Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations [Projection-operator methods for classical transport in magnetized plasmas. Part 1. Nonlinear response and the Burnett equations]

Krommes, John A.

doi:10.1017/s0022377818000892

Title: Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations [Projection-operator methods for classical transport in magnetized plasmas. Part 1. Nonlinear response and the Burnett equations]

Journal Article · 08 November 2018 · Journal of Plasma Physics

DOI: https://doi.org/10.1017/s0022377818000892 · OSTI ID:1558767

^[1]

Princeton Univ., Princeton, NJ (United States). Plasma Physics Lab

The time-independent projection-operator formalism 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). Here, 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.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC02-09CH11466

OSTI ID:: 1558767

Journal Information:: Journal of Plasma Physics, Journal Name: Journal of Plasma Physics Journal Issue: 6 Vol. 84; ISSN 0022-3778

Publisher:: Cambridge University PressCopyright Statement

Country of Publication:: United States

Language:: English

References (43)

Transport Equations of Plasma in a Curvilinear Magnetic Field Mikhailovskii, A. B.; Tsypin, V. S. Beiträge aus der Plasmaphysik, Vol. 24, Issue 4 https://doi.org/10.1002/ctpp.19840240404	journal	January 1984
Renormalized kinetic theory of nonequilibrium many-particle classical systems Rose, Harvey A. Journal of Statistical Physics, Vol. 20, Issue 4 https://doi.org/10.1007/BF01011780	journal	April 1979
Kinetic theory of nonlinear viscous flow in two and three dimensions Ernst, M. H.; Cichocki, B.; Dorfman, J. R. Journal of Statistical Physics, Vol. 18, Issue 3 https://doi.org/10.1007/BF01018092	journal	March 1978
Failures of the Burnett and super-Burnett equations in steady state processes Struchtrup, H. Continuum Mechanics and Thermodynamics, Vol. 17, Issue 1 https://doi.org/10.1007/s00161-004-0186-0	journal	April 2005
Nonanalytic dispersion relations in classical fluids Ernst, M. H.; Dorfman, J. R. Physica, Vol. 61, Issue 2 https://doi.org/10.1016/0031-8914(72)90065-1	journal	September 1972
Nonlinear transport equations in statistical mechanics Brey, J. J.; Zwanzig, R.; Dorfman, J. R. Physica A: Statistical Mechanics and its Applications, Vol. 109, Issue 3 https://doi.org/10.1016/0378-4371(81)90004-2	journal	December 1981
Fundamental statistical descriptions of plasma turbulence in magnetic fields Krommes, John A. Physics Reports, Vol. 360, Issue 1-4 https://doi.org/10.1016/S0370-1573(01)00066-7	journal	April 2002
Beyond the Navier–Stokes equations: Burnett hydrodynamics Garciacolin, L.; Velasco, R.; Uribe, F. Physics Reports, Vol. 465, Issue 4 https://doi.org/10.1016/j.physrep.2008.04.010	journal	August 2008
Anomalous transport due to long-lived fluctuations in plasma Part 1. A general formalism for two-time fluctuations Krommes, John A.; Oberman, Carl Journal of Plasma Physics, Vol. 16, Issue 2 https://doi.org/10.1017/S002237780002016X	journal	October 1976
Anomalous transport due to long-lived fluctuations in plasma Part 2. Hydrodynamic contributions to transport in two-dimensional, strongly magnetized systems Krommes, John A.; Oberman, Carl Journal of Plasma Physics, Vol. 16, Issue 2 https://doi.org/10.1017/S0022377800020171	journal	October 1976
A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’s Plasma Turbulence and the resonance-broadening theory of Dupree and Weinstock Krommes, John A. Journal of Plasma Physics, Vol. 81, Issue 6 https://doi.org/10.1017/S0022377815000756	journal	September 2015
Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics Krommes, John A. Journal of Plasma Physics, Vol. 84, Issue 4 https://doi.org/10.1017/S0022377818000582	journal	July 2018
An introduction to the physics of the Coulomb logarithm, with emphasis on quantum-mechanical effects Krommes, J. A. Journal of Plasma Physics, Vol. 85, Issue 1 https://doi.org/10.1017/S0022377818001319	journal	January 2019
The invariant theory of isotropic turbulence Robertson, H. P. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 36, Issue 2 https://doi.org/10.1017/S0305004100017199	journal	April 1940
Regularization of Grad’s 13 moment equations: Derivation and linear analysis Struchtrup, Henning; Torrilhon, Manuel Physics of Fluids, Vol. 15, Issue 9 https://doi.org/10.1063/1.1597472	journal	August 2003
A drift ordered short mean free path description for magnetized plasma allowing strong spatial anisotropy Catto, Peter J.; Simakov, Andrei N. Physics of Plasmas, Vol. 11, Issue 1 https://doi.org/10.1063/1.1632496	journal	January 2004
Plasma Fluctuations and Convective Modes Kent, A. Physics of Fluids, Vol. 12, Issue 1 https://doi.org/10.1063/1.1692266	journal	January 1969
Derivations of Hierarchies for N ‐Particle Systems and Vlasov Systems by Means of the Functional Calculus Nakayama, Toshio; Dawson, John M. Journal of Mathematical Physics, Vol. 8, Issue 3 https://doi.org/10.1063/1.1705231	journal	March 1967
Test Particles in a Completely Ionized Plasma Rostoker, Norman; Rosenbluth, M. N. Physics of Fluids, Vol. 3, Issue 1 https://doi.org/10.1063/1.1705998	journal	January 1960
Convergent Classical Kinetic Equation for a Plasma Frieman, E. A.; Book, D. L. Physics of Fluids, Vol. 6, Issue 12 https://doi.org/10.1063/1.1711012	journal	January 1963
Dynamics of Nonlinear Stochastic Systems Kraichnan, Robert H. Journal of Mathematical Physics, Vol. 2, Issue 1 https://doi.org/10.1063/1.1724206	journal	January 1961
Kolmogorov's Hypotheses and Eulerian Turbulence Theory Kraichnan, Robert H. Physics of Fluids, Vol. 7, Issue 11 https://doi.org/10.1063/1.2746572	journal	January 1964
Nonequilibrium gyrokinetic fluctuation theory and sampling noise in gyrokinetic particle-in-cell simulations Krommes, John A. Physics of Plasmas, Vol. 14, Issue 9 https://doi.org/10.1063/1.2759879	journal	September 2007
Plasma equilibrium in a magnetic field with stochastic regions Krommes, John A.; Reiman, Allan H. Physics of Plasmas, Vol. 16, Issue 7 https://doi.org/10.1063/1.3159607	journal	July 2009
Theory of nonlinear transport in Burnett order Wong, C. K.; McLennan, J. A.; Lindenfeld, M. The Journal of Chemical Physics, Vol. 68, Issue 4 https://doi.org/10.1063/1.435925	journal	February 1978
Long time behavior of the Burnett transport coefficients Brey, J. J. The Journal of Chemical Physics, Vol. 79, Issue 9 https://doi.org/10.1063/1.446374	journal	November 1983
Two new proofs of the test particle superposition principle of plasma kinetic theory Krommes, John A. Physics of Fluids, Vol. 19, Issue 5 https://doi.org/10.1063/1.861523	journal	January 1976
Cross-field energy transport by plasma waves Rosenbluth, Marshall N.; Liu, C. S. Physics of Fluids, Vol. 19, Issue 6 https://doi.org/10.1063/1.861547	journal	January 1976
Transport equations and gradient instabilities in a high pressure collisional plasma Mikhailovskii, A. B.; Tsypin, V. S. Plasma Physics, Vol. 13, Issue 9 https://doi.org/10.1088/0032-1028/13/9/009	journal	September 1971
Theory of the Dynamics of Simple Fluids for Large Spatial Gradients and Long Memory Piccirelli, Robert A. Physical Review, Vol. 175, Issue 1 https://doi.org/10.1103/PhysRev.175.77	journal	November 1968
The Radiation Theories of Tomonaga, Schwinger, and Feynman Dyson, F. J. Physical Review, Vol. 75, Issue 3 https://doi.org/10.1103/PhysRev.75.486	journal	February 1949
Decay of the Velocity Autocorrelation Function Alder, B. J.; Wainwright, T. E. Physical Review A, Vol. 1, Issue 1 https://doi.org/10.1103/PhysRevA.1.18	journal	January 1970
Theory of Nonlinear Transport Processes: Nonlinear Shear Viscosity and Normal Stress Effects Kawasaki, Kyozi; Gunton, James D. Physical Review A, Vol. 8, Issue 4 https://doi.org/10.1103/PhysRevA.8.2048	journal	October 1973
Statistical Dynamics of Classical Systems Martin, P. C.; Siggia, E. D.; Rose, H. A. Physical Review A, Vol. 8, Issue 1 https://doi.org/10.1103/PhysRevA.8.423	journal	July 1973
On the Theory of the Brownian Motion II Wang, Ming Chen; Uhlenbeck, G. E. Reviews of Modern Physics, Vol. 17, Issue 2-3 https://doi.org/10.1103/RevModPhys.17.323	journal	April 1945
The Distribution of Velocities in a Slightly Non-Uniform Gas Burnett, D. Proceedings of the London Mathematical Society, Vol. s2-39, Issue 1 https://doi.org/10.1112/plms/s2-39.1.385	journal	January 1935
The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas Burnett, D. Proceedings of the London Mathematical Society, Vol. s2-40, Issue 1 https://doi.org/10.1112/plms/s2-40.1.382	journal	January 1936
Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems Kubo, Ryogo Journal of the Physical Society of Japan, Vol. 12, Issue 6 https://doi.org/10.1143/JPSJ.12.570	journal	June 1957
Transport, Collective Motion, and Brownian Motion Mori, Hazime Progress of Theoretical Physics, Vol. 33, Issue 3 https://doi.org/10.1143/PTP.33.423	journal	March 1965
Plasma Kinetic Theory Weyssow, B. Fusion Science and Technology, Vol. 53, Issue 2T https://doi.org/10.13182/fst08-a1692	journal	February 2008
Plasma Kinetic Theory Weyssow, B. Fusion Science and Technology, Vol. 57, Issue 2T https://doi.org/10.13182/fst10-a9397	journal	February 2010
Plasma Kinetic Theory Weyssow, B. Fusion Science and Technology, Vol. 61, Issue 2T https://doi.org/10.13182/fst12-a13494	journal	February 2012
An introduction to the physics of the Coulomb logarithm with emphasis on quantum-mechanical effects Krommes, J. A. arXiv https://doi.org/10.48550/arxiv.1806.04990	text	January 2018

Cited By (1)

Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics Krommes, John A. Journal of Plasma Physics, Vol. 84, Issue 4 https://doi.org/10.1017/s0022377818000582	journal	July 2018

Similar Records

Nonlinear viscosity and its role in drift-Alfven modes

Journal Article · 2005 · Physics of Plasmas · OSTI ID:20782394

Tsypin, V S; Mikhailovskii, A B; Shirokov, M S; +3 more

Comment on 'On higher order corrections to gyrokinetic Vlasov-Poisson equations in the long wavelength limit' [Phys. Plasmas 16, 044506 (2009)]

Journal Article · 2009 · Physics of Plasmas · OSTI ID:21371282

Parra, Felix I; Catto, Peter J

Closure and transport theory for high-collisionality electron-ion plasmas

Journal Article · 2013 · Physics of Plasmas · OSTI ID:22130425

Ji, Jeong-Young; Held, Eric D.

Related Subjects

70 PLASMA PHYSICS AND FUSION TECHNOLOGY
plasma nonlinear phenomena

Title: Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations [Projection-operator methods for classical transport in magnetized plasmas. Part 1. Nonlinear response and the Burnett equations]

Citation Formats

References (43)

Cited By (1)

Similar Records

Related Subjects