Klimontovich Equation Research Articles

The continuum model in astrophysics

The role of the Vlasov equation in the continuum model of cosmology is reviewed. Within this model it is shown that criteria for the validity of the Vlasov equation are not satisfied in galaxy kinetic theory. It is further shown that the Klimontovich equation, referred to in this area of study as the Vlasov equation, is neither the Vlasov equation nor representative of the continuum limit. It is observed that omission of discrete terms in the Klimontovich picture does not lead to the fluid limit of the hierarchy

Green's function method in classical electrodynamics

A perturbation theory for solution of the Vlasov and Klimontovich equations is formulated on the basis of the method of Green's functions.

Acceleration of plasma species by electromagnetic fluctuations in an inhomogeneous plasma

A momentum equation for an arbitrary particle species is derived from the Klimontovich equation when the wave amplitude and the background plasma parameters are changing slowly in space and time. The derivation of the momentum equation presented here does not rely on many assumptions adopted by earlier works such as ISENBERG and HOLLWEG (1982). This work is an extension of Isenberg and Hollweg's work.

Mode coupling from linear and nonlinear kinetic equations

The calculation of mode coupling contributions to equilibrium time correlation functions from the nonlinear Boltzmann equation is reconsidered. It is suggested that the use of a nonlinear kinetic equation is not appropriate in this context, but instead such calculations should be reinterpreted in terms of the Klimontovich equation for the microscopic phase space density. For hard spheres the Klimontovich equation is formally similar to the nonlinear Boltzmann equation, and this similarity is exploited to explain the successful calculation of mode coupling effects from the latter. The relationship of the Klimontovich formulation to the linear ring approximation is also established.

A retarded time superposition principle and the relativistic collision operator

A retarded time superposition principle is formulated and proved for the two particle correlation function in a multi-species relativistic, and fully electro-magnetic, plasma. This principle is used to obtain the relativistic collision operator. Starting from the relativistic Klimontovich equation, the statistical (Liouville) average of the Klimontovich equation yields an expression for the collision operator in terms of the two-time two-point correlation function for two particles, G12(1, t12, t2). It is proved that G12(1, t12, t2) can be written in a retarded time superposition form in terms of the self-correlation W11(1, t12, t2) and the discreteness response function P(1, t12, t2). The equation for the pair correlation function G12(1, t12, t2), excluding triplet or higher-order correlations, is thus replaced by the simpler equation for P(1, t12, t2). This is the test particle problem which relates P(1, t12, t2) to the discreteness source term W11(1, t12, t2). The equations for P(1, t12, t2). and W11(1, t12, t2) are solved for stationary, homogeneous plasmas without external fields. With these solutions, the collision operator is expressed in terms of the relativistic dielectric properties of the plasma.

Geometrical Effects on Turbulent Diffusion of Toroidal Plasma

Effects of toroidal geometry on the particle diffusion of a turbulent plasma are considered. Using the fluid-like equations obtained by appropriate average of the Klimontovich equation, we derived a formula for the diffusion flux across magnetic surface. The formula is applicable to both turbulent and quiescent plasmas, and reproduces the Pfirsch-Schluter diffusion in the latter case. The geometrical effect on the turbulent diffusion is found to become substantial when the ratio of the antisymmetric to symmetric part (with respect to the wavenumber parallel to the magnetic field) of the dissipative part of the susceptibility exceeds some critical value. As a specific example, we treat the case of drift wave turbulence. It is shown that the geometrical effect is insignificant unless the perpendicular wavelength is sufficiently large as compared with the ion Larmor radius.

Direct interaction approximation and plasma turbulence theory

The authors show that a generalization of the Kraichnan Direct Interaction Approximation for fluid turbulence to the Vlasov or Klimontovich equations for plasmas contains most previously proposed 'strong' turbulence theories with the addition of further ad hoc approximations. This comparison depends on an exact factorization of the DIA into screened and unscreened quantities. The usual perturbed orbit or renormalized weak turbulence corrections to the unscreened Green's function are shown to inadequately account for the self-consistent nature of the turbulent electric fields. A direct prescription for the calculation of the unscreened correlation function which determines the Fokker-Planck friction coefficient is carried through in an approximation which neglects mode coupling effects. No enhanced noise due to phase-space clumps is found; in the discrete particle theory a turbulence renormalized (broadened) expression for the discrete particle noise is obtained.

A new hierarchy system on the basis of a ?Master? Boltzmann equation for microscopic density

It is shown that Boltzmann's equation written in terms of microscopic density (namely the unaveraged Boltzmann function) has a wider range of validity as well as finer resolvability for fluctuations than the conventional Boltzmann equation governing Boltzmann's function. In fact the new Boltzmann equation for ideal gases has implications as a microscopically exact continuity equation like Klimontovich's equation for plasmas, and can be derived without invoking any statistical concepts, e.g., distribution functions, or molecular chaos. The Boltzmann equation in the older formalism is obtained by averaging this equation only under a restricted condition of the molecular chaos. The new Boltzmann equation is seen to contain information comparable with Liouville's equation, and serves as a master kinetic equation. A new hierarchy system is formulated in a certain parallelism to the BBGKY hierarchy. They are shown to yield an identical one-particle equation. The difference between the two hierarchy systems first appears in the two-particle equation. The difference is twofold. First, the present formalism includes thermal fluctuations that are missing in the BBGKY formalism. Second, the former allows us to formulate multi-time correlations as well, whereas the latter is restricted to simultaneous correlation. These two features are favorably utilized in deriving the Landau-Lifshitz fluctuation law in a most straightforward manner. Also, equations describing the nonequilibrium interaction between thermal and fluid-dynamical fluctuations are derived.

Dynamic Screening Model of the Electric Microfield Distribution

Abstract Starting from the linearized Klimontovich equations and the pair approximation we study the probability distribution of the electric microfield. We deduce the separation of the total microfield distribution into a high and a low frequency component. We further prove that each of these com-ponents can be calculated from a model of uncorrelated, dynamically screened particles. The low frequency component, which is of prominent interest for applications, is evaluated. The results show devations from all earlier theories.

Hydrodynamic Fluctuations

A set of hydrodynamic‐like equations describing the actual (fluctuating) variables for a dense system are derived from the Klimontovich equations. The equations have the same structure as the hydrodynamic equations but, in addition, have stochastic driving terms that have their origin in the pressure tensor and heat flux vector. The stochastic terms are evaluated in the dilute gas case.

An Exact Expression for Longitudinal Dielectric Permeability of Quiescent or Turbulent Plasmas

On the basis of the Klimontovich equation, the longitudinal dielectric permeability \(\tilde{\varepsilon}\) is obtained in the form

Derivations of Hierarchies for N-Particle Systems and Vlasov Systems by Means of the Functional Calculus

The methods of the functional calculus are applied to the Klimontovich equation and to the Vlasov equations. The equation for the generating functional is derived for both cases. By taking moments of these equations, the BBGKY hierarchy is obtained for the case of the Klimontovich equation, and a similar hierarchy is obtained for the case of the Vlasov equation. The two hierarchies are identical, except for some terms involving individual particle interactions. Methods of solving the Vlasov hierarchy are discussed and it is shown how quasi-linear theory can be obtained.