Generalized Fokker-Planck Equation Research Articles

High-Frequency Conductivity of a Plasma

The high-frequency conductivity of a fully ionized plasma is calculated from the Bogoliubov—Born—Green—Kirkwood—Yvon equations for a plasma subjected to an applied field. These equations are used in the plasma limit but with the further approximation that those terms which lead to a screened Coulomb potential are replaced by a Coulomb potential with a range of the Debye length. This leads to a generalized Fokker—Planck equation, i.e., the collisional term has a more general space and time dependence than the usual Fokker—Planck equation. The equation is linearized about equilibrium and is solved by assuming that the effect of the field dominates the effect of the collisions. The effects of electron—electron as well as electron—ion collisions are considered and the results are calculated to second order in the wavenumber.

On the Structure of Generalized Fokker-Planck Equation of a High Temperature Plasma

The structures of the generalized Fokker-Planck equation derived in the previous pub­ lication (quoted as I) are examined in detail by considering the motion of electron beam in a high temperature electron plasma. The dynamical shielding factor provides the basis of a unified theory of the plasma oscillations. and the static shielding of the particles inter­ action in the plasma. The friction and diffusion coefficient are cakulated by taking into consideration the non-local space-time correlation effect. It is shown that the non-local correlation effect substantially modifies the contributions of the plasmon emission process. Rough estimation of the friction coefficient shows that the non-local correlation effect may increase the amount of the frictional drag at higher velocity. This effect may be essential to resolve the Langmuir paradox.

On the Kinetic Equation for a High Temperature Plasma

Effects of binary and ternary correlations in a high temperature plasma are examined in detail on the basis of the so-called B-B-G-K-Y equation for a system of charged particles. It is shown that the effects of ternary correlation are essential in giving rise to shielding of the interaction between particles and have important influences upon disturbances of the binary correlations. The effects of binary correlation can be divided into the correction terms of the self-consistent field of the Boltzmann-Vlasov equation and the terms representing the collision effects between particles through the shielded interaction. The effects of the collision terms are investigated in detail by deriving a generalized Fokker-Planck equation which can be reduced to the equation derived by Tchen for a special case. The effects of the correction terms of the self-consistent field are examined by deriving an equation of motion of the density fluctuation which describes the longitudinal plasma oscillations. The correction terms give rise to a shift of the κ2-term of the plasma frequency and a damping factor. The frequency of the plasma oscillations is determined to be given by ω2 = ωp2+(1+δ)(3κT/m)k2 where the shift due to the binary correlations δ is given as follows, δ= 1.05{(e2/κT)3n}1/2. The damping factor γ is determined to be γ= 0.028{(e2/κT)3n}1/2(k/kd)2·ωp at the limit of k →0. It is shown that the damping factor due to the long-range binary correlations predominates over the Landau damping in the range of small value of k. On the basis of present investigation, it is concluded that the equation derived by Balescu does not involve any information concerning the plasma oscillations. Since the plasma oscillations are nothing but the appearance of periodical spatial inhomogeneity in the system, it is evident that the assumption of the spatial homogeneity introduced by Temko, Tchen and Balescu in their derivations of the fundamental equations rules out the possible occurrence of the plasma oscillation.

On the transport coefficients in the high temparature plasma (I)

The frictional force and diffusion coefficients are calculated for high temperature plasma on the basis of the generalized Fokker-Planck equation which was derived by one of the authors (Y. H. I.). The collective effects, which are very essential in the system of charged particles, are accounted by the dynamical shielding factor in a very natural way. The contributions of the excitat ion of plasma wave to the transport coefficients appear as the resonance occuring in the medium. The results reduce to Bohm-Pines' and Klimontovich's formula for the special case of a bean moving with uniform velocity.

Kinetic Equation for a Plasma with Unsteady Correlations

As a generalization of the Boltzmann equation, the kinetic equation for a plasma is derived in the form of a generalized Fokker-Planck equation, by considering unsteady correlations, including non-Markovian and nonlinear behavior. Both the binary and ternary correlations are used for many kinds of particles with different temperatures. The coefficients of the kinetic equation depend on the law of interaction for a pair of particles and are influenced by relaxation. The effective potential of friction consists of two parts: the static part corresponds to the Debye potential and is isotropic, the dynamical part is axially symmetrical about the direction of motion, and causes a dynamical friction. The results show that the friction is proportional to velocity for slow particles, and inversely proportional to the square of velocity for fast particles. This tendency of the fast particles to overcome repulsion is a property connected with the "run-away" of electrons. A criterion for maximum friction is derived. The triplet interaction, which mainly affects the shielding phenomena, assures the convergence of the coefficients in case of distant interaction. Since the length scales of interaction are well determined in this way, the kinetic equation can be expected to be valid over a longer range than does the Boltzmann equation. The large scale agrees with the Debye radius, when the shielding term is linearized, as should be expected. When time relaxation is left aside and linearization is made, the kinetic equation degenerates to the classical Fokker-Planck equation with convergent coefficients.

Sur l'extension de la thermodynamique aux phénomènes irreversibles liés aux degrés de liberté internes

Synopsis The thermodynamics of irreversible processes is applied to systems with internal degrees of freedom. By introducing an internal configuration space and assuming continuous motion therein, the second principle of thermodynamics can be stated in the following way: the entropy creation per unit volume of the internal configuration space is essentially positive definite. Explicit evaluation of this entropy creation then at once leads to differential equations of diffusion type for the motion in configuration space. For ideally dilute systems, these equations reduce to generalized FokkerPlanck equations. As a consequence of Onsager reciprocal relations i't is shown that the phenomenological friction tensor in configuration space must be symmetric. The formalism thus obtained can be applied to a great variety of irreversible phenomena related to internal degrees of freedom. Finally some very simple applications are discussed as an illustration.

The Excluded Volume of Polymer Chains

The ``random flight'' distribution function for the relative positions of two segments of a linear polymer chain is based on the assumption that the space occupied by a single polymer segment is relatively small and may be neglected. In this paper, the space requirements of a real polymer chain are taken into account. A generalized Fokker-Planck equation is formulated, and the distribution function, the average square distance, and the average inverse distance between two chain segments are calculated.

Brownian Motion in a Gas of Noninteracting Molecules

The Brownian motion of a hard spherical particle in a gas whose molecules interact only with the particle but not with each other is considered from three points of view. (a) The generalized Fokker-Planck equation for the process is derived by the classical method of Lord Rayleigh. (b) The hypothesis of the phenomenonlogical theory of Brownian motion that the distribution of changes of momentum of the particle is normal is investigated with some mathematical rigor. (c) The formula for the friction constant recently given by Kirkwood is applied to this case. It is found that the plateau value in this formula exists in this case and that the value obtained agrees with Rayleigh's method. The hypothesis of the phenomenonological theory is found to be justified for times long enough for many collisions to have taken place. The theoretical formula for the friction constant is compared with Millikan's data for the friction constant for oil droplets in air at low pressures. It is found to agree with Millikan's data with respect to the form of the dependence on density and temperature. The theoretical values for hard spheres are somewhat lower than the data for oil droplets.