Narrow Results

We present an alternative and simple method for solving kinetic equations which is based on a spline expansion of the one-body phase space occupation. The usefulness of the method is illustrated by studying the dynamic evolution of a hot and compressed infinite slab of nuclear matter, in the framework of the Vlasov approach.

The problem of constructing self-consistent stationary particle distributions in four-dimensional phase space is considered for an azimuthally symmetric charged particle beam in a longitudinal magnetic field. In the general case of a longitudinally nonuniform beam, it is assumed that the magnetic field and the radius of the beam cross-section can slowly vary in the axial direction. The simplest case of a longitudinally uniform beam is studied in more detail. The approach applied here is to analyze the particle density in the space of integrals of motion. The relations between this density, the phase density, and the density in the configuration space are obtained. The set of admissible values of integrals of motion for a radially confined beam is examined. The construction of new self-consistent distributions consists in the specifying of some function defined on this set. Wide classes of new distributions are found. In particular cases, some of these distributions are identical to those known before, for example, the Kapchinsky-Vladimirsky distribution.

The Wigner function is shown related to the quantum dielectric function derived from the quantum Vlasov equation (QVE), with and without a magnetic field, using a standard method in plasma physics with linear perturbations and a self-consistent mean field interaction via Poisson's equation. A finite-limit-of-integration Wigner function, with oscillatory behavior and negative values for free particles, is proposed. In the classical regimes, where the problem size is huge compared to the particle wavelength, these limits go to infinity, and for free particles, the Wigner function becomes a positive delta function as expected. For the harmonic oscillator potential, there is no distinction between finite and infinite limits of integration when these are larger than the eigenfunction localization length.

This paper deals with the design of a nonequilibrium statistical mechanics theory developed to model large systems of interacting individuals. Interactions being ruled, not only by laws of classical mechanics, but also by some intelligent or organized behaviour. The paper technically shows, also by reasoning on specific examples, how the theory can be applied to model large complex systems in natural and applied sciences.

We prove that, for a smooth two-body potential, the quantum mean-field approximation to the nonlinear Schrödinger equation of the Hartree type is stable at the classical limit h → 0, yielding the classical Vlasov equation.

We consider a Hamiltonian system given by a charged particle under the action of a constant electric field and interacting with a medium, which is described as a Vlasov fluid. We assume that the action of the charged particle on the fluid is negligible and that the latter has one-dimensional symmetry. We prove that if the singularity of the particle/medium interaction is integrable and the electric field intensity is large enough, then the particle escapes to infinity with a quasi-uniformly accelerated motion. A key tool in the proof is a new estimate on the growth in time of the fluid particle velocity for one-dimensional Vlasov fluids with bounded interactions.

This work is devoted to the modeling of space charge dominated particle beams in the paraxial approximation with several types of external focusing fields (uniform, periodic and alternating gradient). A solid mathematical background for numerical beam simulation is established. The Kapchinsky–Vladimirsky (KV) distribution which can be matched exactly or numerically to the focusing channel is first studied. Then the matched KV beam is used to approximately match arbitrary beams. Moreover, Waterbag and Maxwell–Boltzmann beams are studied to give analytical solution for code validation. Finally, numerical simulations are presented in different configurations.

In this paper we prove the existence and uniqueness of classical solution for a system of PDEs recently developed in Refs. 60, 8, 10 and 11 to modelize the nonlinear gyrokinetic turbulence in magnetized plasma. From the analytical and numerical point of view this model is very promising because it allows to recover kinetic features (wave–particle interaction, Landau resonance) of the dynamic flow with the complexity of a multi-fluid model. This model, called the gyro-water-bag model, is derived from two-phase space variable reductions of the Vlasov equation through the existence of two underlying invariants. The first one, the magnetic moment, is adiabatic and the second, a geometric invariant named "water-bag", is exact and is just the direct consequence of the Liouville theorem.

In this paper, we build a two-scale macro–micro decomposition of the Vlasov equation with a strong magnetic field. This consists in writing the solution of this equation as a sum of two oscillating functions with circumscribed oscillations. The first of these functions has a shape which is close to the shape of the two-scale limit of the solution and the second one is a correction built to offset this imposed shape. The aim of such a decomposition is to be the starting point for the construction of two-scale asymptotic-preserving schemes.

This paper is devoted to the approximation of the linear Boltzmann equation by fractional diffusion equations. Most existing results address this question when there is no external acceleration field. The goal of this paper is to investigate the case where a given acceleration field is present. The main result of this paper shows that for an appropriate scaling of the acceleration field, the usual fractional diffusion equation is supplemented by an advection term. Both the critical and supercritical case are considered.

In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker–Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulomb interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. The existence of weak and strong solutions to the kinetic and fluid equations is also obtained. We emphasize that the existence of global weak solution with the needed free energy dissipation for the kinetic model is established.

We consider Vlasov-type scaling for the Glauber dynamics in continuum with a positive integrable potential, and construct rescaled and limiting evolutions of correlation functions. Convergence to the limiting evolution for the positive density system in infinite volume is shown. Chaos preservation property of this evolution gives a possibility to derive a nonlinear Vlasov-type equation for the particle density of the limiting system.

We prove in the cases of spherical, plane and hyperbolic symmetry a local in time existence theorem and continuation criteria for cosmological solutions of the Einstein–Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. This system describes the evolution of self-gravitating collisionless matter and scalar waves within the context of general relativity. In the case where the only source is a scalar field it is shown that a global existence result can be deduced from the general theorem.

In this note, we show that small-data asymptotically flat spherically-symmetric solutions to the Einstein–Vlasov system for massless particles are causally geodesically complete and decay appropriately. This extends previous results of Rein and Rendall in the massive case.

We study a mathematical model for sprays which takes into account particle break-up due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier–Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system.

We consider in this paper a spray constituted of an incompressible viscous gas and of small droplets which can breakup. This spray is modeled by the coupling (through a drag force term) of the incompressible Navier–Stokes equation and of the Vlasov–Boltzmann equation, together with a fragmentation kernel. We first show at the formal level that if the droplets are very small after the breakup, then the solutions of this system converge towards the solution of a simplified system in which the small droplets produced by the breakup are treated as part of the fluid. Then, existence of global weak solutions for this last system is shown to hold, thanks to the use of the DiPerna–Lions theory for singular transport equations, and a compactness lemma specifically tailored for our study.

Generation of plasma perturbations because of collisionless super–Alfvénic dense plasma clouds expanding through and decelerating in an ambient plasma background is studied. Using an universal hybrid kinetic-hydrodynamical description, the calculations are made for a 2D and 3D expansion of a spherical cloud into an initially uniform background with uniform magnetic field.

The solutions of the Wigner-transformed time-dependent Hartree–Fock–Bogoliubov equations are studied in the constant-Δ approximation, in spite of the fact that this approximation is known to violate both local and global particle-number conservation. As a consequence of this symmetry breaking, the longitudinal response function given by this approximation contains spurious contributions. A simple prescription for restoring both broken symmetries and removing the spurious strength is proposed. It is found that the semiclassical analogue of the quantum single-particle spectrum, has an excitation gap of 2Δ, in agreement with the quantum result. The effects of pairing correlations on the density response functions of three one-dimensional systems of different size are shown.

We investigate the linear and the nonlinear responses to the external field in the mean-field interaction systems whose dynamics is described by the Vlasov equation in the limit of large population. A nonlinear response formula is derived by the asymptotic-transient decomposition, which is developed for analyzing the nonlinear Landau damping and plasma waves. The formula gives a self-consistent equation for the asymptotic state, which is in general not in thermal equilibrium but in a long-lasting nonequilibrium quasistationary state due to the long-range nature of the interaction. This formula is useful to compute the critical exponent analytically for the nonlinear response at the critical point, denoted by δ. Further advantage of this formula is that it includes the linear response formula obtained from the naively linearized equation, and unifies the nonlinear and the linear responses, where the latter induces another critical exponent γ for the zero-field susceptibility. Interestingly, these critical exponents for the quasistationary states differ from ones in thermal equilibria, but the Widom’s scaling relation holds true even for the non-classical critical exponents. We discuss universality of the critical exponents and the scaling relation for spatially periodic one-dimensional systems.