Molecular random walk and a symmetry group of the Bogoliubov equation

We consider the statistics of molecular random walks in fluids using the Bogoliubov equation for the generating functional of the distribution functions. We obtain the symmetry group of this equation and its solutions as functions of the medium density. It induces a series of exact relations between the probability distribution of the total path of a walking test particle and its correlations with the environment and consequently imposes serious constraints on the possible form of the path distribution. In particular, the Gaussian asymptotic form of the distribution is definitely forbidden (even for the Boltzmann-Grad gas), but the diffusive asymptotic form with power-law tails (cut off by the ballistic flight length) is allowed.     

On the periodic Wigner-Poisson-Fokker-Planck system

This paper is concerned with the existence and uniqueness analysis of global classical solutions of a diffusive quantum evolution equation with nonlinear coupling to the Poisson equation. The main technical difficulty in the existence proof is to show that the quantum Fokker-Planck term is a semigroup-generator in a weighted L²-space. The potential term is then a Lipschitz perturbation of it.     

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.     

Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics

In this paper we consider a two-dimensional lattice gas under Kawasaki dynamics, i.e., particles hop around randomly subject to hard-core repulsion and nearest-neighbor attraction. We show that, at fixed temperature and in the limit as the particle density tends to zero, such a gas evolves in a way that is close to an ideal gas, where particles have no interaction. In particular, we prove three theorems showing that particle trajectories are non-superdiffusive and have a diffusive spread-out property. We also consider the situation where the temperature and the particle density tend to zero simultaneously and focus on three regimes corresponding to the stable, the metastable and the unstable gas, respectively.     

NONLINEAR DIFFUSIVE PHENOMENA OF NONLINEAR HYPERBOLIC SYSTEMS

The authors study the nonlinear hyperbolic system which describes the motion of isentropic gas flow with external friction acting on it, such as a flow through porous media, and show the nonlinear diffusive phenomena for the large time behavior of solutions for this system by proving that the solutions tend to those of a nonlinear diffusion equation time-asymptotically.     

Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.     

L6 BOUND FOR BOLTZMANN DIFFUSIVE LIMIT

We consider diffusive limit of the Boltzmann equation in a periodic box. We establish L⁶ estimate for the hydrodynamic part Pf of particle distribution function, which leads to uniform bounds global in time.     

The quantum drift-diffusion model: Existence and exponential convergence to the equilibrium

This work is devoted to the analysis of the quantum drift-diffusion model derived by Degond et al. in [7]. The model is obtained as the diffusive limit of the quantum Liouville–BGK equation, where the collision term is defined after a local quantum statistical equilibrium. The corner stone of the model is the closure relation between the density and the current, which is nonlinear and nonlocal, and is the main source of the mathematical difficulties. The question of the existence of solutions has been open since the derivation of the model, and we provide here a first result in a one-dimensional periodic setting. The proof is based on an approximation argument, and exploits some properties of the minimizers of an appropriate quantum free energy. We investigate as well the long time behavior, and show that the solutions converge exponentially fast to the equilibrium. This is done by deriving a non-commutative logarithmic Sobolev inequality for the local quantum statistical equilibrium.     

Exact multiplicity of solutions to a diffusive logistic equation with harvesting

An Ambrosetti-Prodi type exact multiplicity result is proved for a diffusive logistic equation with harvesting. We show that a modified diffusive logistic mapping has exactly either zero, or one, or two pre-images depending on the harvesting rate. It implies that the original diffusive logistic equation with harvesting has at most two positive steady state solutions.     

Quantum Mechanics on S 1 Based on Dirac Formalism

We deal with quantum mechanics on S 1 (embedded to R 2 ) based on Dirac formalism. To conclude that the dynamical system is a quantum mechanical system we have to show selfadjointness of the dynamical variables. First we show selfadjointness of the momentum operators and then study their spectra. We find that the continuous spectrum of each momentum operator coincides with the set of all real numbers. Second we discuss selfadjointness of the Hamiltonian for a free quantum mechanical particle moving on S 1 together with its spectrum. We see that the spectrum of the Hamiltonian consists of eigenvalues only. Finally we apply these results to the Cauchy problem for the Schrödinger equation for a free quantum mechanical particle moving on S 1 .     

The number of energy levels of a particle in a uniform MQW structure

A recently obtained multilayer equation that gives the eigenvalues of the energy of a quantum particle in an arbitrary one-dimensional piecewise constant potential field is studied. In particular, this equation can be used to calculate the eigenvalues of the particle’s energy in an MQW structure, in which the potential takes only two different values in the various layers. A formula is analyzed that was obtained earlier by the author for the number of energy levels in a uniform MQW structure, i.e., in a structure with potential wells and walls of constant widths. The equation is substantiated for all uniform MQW structures. It is proved that the number of energy levels in a uniform MQW structure increases indefinitely with unlimited growth of the number of potential wells. The existence of uniform MQW structures with an arbitrarily large prescribed number of potential wells and with a single energy level is proved.     

Long-time correlations for a hard-sphere gas at equilibrium

It has been known since Lanford that the dynamics of a hard-sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simpler than the one devised in Bodineau et al which was specific to the 2D case.     

General-covariant quantum mechanics in Riemannian space-time III. The Dirac particle

A general covariant analog of standard nonrelativistic quantum mechanics with relativistic corrections is constructed for the Dirac particle in a normal geodesic frame in general Riemannian space-time. Not only the Pauli equation with a Hermitian Hamiltonian and the pre-Hilbert structure of the space of its solutions, but also matrix elements of the Hermitian operators of momentum, (curvilinear) spatial coordinates, and spin of the particle, are deduced, as a general-covariant asymptotic approximation in c^–2 (c is the velocity of light), to their naturally determined general-relativistic pre-images. It is shown that the Pauli equation Hamiltonian, generated by the Dirac equation, is unitary-equivalent to the energy operator generated by the metric energymomentum tensor of the spinor field. Commutation and other properties of the observables associated with variation in the geometrical background of quantum mechanics are briefly discussed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 122–132, January, 1996.     

Derivation of quantum kinetic equation from the Liouville-von Neumann equation

A quantum Markov kinetic equation is derived from the Liouville—von Neumann equation in the framework of second-order kinetic perturbation theory. The proposed method can be used to obtain an equation that describes the quantum diffusion of a particle in a crystal.Moscow Aviation Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No.1, pp. 33–43, July, 1994.     

ASYMPTOTIC LIMITS FOR QUANTUM TRAJECTORY MODELS

ABSTRACT

The semi-classical and the inviscid limit in quantum trajectory models given by a one-dimensional steady-state hydrodynamic system for quantum fluids are rigorously performed. The model consists of the momentum equation for the particle density in a bounded domain, with prescribed current density, and the Poisson equation for the electrostatic potential. The momentum equation can be written as a dispersive third-order differential equation which may include viscous terms. It is shown that the semi-classical and inviscid limit commute for sufficiently small data (i.e. current density) corresponding to subsonic states, where the inviscid non-dispersive solution is regular. In addition, we show that these limits do not commute in general. The proofs are based on a reformulation of the problem as a singular second-order elliptic system and on elliptic and W ^1,1 estimates.     

Nonequilibrium processes in a polyatomic gas of moderate density in an external magnetic field

The dissipative and relaxation processes in a polyatomic gas of moderate density in an external magnetic field are considered using the methods of quantum kinetic theory. The momentum and the energy fluxes, as well as the transport coefficients; are obtained in the dissipative Navier-Stokes approximation by taking the first virial correction into account. The tensor coefficient of the shear viscosity in a gas of diamagnetic molecules is obtained as a function of the effective cross sections of binary and ternary collisions and of the value of the magnetic field. The Bloch equation for magnetization in a spatially homogeneous gas of linear diamagnetic molecules is derived. The relaxation times and the shift in the Larmor frequency due to intermolecular collisions are obtained.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 307–319, May, 1996.Translated by V. I. Serdobol'skii.     

A level-set method for convective–diffusive particle deposition

This article presents a fixed-mesh approach to model convective–diffusive particle deposition onto surfaces. The deposition occurring at the depositing front is modeled as a first order reaction. The evolving depositing front is captured implicitly using the level-set method. Within the level-set formulation, the particle consumed during the deposition process is accounted for via a volumetric sink term in the species conservation equation for the particles. Fluid flow is modeled using the incompressible Navier–Stokes equations. The presented approach is implemented within the framework of a finite volume method. Validations are made against solutions of the total concentration approach for one- and two-dimensional depositions with and without convective effect. The presented approach is then employed to investigate deposition on single- and multi-tube arrays in a cross-flow configuration.     

Quantum automaton in a 1-D box

A simple cellular automaton is conjectured based on the Dirac wave equation and a diffusion confinement which attempts to emulate quantum behaviour of a particle in a 1-D box. Some features of quantum behavior such as the collapse of the wave function upon measurement, the wave-like nature of particles, and the role of virtual and non-local interactions become evident after completing a series of computations and analyzing the results. A pronounced correspondence between an automaton and a quantum particle in a 1-D box is explicated and shows promise for explaining some of the cloudy conceptual difficulties enmeshed in present quantum space-time pictures of reality.     

Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation

We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section. © 2000 John Wiley & Sons, Inc.     

Ballistic motion of a tracer particle coupled to a Bose gas

We study the motion of a heavy tracer particle weakly coupled to a dense interacting Bose gas exhibiting Bose–Einstein condensation. In the so-called mean-field limit, the dynamics of this system approaches one determined by nonlinear Hamiltonian evolution equations. We derive the effective dynamics of the tracer particle, which is described by a non-linear integro-differential equation with memory, and prove that if the initial speed of the tracer particle is below the speed of sound in the Bose gas the motion of the particle approaches an inertial motion at constant velocity at large times.