Klimontovich Equation Research Articles

Vlasov-Klimontovich Equation in Action

This paper outlines a variety of possible applications of the Vlasov equation and its generalization, i.e., the Klimontovich equation, in various areas of many-body physics. In particular, these equations are shown to be used in relativistic plasma physics, the theory of semi-classical Bloch electrons, and the metriplectic description of dissipative processes.

On the Klimontovich description of complex (dusty) plasmas

AbstractThe numerous idealizations that are involved in the exact microscopic statistical description of complex (dusty) plasmas are discussed in detail. The two prevailing approaches in the Klimontovich description of dusty plasmas are reviewed in a pedagogical manner. The continuous phase space approximation is introduced, within which the more rigorous treatment should collapse to the more heuristic treatment. The plasma Klimontovich equations are shown to be identical, but there are marked differences in the dust Klimontovich equations whose origin is analysed in depth.

Quasilinear theory for inhomogeneous plasma

This paper presents quasilinear theory (QLT) for a classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic and gravitational effects are subsumed. A Fokker–Planck equation for the dressed ‘oscillation-centre’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum and energy, and it also satisfies the$\smash {H}$-theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave–plasma system. The action of non-resonant waves is also conserved, unlike in the standard version of QLT. Dewar's oscillation-centre QLT of electrostatic turbulence (Phys. Fluids, vol. 16, 1973, p. 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed.

A Note on the Entropy Force in Kinetic Theory and Black Holes.

The entropy force is the collective effect of inhomogeneity in disorder in a statistical many particle system. We demonstrate its presumable effect on one particular astrophysical object, the black hole. We then derive the kinetic equations of a large system of particles including the entropy force. It adds a collective therefore integral term to the Klimontovich equation for the evolution of the one-particle distribution function. Its integral character transforms the basic one particle kinetic equation into an integro-differential equation already on the elementary level, showing that not only the microscopic forces but the hole system reacts to its evolution of its probability distribution in a holistic way. It also causes a collisionless dissipative term which however is small in the inverse particle number and thus negligible. However it contributes an entropic collisional dissipation term. The latter is defined via the particle correlations but lacks any singularities and thus is large scale. It allows also for the derivation of a kinetic equation for the entropy density in phase space. This turns out to be of same structure as the equation for the phase space density. The entropy density determines itself holistically via the integral entropy force thus providing a self-controlled evolution of entropy in phase space.

Causal kinetic equation of non-equilibrium plasmas

Abstract. Statistical plasma theory far from thermal equilibrium is subject to Liouville's equation, which is at the base of the BBGKY hierarchical approach to plasma kinetic theory, from which, in the absence of collisions, Vlasov's equation follows. It is also at the base of Klimontovich's approach which includes single-particle effects like spontaneous emission. All these theories have been applied to plasmas with admirable success even though they suffer from a fundamental omission in their use of the electrodynamic equations in the description of the highly dynamic interactions in many-particle conglomerations. In the following we extend this theory to taking into account that the interaction between particles separated from each other at a distance requires the transport of information. Action needs to be transported and thus, in the spirit of the direct-interaction theory as developed by Wheeler and Feynman (1945), requires time. This is done by reference to the retarded potentials. We derive the fundamental causal Liouville equation for the phase space density of a system composed of a very large number of charged particles. Applying the approach of Klimontovich (1967), we obtain the retarded time evolution equation of the one-particle distribution function in plasmas, which replaces Klimontovich's equation in cases when the direct-interaction effects have to be taken into account. This becomes important in all systems where the distance between two points |Δq| ∼ ct is comparable to the product of observation time and light velocity, a situation which is typical in cosmic physics and astrophysics.

The photon kinetic approach toward terahertz radiation

A one dimensional spatiotemporal model for the femtosecond laser pulse interaction with air is developed in which the laser pulse propagation as well as the plasma formation and evolution accompanied with the THz radiation are investigated numerically. The photon kinetic approach based on the Klimontovich kinetic equation for the photons is used to describe the ultrashort laser pulse propagation while the Ammosov–Delone–Krainov approach accompanied with the plasma fluid equations is used to investigate the plasma formation and evolution, respectively. It is noteworthy that the model main focus was on the investigation of the long standing disagreement on the mechanisms of air-plasma THz radiation. Therefore, the three mechanisms of plasma current radiation, four-wave mixing by third order nonlinear susceptibility, and the interaction with the self exited plasma wave were inserted into the model, and their contributions to the THz emission were computed and compared. Our obtained results show that the interaction with the plasma wave plays no role in THz generation, while the nonlinear polarization and the plasma current are the two competing mechanisms, which both have significant roles in air plasma THz radiation.

Functional integral approach to the kinetic theory of inhomogeneous systems

We present a derivation of the kinetic equation describing the secular evolution of spatially inhomogeneous systems with long-range interactions, the so-called inhomogeneous Landau equation, by relying on a functional integral formalism. We start from the BBGKY hierarchy derived from the Liouville equation. At the order ${1/N}$, where $N$ is the number of particles, the evolution of the system is characterised by its 1-body distribution function and its 2-body correlation function. Introducing associated auxiliary fields, the evolution of these quantities may be rewritten as a traditional functional integral. By functionally integrating over the 2-body autocorrelation, one obtains a new constraint connecting the 1-body DF and the auxiliary fields. When inverted, this constraint allows us to obtain the closed non-linear kinetic equation satisfied by the 1-body distribution function. This derivation provides an alternative to previous methods, either based on the direct resolution of the truncated BBGKY hierarchy or on the Klimontovich equation. It may turn out to be fruitful to derive more accurate kinetic equations, e.g., accounting for collective effects, or higher order correlation terms.

Electromagnetic fluctuations in magnetized plasmas. I. The rigorous relativistic kinetic theory

Using the system of the Klimontovich and Maxwell equations, the general linear fluctuation theory for magnetized plasmas is developed. General expressions for the electromagnetic fluctuation spectra (electric and magnetic fields) from uncorrelated plasma particles in plasmas with a uniform magnetic field are derived, which are covariantly correct within the theory of special relativity. The general fluctuation spectra hold for plasmas of arbitrary composition, arbitrary momentum dependences of the plasma particle distribution functions, and arbitrary orientations of the wave vector with respect to the uniform magnetic field. Moreover, no restrictions on the values of the real and the imaginary parts of the frequency are made. The derived fluctuation spectra apply to both non-collective fluctuations and collective plasma eigenmodes in magnetized plasmas. In the latter case, kinetic equations for the components of fluctuating electric and magnetic fields in magnetized plasmas are derived that include the effect of spontaneous emission and absorption. In the limiting case of an unmagnetized plasmas, the general fluctuation spectra correctly reduce to the unmagnetized fluctuation spectra derived before.

Finite-Ncorrections to Vlasov dynamics and the range of pair interactions

We explore the conditions on a pair interaction for the validity of the Vlasov equation to describe the dynamics of an interacting N-particle system in the large N limit. Using a coarse graining in phase space of the exact Klimontovich equation for the N-particle system, we evaluate, neglecting correlations of density fluctuations, the scalings with N of the terms describing the corrections to the Vlasov equation for the coarse-grained one-particle phase space density. Considering a generic interaction with radial pair force F(r), with F(r)∼1/r(γ) at large scales, and regulated to a bounded behavior below a "softening" scale ɛ, we find that there is an essential qualitative difference between the cases γ<d and γ>d, i.e., depending on the the integrability at large distances of the pair force. In the former case, the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter ɛ, while for γ>d the amplitude of these terms is directly regulated by ɛ, and thus by the small scale properties of the interaction. This corresponds to a simple physical criterion for a basic distinction between long-range (γ≤d) and short-range (γ>d) interactions, different from the canonical one (γ≤d+1 or γ>d+1) based on thermodynamic analysis. This alternative classification, based on purely dynamical considerations, is relevant notably to understanding the conditions for the existence of so-called quasistationary states in long-range interacting systems.

Dynamic Finite Size Effects in Spiking Neural Networks

We investigate the dynamics of a deterministic finite-sized network of synaptically coupled spiking neurons and present a formalism for computing the network statistics in a perturbative expansion. The small parameter for the expansion is the inverse number of neurons in the network. The network dynamics are fully characterized by a neuron population density that obeys a conservation law analogous to the Klimontovich equation in the kinetic theory of plasmas. The Klimontovich equation does not possess well-behaved solutions but can be recast in terms of a coupled system of well-behaved moment equations, known as a moment hierarchy. The moment hierarchy is impossible to solve but in the mean field limit of an infinite number of neurons, it reduces to a single well-behaved conservation law for the mean neuron density. For a large but finite system, the moment hierarchy can be truncated perturbatively with the inverse system size as a small parameter but the resulting set of reduced moment equations that are still very difficult to solve. However, the entire moment hierarchy can also be re-expressed in terms of a functional probability distribution of the neuron density. The moments can then be computed perturbatively using methods from statistical field theory. Here we derive the complete mean field theory and the lowest order second moment corrections for physiologically relevant quantities. Although we focus on finite-size corrections, our method can be used to compute perturbative expansions in any parameter.

Kinetic theory of long-range interacting systems with angle–action variables and collective effects

We develop a kinetic theory of systems with long-range interactions taking collective effects and spatial inhomogeneity into account. Starting from the Klimontovich equation and using a quasilinear approximation, we derive a Lenard–Balescu-type kinetic equation written in angle–action variables. We confirm the result obtained by Heyvaerts [Heyvaerts, Mon. Not. R. Astron. Soc. 407, 355 (2010)] who started from the Liouville equation and used the BBGKY hierarchy truncated at the level of the two-body distribution function (i.e., neglecting three-body correlations). When collective effects are ignored, we recover the Landau-type kinetic equation obtained in our previous papers [P.H. Chavanis, Physica A 377, 469 (2007); J. Stat. Mech., P05019 (2010)]. We also consider the relaxation of a test particle in a bath of field particles. Its stochastic motion is described by a Fokker–Planck equation written in angle–action variables. We determine the diffusion tensor and the friction force by explicitly calculating the first and second order moments of the increment of action of the test particle from its equations of motion, taking collective effects into account. This generalizes the expressions obtained in our previous works. We discuss the scaling with N of the relaxation time for the system as a whole and for a test particle in a bath.

Kinetic theory of two-dimensional point vortices with collective effects

We develop a kinetic theory of point vortices in two-dimensional hydrodynamics takingcollective effects into account. We first recall the quasilinear theory of Dubin and O’Neil(1988 Phys. Rev. Lett. 60 1286) based on the Klimontovich equation and leading to aLenard–Balescu-type kinetic equation for axisymmetric flows. When collective effects areneglected, it reduces to the Landau-type kinetic equation obtained independently in ourprevious papers (Chavanis 2001 Phys. Rev. E 64 026309; 2008 Physica A 387 1123) formore general flows. We also consider the relaxation of a test vortex in a ‘sea’ (bath) of fieldvortices. Its stochastic motion is described in terms of a Fokker–Planck equation. Wedetermine the diffusion coefficient and the drift term by explicitly calculating the first-and second-order moments of the radial displacement of the test vortex fromits equations of motion, taking collective effects into account. This generalizesthe expressions obtained in our previous papers. We discuss the scaling withN of the relaxation time for the system as a whole and for a test vortex in a bath.

Spontaneous electromagnetic fluctuations in unmagnetized plasmas I: General theory and nonrelativistic limit

Using the system of the Klimontovich and Maxwell equations, general expressions for the electromagnetic fluctuation spectra (electric and magnetic field, charge and current densities) from uncorrelated plasma particles are derived, which are covariantly correct within the theory of special relativity. The general expressions hold for arbitrary momentum dependences of the plasma particle distribution functions and for collective and non-collective fluctuations. In this first paper of a series, the results are illustrated for the important special case of nonrelativistic isotropic Maxwellian particle distribution functions providing in particular the thermal fluctuations of weakly amplified modes and aperiodic modes.

Kinetic theory for low-frequency turbulence in magnetized plasmas including discrete-particle effects

In a recently developed kinetic theory for low-frequency turbulence propagating parallel to the ambient magnetic field [P. H. Yoon, Phys. Plasmas 14, 10230 (2007); P. H. Yoon and T.-M. Fang, Phys. Plasmas 14, 102303 (2007); P. H. Yoon and T.-M. Fang, Plasmas Phys. Control. Fusion 50, 085007 (2008)], the effects arising from the discrete particle nature, or the spontaneous thermal effects were ignored. Such an assumption thus limited the theory to purely collisionless “Vlasov” regime. In the present paper the previous formalism is generalized to include discrete-particle effects by reformulating the problem on the basis of the Klimontovich equation. The new terms that appear in the particle and wave kinetic equations as a result of the new formalism are the spontaneous drag term associated with the particles and the spontaneous emission and scattering terms associated with the waves.

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.

Kinetic theory with angle–action variables

We present a kinetic theory for one-dimensional inhomogeneous systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory valid at order 1 / N in a proper thermodynamic limit N → + ∞ , we obtain a closed kinetic equation describing the relaxation of the distribution function of the system as a whole due to resonances between different orbits. This equation is written in angle–action variables. It conserves mass and energy and increases the Boltzmann entropy ( H -theorem). Using a thermal bath approximation, we derive a Fokker–Planck equation describing the relaxation of a test particle towards the Boltzmann distribution under the combined effects of diffusion and friction. We mention some analogies with the kinetic theory of point vortices in two-dimensional hydrodynamics. We also stress the limitations of our approach and the connection with recent works.

Kinetic theory for metallic clusters II. Klimontovich equation approach

The classical Klimontovich equation for the microsocopic phase space density is given a Weyl quantization. As an example of its application, a simple nonequilibrium jelliumtype model for metallic clusters in strong fields is considered and the corresponding quantum kinetic equation is discussed. The treatment of collective excitations in such an inhomogeneous plasma is described briefly.

Description of diffusion in fractal media on the basis of the Klimontovich kinetic equation in fractal space

Description of diffusion in fractal media on the basis of the Klimontovich kinetic equation in fractal space

Solution to the initial value problem for a high-gain FEL via Van Kampen's method

Using Van Kampen's normal mode expansion, we solve the initial value problem for a high-gain free-electron laser described by the three-dimensional Maxwell–Klimontovich equations. An expression of the radiation spectrum is given for the process of coherent amplification and self-amplified spontaneous emission. It is noted that the input coupling coefficient for either process increases with the initial beam energy spread. The effective start-up noise is identified as the coherent fraction of the spontaneous undulator radiation in one field gain length, and is larger with increasing energy spread and emittance mainly because of the increase in gain length.

Nonlinear harmonic generation of coherent amplification and self-amplified spontaneous emission

Nonlinear harmonic generation in a uniform planar undulator is analyzed using the three-dimensional Maxwell–Klimontovich equations that include both even and odd harmonic emissions. After a certain stage of exponential growth, the dominant nonlinear harmonic interaction is caused by strong bunching at the fundamental. As a result, gain length, transverse profile, and temporal structure of these harmonic radiations are eventually determined by those of the fundamental. Transversely coherent third-harmonic radiation power is found to approach one percent of the fundamental power level for current high-gain FEL projects, while the power of the second-harmonic radiation is less but still significant for relatively low-energy FEL experiments.