Structural evolution of the Tropical Pacific climate network

An attempt is made to develop an equilibrium kinetic equation for a weakly non ideal inhomogeneous gravitational system utilizing the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations. It is shown that the pair correlational function explicitly depends upon the nature of binary interaction between particles. The corresponding kinetic equation containing pair correlation corrections is devoid of the degeneracy present in the collisionless Boltzmann equation with respect to the nature of the two particle interactions, unlike the Vlasov equation that cannot recognize the nature of two particle interaction. A net effect of the particle correlations can be realized only if the spatial symmetry of the correlation interaction is broken due to a spatial inhomogeneity. Such an inhomogeneity is inherently present in a bulk gravitational system in view of the unshielded long range nature of the two-particle interactions. In a finite gravitational system, the effects of pair correlations in the first order kinetic equation can be expressed in terms of the macroscopic gravitational potential to obtain a modified Boltzmann distribution that includes the effects of correlations.     

Verallgemeinerung der Methode der singulären Normallösungen für isotherme irreversible Vlasov-Systeme

Irreversible Vlasov systems, i.e. systems governed by a Vlasov-type kinetic equation including entropy-producing collision terms, are treated by the techniques of singular normal modes and singular integral equations using a new indirect method which renders possible a straightforward generalization of the Case formalism as developed originally for collision-free Vlasov plasmas. This method is in contrast to a more complex method given by the present authors for the first application of the singular normal mode expansion to irreversible Vlasov systems (1970). The linearized Vlasov operator supplemented by complete Bhatnagar-Gross-Krook collision integrals as the most important model collision terms is analyzed in detail for a nonrelativistic, nondegenerate, stationary electron gas with neutralizing positive ions and neutral particles without a magnetic field at constant temperature, generalizations for more complex irreversible Vlasov systems being possible. The key of the indirect method given is the introduction of a transformed electron distribution function containing as an additive term an integral over the usual distribution function.     

Ion acceleration during adiabatic plasma expansion: Renormalization group approach

The renormalization group approach is applied to derive an exact solution to self-consistent Vlasov kinetic equations for plasma particles in the quasineutral approximation. The solution obtained describes the one-dimensional adiabatic expansion into vacuum of a plasma bunch with arbitrary initial velocity distributions of the electrons and ions. The ion acceleration is investigated for both a Maxwellian two-temperature initial electron distribution and a super-Gaussian initial electron distribution.     

Analytic solutions to the Vlasov equations for expanding plasmas

The renormalization-group approach is applied to derive an exact solution to the self-consistent Vlasov kinetic equations for plasma particles in the quasineutral approximation. The solutions obtained describe three-dimensional adiabatic expansion of a plasma bunch with arbitrary initial velocity distributions of the electrons and ions. The solution found is illustrated by the examples on ion acceleration in a plasma with hot electrons and in a plasma with light impurity ions.     

Stationary self-consistent distributions for a charged particle beam in the longitudinal magnetic field

A review of analytical solutions of the Vlasov equation for a beam of charged particles is given. These results are analyzed on the basis of a unified approach developed by the authors. In the context of this method, a space of integrals of motion is introduced in which the integrals of motion of particles are considered as coordinates. In this case, specifying a self-consistent distribution is reduced to defining a distribution density in this space. This approach allows us to simplify the construction and analysis of different self-consistent distributions. In particular, it is possible, in some cases, to derive new solutions by considering linear combinations of well-known solutions. This approach also makes it possible in many cases to give a visual geometric representation of self-consistent distributions in the space of integrals of motion.     

Coarse graining of nonbonded degrees of freedom

We investigate the physical meaning of coarse-grained beads generated by coarse graining of nonbonded particles such as solvent molecules in a solution. Starting from the partition function, we analytically coarse grain an N-particle fluid to a system containing N-2 of the original particles plus a bead representing the two remaining particles. As a direct consequence of the lack of bonding interactions, the resulting effective potential becomes independent of the bead coordinates, i.e., ideal-gas-like, in the thermodynamic limit. Thus, there are no conservative forces between coarse-grained beads representing assemblies of nonbonded molecules nor between these beads and any other species in the system.     

Kinetic description of vacuum particle production in collisions of ultrarelativistic nuclei

The dynamics of partons that emerge as the result of quantum tunneling in a spatially uniform time-dependent field is studied under conditions prevalent in ultrarelativistic heavy-ion collisions. A self-consistent set of coupled equations that consists of the renormalized Maxwell equation and the Vlasov kinetic equation that involves a source and which is derived on a dynamical basis is solved numerically. The time dependence of the distributions of internal fields and currents for bosons and fermions is investigated within this back-reaction mechanism, and their momentum spectra are constructed. Clear evidence that oscillations in the time dependence of parton distributions in phase-space cells are of a stochastic character is obtained, and a significant irregularity in the momentum distribution on large time scales is found. If the influence of the back reaction is disregarded, these effects disappear completely, the oscillations becoming regular. A possible thermalization scenario for such a quasiparticle plasma is considered in the relaxation-time approximation. A locally equilibrium state is described within the two-component thermodynamics of particles and antiparticles. The possibility of introducing temperature under conditions of a strong vacuum polarization is discussed.     

Systematic derivation of classical kinetic equations for a three component ionized system

A set of general kinetic classical equations is derived for the correlations between particles and/or fields in an ionized three component system. External electric and magnetic fields may exist as well as the induced fields. In the lowest order the reversible Vlasov equation and the equivalent one oscillator equation result. In the first-order a Fokker-Planck type equation is obtained for both the one-particle and one-oscillator distribution functions.     

A nonequilibrium equation of state in heavy-ion collisions at intermediate energies

A modified hydrodynamic approach using a nonequilibrium equation of state is used to describe heavy-ion collisions at intermediate energies. The calculated energy spectra of protons produced in heavyion collisions are compared to experimental data and the results from calculations based on solving the Vlasov–Uehling–Uhlenbeck (VUU) kinetic equation.     

Fokker-Planck modeling for particle slowing down and thermalization in a Maxwellian plasma

In this paper the Fokker-Planck equation, which takes into account the medium absorption and the Maxwellian behavior of the field particles at low energy for Coulomb interactions is obtained. The analytical solution of the stationary distribution function is obtained in both angle and velocity variables. In particular the electron distribution for electron-ion collisions has been obtained using this diffusion approximation for beam and isotropic sources. If the absorption is neglected the solution recovers the classical stationary Maxwell distribution. For low absorption rates the solution shows a typical slowing down spectrum for high energy and a Maxwellian-like distribution at thermal energy. For moderate and high absorption rates the test particles do not reach the thermal equilibrium and the Maxwell distribution at low energies does not appear.Received: 1 December 2003, Published online: 16 March 2004PACS: 51.10. + y Kinetic and transport theory of gases - 41.75.-i Charged-particle beams - 52.65.Ff Fokker-Planck and Vlasov equation     

Particle dynamics during adiabatic expansion of a plasma bunch

The renormalization-group approach is used to obtain an exact solution to the self-consistent Vlasov kinetic equations for plasma particles in the quasi-neutral approximation. This solution describes the one-dimensional adiabatic expansion of a plasma bunch into a vacuum for arbitrary initial particle velocity distributions. Ion acceleration is studied for two-temperature Maxwellian and super-Gaussian initial electron distributions, which predetermine distinctly different ion spectra. The solution found is used to describe the acceleration of ions of two types. The relative acceleration efficiency of light and heavy ions as a function of atomic weights and number densities is analyzed. The solutions obtained are of practical importance in describing ion acceleration during the interaction of an ultrashort laser pulse with nanoplasma, for example, cluster plasma or plasma produced when thin foils are irradiated by a laser.     

Spectral coarse graining and synchronization in oscillator networks

Coarse graining techniques offer a promising alternative to large-scale simulations of complex dynamical systems, as long as the coarse-grained system is truly representative of the initial one. Here, we investigate how the dynamical properties of oscillator networks are affected when some nodes are merged together to form a coarse-grained network. Moreover, we show that there exists a way of grouping nodes preserving as much as possible some crucial aspects of the network dynamics. This coarse graining approach provides a useful method to simplify complex oscillator networks, and more generally, networks whose dynamics involves a Laplacian matrix.     

Method of projection operators in the theory of irreversible processes

The Zwanzig-Nakajima projection-operator method is extended to the case of time-dependent nonlinear projection operators. A method employing these operators is proposed for constructing the non-Markovian kinetic equation for a single-particle, time-dependent distribution function. An important assumption is made in the derivation of the equation regarding the factorization of the initial nonequilibrium distribution of the multi-particle system. An approximate kinetic equation is obtained for a slightly nonequilibrium system which asymptotically approaches the equilibrium canonical distribution at a fixed temperature. The effects of the self-consistent Vlasov field and the non-Markovian Fokker-Planck collision operator with a microscopic parameter appear in this equation. This is the first derivation of such an equation.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

Effect of noise on a quantum bound state

The evolution of a quantum wave function on a lattice with time-dependent random disorder is studied. It is found that the bound state wave function of a quantum system exhibits intrinsic instability with respect to noisy disorder in the statistical sense. Under the general scheme of time coarse graining, we show nonperturbatively that the stationary attractive potential is completely washed out at large time scales by the existence of the noisy disorder to any order of density correlations.     

Self-consistent chaotic transport in fluids and plasmas

Self-consistent chaotic transport is the transport of a field F by a velocity field v according to an advection-diffusion equation in which there is a dynamical constrain between the two fields, i.e., O(F,v)=0 where O is an integral or differential operator, and the Lagrangian trajectories of fluid particles exhibit sensitive dependence on initial conditions. In this paper we study self-consistent chaotic transport in two-dimensional incompressible shear flows. In this problem F is the vorticity zeta, the corresponding advection-diffusion equation is the vorticity equation, and the self-consistent constrain is the vorticity-velocity coupling z nabla xv=zeta. To study this problem we consider three self-consistent models of intermediate complexity between the simple but limited kinematic chaotic advection models and the approach based on the direct numerical simulation of the Navier-Stokes equation. The first two models, the vorticity defect model and the single wave model, are constructed by successive simplifications of the vorticity-velocity coupling. The third model is an area preserving self-consistent map obtained from a space-time discretization of the single wave model. From the dynamical systems perspective these models are useful because they provide relatively simple self-consistent Hamiltonians (streamfunctions) for the Lagrangian advection problem. Numerical simulations show that the models capture the basic phenomenology of shear flow instability, vortex formation and relaxation typically observed in direct numerical simulations of the Navier-Stokes equation. Self-consistent chaotic transport in electron plasmas in the context of kinetic theory is also discussed. In this case F is the electron distribution function in phase space, the corresponding advection equation is the Vlasov equation and the self-consistent constrain is the Poisson equation. This problem is closely related to the vorticity problem. In particular, the vorticity defect model is analogous to the Vlasov-Poisson model and the single wave model and the self-consistent map apply equally to both plasmas and fluids. Also, the single wave model is analogous to models used in the study of globally coupled oscillator systems. (c) 2000 American Institute of Physics.     

Phase particle integrations of the nuclear vlasov equation

The non-linear Vlasov equation, describing the time development of the one-body Wigner distribution function, is integrated as an initial value problem by following the trajectories of notional phase particles which evolve along its hamiltonian characteristic curves. Initial conditions are generated by galilean transformations of the self-consistent solutions of the static equation obtained assuming a Thomas-Fermi form for the Wigner distribution.Fusion, deep inelastic collisions and fragmentation are all exhibited depending on the bombarding energy per nucleon, in qualitative agreement with the results of comparable TDHF calculations. A simple criterion for determining the boundaries between these phenomena, based on classical penetration of the collective mean field by the phase particles, is found not to be accurate, presumably due to an inadequate allowance for the effects of self-consistency.     

Normal modes of a relativistic quantum plasma; The one-component plasma

The normal modes of a relativistic electron gas are studied on the basis of the Boltzmann-Vlasov kinetic equation via a projection operator formalism. A general framework is constructed in which the fully relativistic Vlasov self-consistent force term appears as a symmetric operator acting in the Hilbert space of one-particle states. The plasma-dynamical equations are obtained by projecting onto the subspace consisting of the charge, energy and momentum densities, plus the nonconserved current density. The eigenmodes of these equations include two transverse and two longitudinal plasma modes, and one damped heat mode. They are explicitly calculated up to second order in the wave vector and to first order in the collision frequency.     

Inertial Alfven waves in an inhomogeneous bi-Maxwellian plasma

The Vlasov kinetic equation is solved using gyrokinetic theory and the dielectric tensor for non-relativistic, magnetized, bi-Maxwellian plasmas is calculated. A generalized dispersion relation for kinetic Alfven waves is derived taking into account the density inhomogeneity and temperature anisotropy. The modified dispersion relation thus obtained is then used to examine the propagation characteristics of the kinetic Alfven waves in the inertial regime. The importance of density inhomogeneity and temperature anisotropy for Solar corona is highlighted. The growth rate of the inertial Alfven wave proves that density inhomogeneity acts as a source of free energy.