scholarly journals Comment on FTI Method and Transport Coefficient Definitions for Charged Particle Swarms in Gases

1995 ◽  
Vol 48 (4) ◽  
pp. 677 ◽  
Author(s):  
RE Robson
Keyword(s):  
Charged Particle ◽  
Path Integral ◽  
Transport Coefficients ◽  
Space Distribution ◽  
Particle Swarms ◽  
Integral Methods ◽  
Time Integral ◽  
Definition Of ◽  
Boltzmann's Equation ◽  
Boltzmann’S Equation

The kinetic theory of charged particle swarms in gases is based upon solution of the space and time dependent Boltzmann's equation for the phase space distribution function f(r, c, t). Hydrodynamic transport coefficients are defined in connection with a density gradient expansion (DGE) of f(r, c, t) and it is believed that these are the quantities measured in experiment. On the other hand, Ikuta and coworkers start with the spatially independent form of the Boltzmann equation, which they solve iteratively as in path-integral methods, and define transport coefficients in terms of the 'starting rate distribution', rather than f itself. Ikuta's procedure has come to be known as the 'flight time integral' (FTI) method and the discrepancies between numerical calculations based upon this and the more commonly known DGE procedure have generated a deal of controversy in recent times. The purpose of this paper is to point out that the respective definitions of the transverse diffusion coefficient DT coincide only for light swarm particles undergoing collisions for which the differential cross section is isotropic, and that the particular technique used for solving Boltzmann's equation, be it a path-integral or a multi-term method, has nothing to do with the numerical discrepancies which are observed when scattering is anisotropic. In particular, it is shown that Ikuta's definition of DT is inconsistent with even the well established result for constant collision frequency.

scholarly journals COORDINATE-INVARIANT PATH INTEGRAL METHODS IN CONFORMAL FIELD THEORY

2006 ◽  
Vol 21 (17) ◽  
pp. 3525-3563 ◽  
Author(s):  
ANDRÉ VAN TONDER
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Path Integral ◽  
Direct Calculation ◽  
Conformal Anomaly ◽  
Operator Formalism ◽  
Conformal Field ◽  
Integral Methods ◽  
Change Of Variables ◽  
Definition Of

We present a coordinate-invariant approach, based on a Pauli–Villars measure, to the definition of the path integral in two-dimensional conformal field theory. We discuss some advantages of this approach compared to the operator formalism and alternative path integral approaches. We show that our path integral measure is invariant under conformal transformations and field reparametrizations, in contrast to the measure used in the Fujikawa calculation, and we show the agreement, despite different origins, of the conformal anomaly in the two approaches. The natural energy–momentum in the Pauli–Villars approach is a true coordinate-invariant tensor quantity, and we discuss its nontrivial relationship to the corresponding nontensor object arising in the operator formalism, thus providing a novel explanation within a path integral context for the anomalous Ward identities of the latter. We provide a direct calculation of the nontrivial contact terms arising in expectation values of certain energy–momentum products, and we use these to perform a simple consistency check confirming the validity of the change of variables formula for the path integral. Finally, we review the relationship between the conformal anomaly and the energy–momentum two-point functions in our formalism.

Third-order transport coefficients for charged particle swarms

1999 ◽  
Vol 110 (5) ◽  
pp. 2423-2430 ◽  
Author(s):  
Slobodan B. Vrhovac ◽  
Zoran Lj. Petrović ◽  
Larry A. Viehland ◽  
Thalanayar S. Santhanam
Keyword(s):  
Charged Particle ◽  
Transport Coefficients ◽  
Third Order ◽  
Particle Swarms

scholarly journals An Exact Solution of Boltzmann's Equation for a Rigid Sphere Gas

1968 ◽  
Vol 21 (5) ◽  
pp. 543 ◽  
Author(s):  
PI Brooker ◽  
HS Green
Keyword(s):  
Transport Coefficients ◽  
Velocity Distribution Function ◽  
Zero Order ◽  
Rigid Sphere ◽  
Stream Velocity ◽  
Approximation Techniques ◽  
The Third ◽  
Boltzmann's Equation ◽  
Good Agreement ◽  
Boltzmann’S Equation

The second approximation to Boltzmann's equation, in Chapman and Cowling's form, is solved exactly for a rigid sphere gas by using appropriate transformations to reduce the integral equation to a differential equation, which is solved numerically. The values of the transport coefficients, calculated directly from this solution for the velocity distribution function, are in good agreement with those obtained from the usual approximation techniques. A solution for part of the third approximation, with a term of zero order in V, the velocity of a molecule relative to the stream velocity, is also obtained analytically.

scholarly journals An Investigation of the Accuracy of Numerical Solutions of Boltzmann's Equation for Electron Swarms in Gases with Large Inelastic Cross Sections

1979 ◽  
Vol 32 (3) ◽  
pp. 231 ◽  
Author(s):  
Ivan D Reid
Keyword(s):  
Monte Carlo Simulation ◽  
Cross Sections ◽  
Numerical Solutions ◽  
Transport Coefficients ◽  
Distribution Functions ◽  
Monte Carlo Simulation Technique ◽  
Energy Distribution Functions ◽  
Boltzmann's Equation ◽  
Term Approximation ◽  
Boltzmann’S Equation

A Monte Carlo simulation technique has been used to test the accuracy of electron energy distribution functions and transport coefficients calculated using conventional numerical solutions of Boltzmann's equation based on a two-term approximation. The tests have been applied to a number of model gases, some of which have characteristics close to those of real gases, and include cases where the scattering is anisotropic. The results show that, in general, previous application of the numerical solution to real gases has been valid.

scholarly journals Generalizations of Boltzmann's Equation

1974 ◽  
Vol 27 (6) ◽  
pp. 773 ◽  
Author(s):  
GR Anstis
Keyword(s):  
Diffusion Coefficient ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
The Self ◽  
One Dimensional ◽  
Point Particles ◽  
Self Diffusion ◽  
Self Diffusion Coefficient ◽  
Boltzmann's Equation ◽  
Boltzmann’S Equation

A scheme is proposed for systematically generalizing Boltzmann's equation in order to describe the non-equilibrium behaviour of an arbitrarily dense gas. The method avoids the divergences that arise from considering the dynamics of groups of isolated particles by introducing appropriate damping terms. Transport coefficients are obtained from the kinetic equations by using the autocorrelation formulae. For a one-dimensional gas of impenetrable point particles, approximations to the coefficient of self-diffusion may be obtained readily from the proposed generalization. A first correction to Boltzmann's equation yields the self-diffusion coefficient to within 1 % of its exact value.

scholarly journals Heat capacity estimators for random series path-integral methods by finite-difference schemes

2003 ◽  
Vol 119 (23) ◽  
pp. 12119-12128 ◽  
Author(s):  
Cristian Predescu ◽  
Dubravko Sabo ◽  
J. D. Doll ◽  
David L. Freeman
Keyword(s):  
Heat Capacity ◽  
Finite Difference ◽  
Path Integral ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Random Series ◽  
Integral Methods
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