positron plasma

A kinetic equation describing the nonlinear evolution of intense electromagnetic pulses in electron-positron (e-p) plasmas is presented. The modulational instability is analyzed for a relativistically intense partially coherent pulse, and it is found that the modulational instability is inhibited by the spectral pulse broadening. A numerical study for the one-dimensional kinetic photon equation is presented. Computer simulations reveal a Fermi-Pasta-Ulam-type recurrence phenomenon for localized broadband pulses. The results should be of importance in understanding the nonlinear propagation of broadband intense electromagnetic pulses in e-p plasmas in laser-plasma systems as well as in astrophysical plasma settings.

Quasi-Steady-State particle distributions for an equation of the Landau-Fokker-Planck type with sources

The formation of nonequilibrium particle distributions for the power-law interaction potentials V = α/rβ, where 1 ≤ β < 4, is considered. The analytical and numerical studies are based on a one-dimensional (in the velocity space) kinetic equation of the Landau-Fokker-Planck type with energy (particle) sources (sinks) localized in the high-energy range. Fully conservative finite-difference schemes are used for numerical modeling. The resulting asymptotic estimates are confirmed by numerical computations. The results can be used to predict the behavior of intrinsic and extrinsic semiconductors influenced by particle fluxes or electromagnetic radiation.

Quasi steady-state distributions for particles with power-law interaction potentials

The formation of a non-equilibrium steady-state distribution function of particles with the power law interaction potentials , is studied numerically. Consideration is based on the one-dimensional nonlinear kinetic equation of a Landau–Fokker–Planck type in the presence of particle (energy) sources. Non-equilibrium quasi steady-state local distributions exist inside the momentum interval between the energy (particle) source and the bulk (or sink) of the particle distribution and have the form of gradually decreasing functions. Numerical calculations based on the completely conservative difference schemes are accompanied by the analytical consideration and comparison with the experimental results is given. Obtained results can be useful in connection with the development of high-power particle and energy sources and for the prediction of the semiconductors behavior with an intrinsic or extrinsic conductivity under the action of particle fluxes or electromagnetic radiation.

Effects of the velocity dependence of the collision frequency on the Dicke line narrowing

The effect of the velocity (v) dependence of the transport collision frequency νtrv on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance (M) and buffer (Mb) particles, β = Mb/M: it is at a minimum for β ≪ 1 and reaches a maximum for β ≳ 3. A power-law particle interaction potential, U(r) ∝ r−n, is used as an example to show that, compared to νtrv(v) = const (n = 4), the line narrows if νtrv(v) decreases with increasing v (n 4). At β ≳ 3, the line width can increase [compared to νtrv(v) = const] by 5 and 12% for the potentials with n = 6 and n ≳ 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I(Ω) has been found to be weakly sensitive to νtrv(v) at some detuning Ωc of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.

On the collisional transfer of nonequilibrium in the velocity distribution of resonant particles in a laser radiation field

The collisional transfer of nonequilibrium in the velocity distribution of resonant particles in a laser radiation field is investigated theoretically. It is shown numerically that the transfer effect is weak. This makes it possible to use simpler approximate one-dimensional quantum kinetic equations instead of three-dimensional equations to solve spectroscopy and light-induced gas kinetics problems, where it is important to take account of the velocity dependence of the collision frequency. It is shown for anomalous light-induced drift, calculations of which are most sensitive to neglecting the transfer effect, that in a wide range of spectroscopy and light-induced gas kinetics problems the transfer of nonequilibrium can be neglected without risking the loss of important fine details of the phenomena being described.

Exact solutions of the one-dimensional Russo-Smereka kinetic equation

We obtain new classes of invariant solutions of the integrodifferential equations describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid. For all the solutions obtained, trajectories of particle motion in phase space are calculated. The stability of some flows is studied in a linear approximation. In several cases, the construction of solutions reduces to an integrodifferential equation of the second kind, which can be solved by the iteration method.

Two-Dimensional Force-Free Relaxation and Superdiffusion Governed by Fractional Kinetic Equations

Starting from the integral Chapman-Kolmogorov equation and the Langevin equations with a multivariate Levy noise, we consider possible generalization of a one-dimensional fractional kinetic equations. It is assumed that the Levy noise has independent components. As the result, we get fractional Fokker-Planck equation for the probability density function in a phase space and fractional Einstein-Smoluchowski equation for the probability density function in a real space. We consider force-free superdiffusion and relaxation described by fractional equations and carry out numerical simulation of these processes by solving the corresponding Langevin equations. Analytical and numerical results agree quantitatively.

One-dimensional kinetic equations for non-Markovian processes and statistical analysis problems of systems driven by the asymmetrical binary Markovian process

The problem of the number of terms in the one-dimensional kinetic equation for non-Markovian processes is discussed. The comparison of solutions obtained by different kinds of kinetic equations is given. The derivation of the kinetic coefficients, when the system is driven by the asymmetrical binary Markovian process, is presented. Some examples are considered.

Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis

The interaction of a plane monochromatic traveling wave with two-level particles suffering collisions with buffer-gas particles is considered. Collision rates are assumed to be velocity dependent. The collision integral is obtained on the basis of the strong-collision model, generalized to the case of velocity-dependent collision rates (the so-called ``kangaroo'' model). We obtained the exact analytical solution of the problem for arbitrary intensity of radiation, arbitrary ratio of homogeneous and Doppler widths of the absorption line, and arbitrary mass ratio between absorbing- and buffer-gas particles. The obtained analytical solutions of the quantum kinetic equations allowed us to analyze the spectral shape of the strong-field absorption line as well as the probe-field absorption line (the nonlinear part of the work done by the probe field) and the frequency dependence of the light-induced drift (LID) velocity. A comprehensive comparative analysis for the three- and one-dimensional versions of the model is given. On the basis of this analysis, we reach the conclusion that the one-dimensional quantum kinetic equation has quite a wide range of application. We also reveal the conditions for the strongest manifestation of the velocity dependence of the collision rates, which affects most strongly the anomalous LID.

Kinetic theory of nonisothermal binary nucleation: the stage following thermal relaxation

A kinetic theory is constructed for a nonisothermal binary nucleation at the stage following the thermal relaxation of nuclei. The three-dimensional kinetic equation to be solved reaches beyond the framework of the Fokker–Planck approximation even if one of two components has a large value of condensation heat. It is shown that, by successively applying the method of Enskog–Chapman and the method of complete separation of variables to that kinetic equation, one can reduce the problem of constructing the three-dimensional kinetic theory to the well-investigated problem of solving an one-dimensional kinetic equation of first-order phase transition, in the nonstationary case as well as in the stationary one. For the steady state, the main characteristics of nucleation, including the nucleation rate, are found. Theoretical results are numerically evaluated for the nucleation in ethanol–hexanol system and compared with predictions of classical (isothermal) theory and experimental data.

Simulation of heavy ion energy distribution function in the cathode sheath of noble gas mixture glow discharge

A one-dimensional model kinetic equation for the energy distribution function of heavy ions in the cathode sheath of gas mixture glow discharges is suggested, taking into account charge exchange collisions of heavy ions with heavy atoms and momentum transfer collisions of heavy ions with light atoms. An approximate analytical solution of the kinetic equation is found and a procedure of its numerical solution is developed. It is shown that heavy ion–light atom momentum transfer collisions can substantially influence the heavy ion energy distribution function and the cathode sputtering rate in gas mixtures.

Classical model of bosons and fermions.

In a previous work [Phys. Rev. E 48, 4263 (1993)] we have derived a nonlinear one-dimensional kinetic equation for the distribution function of particles obeying an exclusion principle. In the present work, on the same grounds, we extend this kinetics to D-dimensional continuous or discrete space, in order to study the distribution function of particles obeying a generalized exclusion-inclusion Pauli principle (EIP). This exclusion or inclusion principle is introduced into the classical transition rates by means of an inhibition or an enhancement factor, which contains a parameter \ensuremath{\kappa}, whose values range between -1 and +1 and can balance the effect of the full or partial validity of EIP. After deriving the kinetic equation we obtain a general expression of the stationary distribution function, depending on the value we give to the parameter \ensuremath{\kappa}. When we limit ourselves to Brownian particles, we derive exactly for \ensuremath{\kappa}=-1 the Fermi-Dirac (FD) distribution, for \ensuremath{\kappa}=0 the Maxwell-Boltzmann distribution, and for \ensuremath{\kappa}=1 the Bose-Einstein (BE) distribution. When \ensuremath{\kappa} assumes an intermediate value, except zero, between the extreme values -1 and +1, we obtain statistical distributions different from the FD and BE ones. We attribute to the parameter \ensuremath{\kappa} the meaning of the degree of indistinguishability of identical particles, the degree of antisymmetrization, or the symmetrization of the wave function of the particle system.

Moment series expansions for a model kinetic equation

A steady-state, one-dimensional model kinetic equation with a simple external force and a Lorentz collision term is solved by the approximation methods of moment expansions, by a neoclassical type of asymptotic expansion, and by other simple expansions. The very high-order moment expansions are assumed to give the correct solution, and a comparison is given with low-order moment expansions, the neoclassical expansion, and a small external force expansion. All the calculations and expansions are given for the cases in which the three fundamental dimensional physical parameters that characterize the problem are small. These parameters are the collisionality, the mean velocity to which the system is driven, and the magnitude of the spatially varying external force. Each of the expansions has regions of applicability and utility. Even in the very low collisionality case, low-order moment expansions give the correct order of magnitude of the solution, the mean flow velocity within 30%, the temperature quite accurately, the average value of the heat flow poorly, but give the spatially varying part of the heat flow very well. For the neoclassical expansion to be accurate, not only must the collisionality be small, but it also must be sufficiently smaller than other small physical parameters in the system. For parameter values somewhat less than one moment expansions of order five are quite accurate. For large values of the collisionality, the Hilbert–Chapman–Enskog expansion is recovered.

On electric breakdown in plane dielectrics

The relation between the breakdown current and the applied electric field strength has been determined and the critical breakdown current calculated on the basis of the exact solution of the one-dimensional kinetic Boltzmann equation for the electron distribution function in coordinates and momenta. It is demonstrated that the field strength E ∗ is fully controlled by the probability of electron scattering on charged complexes and dipole momenta of atoms.