Narrow Results

We construct and study the stochastic force field generated by a Poisson distribution of sources at finite density, $x_1,x_2,\dots ,$ in ${ℝ}^3$ each of them yielding a long range potential $Q_i\mathrm{\Phi }(x-x_i)$ with possibly different charges $Q_i\in ℝ$. The potential $\mathrm{\Phi }$ is assumed to behave typically as $|x{|}^{-s}$ for large $\left|x\right|$, with $s>1/2$. We will denote the resulting random field as “generalized Holtsmark field”. We then consider the dynamics of one tagged particle in such random force fields, in several scaling limits where the mean free path is much larger than the average distance between the scatterers. We estimate the diffusive time scale and identify conditions for the vanishing of correlations. These results are used to obtain appropriate kinetic descriptions in terms of a linear Boltzmann or Landau evolution equation depending on the specific choices of the interaction potential.

Rumor, as an important form of information dissemination, has always been a research hotspot in the field of complex networks. How to better understand the rules of rumor propagation and establish a practical dissemination model is a significant challenge. To further study the state transfer in information transmission, this paper established the Ignorant-Spreader-Stifler-Transition (ISRT) model, introduced different influence mechanisms and calculate the influence rate accurately by function. Specifically, (1) Based on SIR model, this paper introduces the transition state, considering that transition may awaken spontaneously to spread rumors due to individual cognition. In this paper, the ratio of the current communicator and the degree of doubt of the transition are introduced into the spontaneous arousal function. (2) This paper redefines the propagation probability function and the forgetting probability function, and introduces the time function to describe the rate from the propagator to the restorer. (3) Due to the presence of highly emotional leader propagators in the network who would awaken the immune to spread rumors again, the model added a link from the recovering person to the infected person. Finally, the nonpropagation equilibrium point $E_0$ and propagation equilibrium point $E_1$ are obtained by establishing the mean field equation. The experimental results show that different influencing mechanisms can more accurately locate the stage change of rumor transmission, which provides theoretical support for more effective control of information transmission.

We consider a quantum kinetic equation for $e^{-}{e}^{+}$ plasma created from vacuum under the action of a strong time-dependent linearly polarized electric field. Simplification of the collision integral for photon emission along the polarization direction of the field is discussed.

The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.

We consider the possibilities of the formation of quasistationary distributions of particles over energy with power asymptotics in nonequilibrium systems and dynamical systems with couplings. It is shown that the Tsallis distribution is related to the exact solutions of a kinetic equation of the Boltzmann type and those of covariant kinetic equations of the Vlasov nonlocal statistical mechanics.

We have studied the connection of the power-like solutions of kinetic equations with the eigenfunctions of fractional integro-differential operators and Jackson operators within quantum analysis, and that of nonextensiveness parameters in the framework of the Tsallis thermostatics, with flows in the phase space. It is shown that the processes running in a nonequilibrium nonconservative medium can be described by the solutions of the equation with fractional derivatives or Jackson derivatives for oscillations.

In this paper, we propose an efficient and fast numerical method of finding a stationary solution of large systems of aggregation–fragmentation equations of Smoluchowski type for concentrations of reacting particles. This method is applicable when the stationary concentrations steeply decrease with increasing aggregate size, which is fulfilled for the most important cases. We show that under rather mild restrictions, imposed on the kernel of the Smoluchowski equation, the following numerical procedure may be used: First, a complete solution for a relatively small number of equations (a “seed system”) is generated and then the result is exploited in a fast iterative scheme. In this way the new approach allows to obtain a steady-state solution for rather large systems of equations, by orders of magnitude faster than the standard schemes.

We present a variational formulation for the Kardar–Parisi–Zhang (KPZ) equation that leads to a thermodynamic-like potential for the KPZ as well as for other related kinetic equations. For the KPZ case, with the knowledge of such a potential we prove some global shift invariance properties previously conjectured by other authors. We also show a few results about the form of the stationary probability distribution function for arbitrary dimensions. The procedure used for KPZ was extended in order to derive more general forms of such a functional leading to other nonlinear kinetic equations, as well as cases with density dependent surface tension.

We investigate rescaling transformations for the Vlasov–Poisson and Euler–Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behavior of the solutions and can be applied to other models in kinetic theory (two-dimensional symmetric Vlasov–Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger–Poisson system, nonlinear Schrödinger equation).

The high field limit for the semiconductor Boltzmann equation with Pauli exclusion terms is investigated. The limit problem is shown to have a unique solution for every given density. The proof relies on a linearization procedure together with a continuation argument. The density is finally proven to converge in the high field limit towards the solution of a nonlinear hyperbolic equation.

Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schrödinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.

This paper deals with the competition between the immune system and an agressive host such as tumor. We are particularly interested in the asymptotic behavior of the system when the time goes to infinity: we prove that this competition will absolutely finish either by the suppression of the harmful substances or by the inhibition of the immune system. The model which we introduce is suitable to describe the disease evolution for a long time scale, because it takes the capacity of the body to repair cells damage into account: body cells produce new cells to replace dead ones. We also examine the evolution of the solutions if we assume that the immune system has a normal healthy state that it always tries to reach.

The behaviour of a collisionless plasma is described by the relativistic Vlasov–Maxwell system of equations. A criteria ensuring the existence of smooth solutions was given⁸ by Glassey and Strauss, along with pointwise bounds on the electromagnetic field generated by the particles. In this paper, we obtain an improved bound.

We analyze a reduced 1D Vlasov–Maxwell system introduced recently in the physical literature for studying laser-plasma interaction. This system can be seen as a standard Vlasov equation in which the field is split into two terms: an electrostatic field obtained from Poisson's equation and a vector potential term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson and wave equations are due to the coupling with the Vlasov equation through the charge density. We show global existence of weak solutions in the nonrelativistic case, and global existence of characteristic solutions in the quasi-relativistic case. Moreover, these solutions are uniquely characterized as fixed points of a certain operator. We also find a global energy functional for the system allowing us to obtain L^p-nonlinear stability of some particular equilibria in the periodic setting.

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.

This paper is concerned with the formulation and analysis of particle fragmentation by a mathematical approach of the kinetic theory. We consider a fairly general model which may require a description of the internal configuration of each particle, like internal energy. The fragmentation process is assumed to occur due to the configuration of the corresponding particle; an easy modification would allow one to consider the interaction with some external medium (typically a fluid) but we do not deal with fragmentation processes induced by particles collision here. The proposed model is therefore linear and may be analyzed with the use of correct entropies.

We study the two-scale asymptotics for a charged beam under the action of a rapidly oscillating external electric field. After proving the convergence to the correct asymptotic state, we develop a numerical method for solving the limit model involving two time scales and validate its efficiency for the simulation of long time beam evolution.

This paper is devoted to the study of optimal control problems arising in radiotherapy planning problems. The distribution of the radiative intensity in the patient's body is described by a Boltzmann-integro differential equation with position, angular and energy-dependent scattering and absorption coefficients and an energy loss term. The presented discussion is the last in the series of Refs. 13 and 14 discussing radiotherapy problems using the Boltzmann transport equation. We show the existence, uniqueness and regularity of an optimal control using evolution group theory. We extend existing results in order to treat the important case of energy-dependent scattering coefficients.

We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair-interaction driven master equations. In the spatially homogeneous case, we prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well-known result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.

The paper deals with the analysis of a nonlinear Fokker–Planck equation modeling the mechanical unzipping of double-stranded DNA under the influence of an applied force. The dependent variable is the probability density of unzipping m base pairs. The nonlinear Fokker–Planck equation we propose here is obtained when we couple the model proposed in [D. K. Lubensky and D. R. Nelson, Pulling pinned polymers and unzipping DNA, Phys. Rev. Lett.85 (2000) 1572–1575] with a transcendental equation for the applied force. The resulting model incorporates nonlinear effects in a different way than the usual models in kinetic theory. We show the well-posedness of this model. For that we require a combination of techniques coming from second-order kinetic equations and compensated compactness arguments in conservation laws.

This paper presents a new approach to the modeling of vehicular traffic flows on road networks based on kinetic equations. While in the literature the problem has been extensively studied by means of macroscopic hydrodynamic models, to date there are still not, to the authors' knowledge, contributions tackling it from a genuine statistical mechanics point of view. Probably one of the reasons is the higher technical complexity of kinetic traffic models, further increased in case of several interconnected roads. Here such difficulties of the theory are overcome by taking advantage of a discrete structure of the space of microscopic states of the vehicles, which is also significant in view of including the intrinsic microscopic granularity of the system in the mesoscopic representation.