scholarly journals Evolution Equation and Transport Coefficients of Swarms in Initial Relaxation Processes

1987 ◽  
Vol 40 (3) ◽  
pp. 367 ◽  
Author(s):  
Keiichi Kondo
Keyword(s):  
Evolution Equation ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Relaxation Processes ◽  
Projection Operator Method ◽  
Model Collision ◽  
The Difference ◽  
Dependent Transport

The problem of a swarm approaching the hydrodynamic regime is studied by using the projection operator method. An evolution equation for the density and the related time-dependent transport coefficient are derived. The effects of the initial condition on the transport characteristics of a swarm are separated from the intrinsic evolution of the swarms, and the difference from the continuity equation with time-dependent transport coefficients introduced by Tagashira et al. (1977, 1978) is discussed. To illustrate this method, calculations on the relaxation model collision operator have been carried out. The results are found to agree with the analysis by Robson (1975).

Anomalous transport coefficients in a magnetized turbulent plasma

1982 ◽  
Vol 28 (2) ◽  
pp. 193-214 ◽  
Author(s):  
Qiu Xiaoming ◽  
R. Balescu
Keyword(s):  
Transport Coefficients ◽  
Collision Operator ◽  
Turbulent Plasma ◽  
Anomalous Transport ◽  
Weak Turbulence ◽  
Dissipative Effects ◽  
Dependent Transport ◽  
Two Fluid ◽  
Two Fluid Plasma

In this paper we generalize the formalism developed by Balescu and Paiva-Veretennicoff, valid for any kind of weak turbulence, for the determination of all the transport coefficients of an unmagnetized turbulent plasma, to the case of a magnetized one, and suggest a technique to avoid finding the inverse of the turbulent collision operator. The implicit plasmadynamical equations of a two-fluid plasma are presented by means of plasmadynamical variables. The anomalous transport coefficients appear in their natural places in these equations. It is shown that the necessary number of transport coefficients for describing macroscopically the magnetized turbulent plasma does not exceed the number for the unmagnetized one. The typical turbulent and gyromotion terms, representing dissipative effects peculiar to the magnetized system, which contribute to the frequency-dependent transport coefficients are clearly exhibited.

Projected evolution superoperators and the density operator: theory and applications to inelastic scattering

1982 ◽  
Vol 60 (10) ◽  
pp. 1371-1386 ◽  
Author(s):  
R. E. Turner ◽  
J. S. Dahler ◽  
R. F. Snider
Keyword(s):  
Inelastic Scattering ◽  
Density Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Interaction Picture ◽  
Von Neumann ◽  
First Order ◽  
Density Operators ◽  
Projection Operator Method ◽  
Time Dependent Solution

The projection operator method of Zwanzig and Feshbach is used to construct the time dependent density operator associated with a binary scattering event. The formula developed to describe this time dependence involves time-ordered cosine and sine projected evolution (memory) superoperators. Both Sehrödinger and interaction picture results are presented. The former is used to demonstrate the equivalence of the time dependent solution of the von Neumann equation and the more familiar, frequency dependent Laplaee transform solution. For two particular classes of projection superoperators projected density operators arc shown to be equivalent to projected wave functions. Except for these two special eases, no projected wave function analogs of projected density operators exist. Along with the decoupled-motions approximation, projected interaction picture density operators arc applied to inelastic scattering events. Simple illustrations arc provided of how this formalism is related to previously established results for two-state processes, namely, the theory of resonant transfer events, the first order Magnus approximation, and the Landau–Zener theory.

Unitary approximations to transition probabilities: Generalized impact parameter theory and the time dependent projection operator method

1984 ◽  
Vol 62 (5) ◽  
pp. 446-453
Author(s):  
Ralph Eric Turner
Keyword(s):  
Impact Parameter ◽  
Projection Operator ◽  
Approximation Scheme ◽  
Transition Probabilities ◽  
Translational Motion ◽  
Operator Method ◽  
Classical Trajectory ◽  
Time Dependent ◽  
Projection Operator Method ◽  
The Impact

A general unitary approximation scheme is presented for transition probabilities based upon the application of the time dependent version of the Zwanzig–Feshbach projection operator method to the generalized impact parameter approximation. The overall approximation scheme is based upon the time dependent picture of the binary collision process. To begin with, the translational and internal motions are decoupled with the translational motion being treated in one of three ways, namely, as a straight line trajectory, as a single curved reference classical trajectory, or as a collection of such trajectories, each of which is associated with a pair of internal states. The internal dynamics is then parameterized by the set of given trajectories which are functions of the impact parameter. This motion is then formally solved using the Zwanzig–Feshbach projection operator method. A set of unitary approximations to these solutions are then presented, based upon a time disordered approximation. These latter approximations are applicable to each of the three treatments of the translational motion. As well, the basis used to describe the internal degrees of freedom may be either atomic or molecular (diabatic or adiabatic). Thus this combined approximation scheme provides a systematic theory for testing various effects associated with either the treatment of the translational motion or the internal motion.

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