Kaya Imre

WIGNER METHOD IN QUANTUM STATISTICAL MECHANICS.

The Wigner method of transforming quantum‐mechanical operators into their phase‐space analogs is reviewed with applications to scattering theory, as well as to descriptions of the equilibrium and dynamical states of many‐particle systems. Inclusion of exchange effects is discussed.

OSCILLATIONS IN A RELATIVISTIC PLASMA

The linear oscillations in a hot plasma which is representable by the relativistic Vlasov equation with the self‐consistent fields are investigated. The generalization of Bernsteins method for the relativistic case is used to obtain the formal solution of the linearized problem. Particular attention is given to the case when the system initially is in the relativistic equilibrium state. The dispersion equation is derived and studied for the case when the propagation is along the direction of the unperturbed magnetic field, considering the spatial dispersions explicitly. The asymptotic expansions are developed corresponding to the dispersion relations of the cases studied. It is found that transverse waves propagating along the unperturbed field are Landau damped if ν2 ≥ 1 − Ω2/ω2, ν and Ω being the index of refraction and the gyrofrequency, respectively. In the absence of the external field the cutoff frequency, which is found to be the same for both longitudinal and the transverse modes, is shown to be ...

Comments on “Stability limits for longitudinal waves in ion beam‐plasma interactions”

It is shown that electron dynamics significantly alters the stability limits of the ion‐acoustic waves in a plasma‐ion beam system even when phase velocity is much smaller than the electron thermal velocity.

Correlationless Plasma with Time‐Dependent Background State

A Volterra type (second kind) integral equation is derived for the electric charge density starting from the Vlasov equation linearized with respect to a time‐dependent spatially uniform background state. The kernel of this integral equation vanishes identically in the case when the background state is time‐independent (i.e., in the usual Landau problem), and the inhomogeneous term reduces to the exact solution of the initial value problem in the same case. It is shown that the inhomogeneous term of this integral equation corresponds to the adiabatic approximation; hence, a Neumann series type solution can be constructed for the charge density by iteration of this inhomogeneous term for the case when the background state is varying slowly in time. Approximate solutions of this integral equation are obtained for the small time and asymptotic limits. It appears that for some particular cases it is possible to have nondecaying solutions for the continuum mode. This is in contrast with one of the basic assumptions of the quasi‐linear theory of plasmas.

Correlations in plasmas: I. Ternary correlations

The kinetic stage of an interacting, multi-component system is studied using Bogoliubovs formalism including terms up to the second orde formalism including terms up to the second order in interaction strength λ, but to all orders in nλ, n being the density. It is shown that the second order s-particle distribution function can be expressed in terms of zero and first order binaiy and zero order ternary correlation functionals. This functional form is also shown to be deducible in a prescribed manner from the so-called cluster expansion series of the equilibrium theory. The latter observation provides a means for guessing the structures of the higher order distribution functionals. The equations satisfied by the first order binary and zero order ternary correlation functionals are derived. They constitute a coupled set of equations determining the second order contribution to the kinetic equation. Explicit expressions are found for the correlation functions when the one-particle distribution function is Maxwellian. The results, in this case, are in agreement with those obtained from the equilibrium theory.

Correlations in plasmas: II. Correlation functional representation of the Bogoliubov formalism

The distribution functionals that are encountered in the Bogoliubov formalism for systems in the kinetic stage are replaced by the correlation functionals in a manner similar to the ones that are familiar in the cluster expansion series. The equation satisfied by these functionals is derived for the case when the interaction potential is binary. It is shown that the analytic solution of this equation subject to Bogoliubovs initial condition is of order λs−1 where λ is the interaction parameter and s is the number of particles involved in the correlation functional under consideration. This property enables one to break the hierarchy even if one keeps the terms of all orders in nλ, n being the density, as is the usual procedure for plasmas.

Neutron diffusion in a randomly inhomogeneous multiplying medium with random phase approximation

Neutron diffusion in a randomly inhomogeneous multiplying medium is studied. By making use of a random phase assumption we show that the average neutron density approximately satisfies an integral equation in Fourier space, which is solved using Kummer functions. We used multi-dimensional formulation. In the case of one dimension, we obtain the result of Rosenbluth and Tao for the mean total density for large t. In the three-dimensional case, a closed form of solution is derived for the mean total neutron density. Its asymptotic behavior is also investigated for large t.

PROFILE SHAPE PARAMETERIZATION OF JET ELECTRON TEMPERATURE AND DENSITY PROFILES

The temperature and density profiles of the Joint European Torus (JET) are parametrized using log additive models in the control variables. Predictive error criteria are used to determine which terms in the log linear model to include. The density and temperature profiles are normalized to their line averages (n and T). The normalized ohmic density shape depends primarily on the parameter n/Bt, where Bt is the toroidal magnetic field. Both the low mode (L mode) and the edge localized mode-free (ELM-free) high mode (H mode) temperature profile shapes depend strongly on the type of heating power, with ion cyclotron resonant heating (ICRH) producing a more peaked profile than neutral beam injection (NBI). Given the heating type dependence, the L mode temperature shape is nearly independent of the other control variables. The H mode temperature shape broadens as the effective charge, Zeff, increases. The line average L mode temperature scales as Bt0.96 (power per particle)0.385. The L mode normalized density shape depends primarily on the ratio of line average density, n, to the edge safety factor, q95. As n/q95 increases, the profile shape broadens. The current, Ip, is the most important control variable for the normalized H mode density. As the current increases, the profile broadens and the gradient at the edge sharpens. Increasing the heating power, especially ICRH, or decreasing the average density, peaks the H mode density profile slightly

On the Algebraic Structure of the Cluster Expansion in Statistical Mechanics

The structure of cluster expansion which is widely used in statistical mechanics is studied from an algebraic point of view. In doing this, a commutative algebra is constructed which is generated by partitions of a finite set by regarding them as operators which divide the set into disjoint parts. Physically, these operators correspond to operations which remove interaction among certain clusters of particles. It is shown that the cluster expansion stems from the relation between two basis sets of this algebra; the first set is the set of all partitions and the second is the set of pairwise orthogonal minimal idempotents. This property enables one to demonstrate the equivalence of the product versus cluster properties of the distribution and correlation functions, respectively, in general terms. This is done by constructing a simple representation space for the partition algebra corresponding to the distribution functions. A second application of the partition algebra is considered in the case when correl...

Correlations in plasmas: III. Green's function solutions for correlation functionals in Bogoliubov theory

We use a set of eigendistributions in constructing the formal solution of tho set of integrodifferential equations for the correlation functionals in the context of the Bogoliubov formalism. The equations apply to fully ionized systems interacting through the Coulomb potential, and the method is displayed to an arbitrary order in the interaction strength for the homogeneous case. It is indicated that the method is equally applicable to the slightly inhomogeneous systems for which a perturbation series solution can be developed formally. It is also possible to apply this method in solving the initial-value problem of the BBGKY (Bogoliubov, Born, Green, Kirkwood, Yvon) hierarchy. The eigendistributions employed are those developed by Van Kampen and Case in connection with the linearized Vlasov equation. Although the resulting Greens function is quite complicated, it exhibits the general structure of the solutions.