scholarly journals Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

2018 ◽  
Vol 84 (4) ◽  
Author(s):  
John A. Krommes
Keyword(s):  
Distribution Function ◽  
Projection Operator ◽  
Linear Response ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Transport Equations ◽  
Transport Processes ◽  
Projection Operators ◽  
Covariant Representation ◽  
Classical Transport

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

scholarly journals Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations

2018 ◽  
Vol 84 (6) ◽  
Author(s):  
John A. Krommes
Keyword(s):  
Correlation Functions ◽  
Projection Operator ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Time Correlation ◽  
Chem Phys ◽  
Magnetized Plasmas ◽  
Burnett Equations ◽  
Momentum Exchange ◽  
Time Formalism

The time-independent projection-operator formalism of Breyet al. (PhysicaA, vol. 109, 1981, pp. 425–444) for the derivation of Burnett equations is extended and considered in the context of multispecies and magnetized plasmas. The procedure provides specific formulas for transport coefficients in terms of two-time correlation functions involving both two and three phase-space points. It is shown how to calculate those correlation functions in the limit of weak coupling. The results are used to demonstrate, with the aid of a particular non-trivial example, that the Chapman–Enskog methodology employed by Catto & Simakov (CS) (Phys. Plasmas, vol. 11, 2004, pp. 90–102) to calculate the contributions to the parallel viscosity driven by temperature gradients is consistent with formulas previously derived from the two-time formalism by Brey (J. Chem. Phys., vol. 79, 1983, pp. 4585–4598). The work serves to unify previous work on plasma kinetic theory with formalism usually applied to turbulence. Additional contributions include discussions of (i) Braginskii-order interspecies momentum exchange from the point of view of two-time correlations; and (ii) a simple stochastic model, unrelated to many-body theory, that exhibits Burnett effects. Insights from that model emphasize the role of non-Gaussian statistics in the evaluation of Burnett transport coefficients, including the effects calculated by CS that stem from the nonlinear collision operator. Together, Parts 1 and 2 of this series provide an introduction to projection-operator methods that should be broadly useful in theoretical plasma physics.

Numerical Study of Nonequilibrium Diagram Theory

1972 ◽  
Vol 50 (4) ◽  
pp. 317-335 ◽  
Author(s):  
Gary R. Dowling ◽  
H. Ted Davis
Keyword(s):  
Distribution Function ◽  
Correlation Function ◽  
Kinetic Equation ◽  
Numerical Study ◽  
Pair Correlation ◽  
Transport Coefficients ◽  
Kinetic Equations ◽  
Collision Operator ◽  
Dense Fluid ◽  
Equilibrium Pair

In this paper we numerically analyze the first few diagrams in a Boltzmann-like collision operator that occurs in Severne's exact kinetic equation for the singlet distribution function. A similar analysis was used by Allen and Cole in deriving their singlet and doublet kinetic equations. Our analysis shows that the diagrams neglected by Allen and Cole in their kinetic equations are not negligible and these should be incorporated into dense fluid theories. The Allen–Cole kinetic transport coefficients and equilibrium pair correlation function are presented and calculated for dense argon. These results are not promising.

LINEAR RESPONSE THEORY: A HISTORICAL PERSPECTIVE

1993 ◽  
Vol 07 (13) ◽  
pp. 2397-2467 ◽  
Author(s):  
HUZIO NAKANO
Keyword(s):  
Linear Response ◽  
Variational Principles ◽  
Transport Coefficients ◽  
Linear Response Theory ◽  
Bloch Equation ◽  
Transport Processes ◽  
Nonequilibrium Systems ◽  
Response Theory ◽  
Reciprocity Relations ◽  
Short Period

A survey is given of the development of the linear response theory of transport processes in Japan in a short period between early 1955 and late 1956, immediately after the discovery of the formula for electrical conductivity. Although the article gives a historical account, it also provides sufficient pedagogical material for those who wish to study linear response theory. The conventional theory based on the Boltzmann-Bloch equation is also briefly reviewed for the sake of pedagogical completeness. To clarify the origin of irreversibility, variational principles of transport processes are described including Onsager’s thermostatistical theory, which are well known in conjunction with the reciprocity relations of transport coefficients. These variational principles correspond to various levels of the description of nonequilibrium systems. Contraction of microscopic to a macroscopic information transforms one variational principle into another. A reflection upon the foundation of linear response theory is given through the comparison of two different traditional theories of transport processes, Thomson’s theory of thermoelectricity and Onsager’s thermostatistical theory.

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.

scholarly journals Electron Transport in Argon - Hydrogen Mixtures

1980 ◽  
Vol 33 (6) ◽  
pp. 975 ◽  
Author(s):  
GN Haddad ◽  
RW Crompton
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Cross Section ◽  
Transport Coefficients ◽  
Velocity Distribution Function ◽  
Significant Error ◽  
Cross Section Data ◽  
Section Data ◽  
Hydrogen Mixtures ◽  
Legendre Expansion

The transport coefficients υdr and D⊥/μ have been measured in mixtures of 0.5 % and 4 % hydrogen in argon. All measurements were made at 293 K. It is shown that for these mixtures the use of the solution of the Boltzmann equation based on the two-term Legendre expansion of the velocity distribution function introduces no significant error in the analysis of the transport data. All the experimental data have been predicted to within � 3.5 % using previously published cross section data.

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).

Bootstrap-current computations for a stellarator-reactor plasma

1990 ◽  
Vol 44 (3) ◽  
pp. 431-453 ◽  
Author(s):  
W. D. D'Haeseleer ◽  
W. N. G. Hitchon ◽  
C. D. Beidler ◽  
J. L. Shohet
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Physical Interpretation ◽  
Collision Operator ◽  
Numerical Code ◽  
Parallel Flows ◽  
Bootstrap Current ◽  
The Difference ◽  
Rotational Transform ◽  
The Impact

Numerical results for the bootstrap current in a stellarator-reactor plasma are presented. The distribution function f is computed numerically from a kinetic equation that is averaged over the helical ripple. The parallel flows and the current are obtained as v‖ moments of f. The physics issues embedded in the code are discussed concisely, concentrating on the justification as to why the bootstrap current can be estimated from an averaged scheme. Results are presented for typical stellarator-reactor parameters. The numerical code FLOCS predicts that the momentum-restoring terms in the collision operator have no significant impact on the value of the bootstrap current (the difference being about 10%). The results obtained are related to the equilibrium flows, and a physical interpretation based on the kinetic picture is presented. Finally, an estimate for the impact of J‖ on the rotational transform is given.

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