Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T07:36:29.211Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  09 November 2018

John A. Krommes Open the ORCID record for John A. Krommes [Opens in a new window]
John A. Krommes*
Affiliation:
Princeton University, Plasma Physics Laboratory, P.O. Box 451, MS 28, Princeton, NJ 08543–0451, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The time-independent projection-operator formalism of Brey et al. (Physica A, 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.

Keywords

plasma nonlinear phenomena
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 6 , December 2018 , 905840601
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alder, B. J. & Wainwright, T. E. 1970 Decay of the velocity autocorrelation function. Phys. Rev. A 1, 18–21.Google Scholar
Balescu, R. 1975 Equilibrium and Nonequilibrium Statistical Mechanics. Wiley (reprinted by Krieger Publishing Co., 1991).Google Scholar
Birdsall, C. K. & Langdon, A. B. 1985 Plasma Physics via Computer Simulation. McGraw-Hill.Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. In Reviews of Plasma Physics (ed. Leontovich, M. N.), vol. 1, pp. 205–311. Consultants Bureau.Google Scholar
Brey, J. J. 1983 Long time behavior of the Burnett transport coefficients. J. Chem. Phys. 79, 4585–4598.Google Scholar
Brey, J. J., Zwanzig, R. & Dorfman, J. R. 1981 Nonlinear transport equations in statistical mechanics. Physica A 109, 425–444.Google Scholar
Burnett, D. 1935 The distribution of velocities in a slightly non-uniform gas. Proc. Lond. Math. Soc. 39, 385–430.Google Scholar
Burnett, D. 1936 The distribution of molecular velocities and the mean motion in a non-uniform gas. Proc. Lond. Math. Soc. 40, 382–435.Google Scholar
Catto, P. J. & Simakov, A. N. 2004 A drift ordered short mean free path description for magnetized plasma allowing strong spatial anisotropy. Phys. Plasmas 11, 90–102.Google Scholar
Catto, P. J. & Simakov, A. N. 2005 A new, explicitly collisional contribution to the gyroviscosity and the radial electric field in a collisional tokamak. Phys. Plasmas 12, 114503 (4 pages); 13, 129901(E).Google Scholar
Dawson, J. & Nakayama, T. 1967 Derivations of hierarchies for $N$ -particle systems and Vlasov systems by means of the functional calculus. J. Math. Phys. 8, 553–560.Google Scholar
Dyson, F. J. 1949 The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75, 486–502.Google Scholar
Ernst, M. H., Cichocki, B., Dorfman, J. R., Sharma, J. & van Beijeren, H. 1978 Kinetic theory of nonlinear viscous flow in two and three dimensions. J. Stat. Phys. 18, 237–270.Google Scholar
Ernst, M. H. & Dorfman, J. R. 1972 Nonanalytic dispersion relations in classical fluids: I. The hard-sphere gas. Physica (Utrecht) 61, 157–181.Google Scholar
Frieman, E. A. & Book, D. L. 1963 Convergent classical kinetic equation for a plasma. Phys. Fluids 6, 1700–1706.Google Scholar
García-Colín, L. S., Velasco, R. M. & Uribe, F. J. 2008 Beyond the Navier–Stokes equations: Burnett hydrodynamics. Phys. Rep. 465, 149–189.Google Scholar
Gorban, A. N. & Karlin, I. 2014 Hilbert’s 6th problem: exact and approximate hydroynamic manifold for kinetic equations. Bull. Am. Math. Soc. 51 (2), 186–246.Google Scholar
Grad, H. 1958 Principles of the kinetic theory of gases. In Handbuch der Physik, vol. XII, pp. 205–294. Springer.Google Scholar
Ichimaru, S. 1973 Basic Principles of Plasma Physics – A Statistical Approach. Benjamin.Google Scholar
Kawasaki, K. & Gunton, J. D. 1973 Theory of nonlinear transport processes: nonlinear shear viscosity and normal stress effects. Phys. Rev. A 8, 2048–2064.Google Scholar
Kent, A. & Taylor, J. B. 1969 Plasma fluctuations and convective modes. Phys. Fluids 12, 209–212.Google Scholar
Klimontovich, Y. L. 1967 The Statistical Theory of Non-equilibrium Processes in a Plasma (ed. ter Harr, D.). MIT Press (translated by H. S. H. Massey and O. M. Blunn).Google Scholar
Kraichnan, R. H. 1961 Dynamics of nonlinear stochastic systems. J. Math. Phys. 2, 124–148; 3, 205(E).Google Scholar
Kraichnan, R. H. 1962 The closure problem of turbulence theory. Proc. Symp. Appl. Math. 13, 199–225.Google Scholar
Kraichnan, R. H. 1964 Kolmogorov’s hypotheses and Eulerian turbulence theory. Phys. Fluids 7, 1723–1734.Google Scholar
Krommes, J. A.1975 On renormalized kinetic theories of anomalous transport due to hydrodynamic fluctuations in strongly magnetized plasma. PhD thesis, Princeton University.Google Scholar
Krommes, J. A. 1976 Two new proofs of the test particle superposition principle of plasma kinetic theory. Phys. Fluids 19, 649–655.Google Scholar
Krommes, J. A. 1984 Statistical descriptions and plasma physics. In Handbook of Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), vol. 2, chap. 5.5, pp. 183–268. North-Holland.Google Scholar
Krommes, J. A. 2002 Fundamental statistical theories of plasma turbulence in magnetic fields. Phys. Rep. 360, 1–352.Google Scholar
Krommes, J. A. 2007 Nonequilibrium gyrokinetic fluctuation theory and sampling noise in gyrokinetic particle-in-cell simulations. Phys. Plasmas 14, 090501 (26 pages).Google Scholar
Krommes, J. A. 2015 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. J. Plasma Phys. 81, 205810601 (80 pages).Google Scholar
Krommes, J. A.2018a An introduction to the physics of the Coulomb logarithm, with emphasis on quantum-mechanical effects. J. Plasma Phys., arXiv:1806.04990 (8 pages).Google Scholar
Krommes, J. A. 2018b Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics. J. Plasma Phys. 84, 925840401 (80 pages).Google Scholar
Krommes, J. A. 2018c Projection-operator methods for classical transport in magnetized plasmas. II. Nonlinear response and the Burnett equations – Supplement. J. Plasma Phys. doi:10.1017/S0022377818000892.Google Scholar
Krommes, J. A. & Oberman, C. 1976a Anomalous transport due to long-lived fluctuations in plasma. Part I. A general formalism for two-time fluctuations. J. Plasma Phys. 16, 193–227.Google Scholar
Krommes, J. A. & Oberman, C. 1976b Anomalous transport due to long-lived fluctuations in plasma. Part II. Hydrodynamic contributions to transport in two-dimensional, strongly magnetized systems. J. Plasma Phys. 16, 229–260.Google Scholar
Krommes, J. A. & Parker, J. B. 2018 Statistical closures and zonal flows. In Zonal Jets (ed. Galperin, B. & Read, P.), chap. V.1.1. Cambridge University Press (in press).Google Scholar
Krommes, J. A. & Reiman, A. H. 2009 Plasma equilibrium in a magnetic field with stochastic regions. Phys. Plasmas 16, 072308 (26 pages).Google Scholar
Kubo, R. 1957 Statistical–mechanical theory of irreversible processses. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan 12, 570–586.Google Scholar
Kubo, R. 1959 Some aspects of the statistical–mechanical theory of irreversible processes. In Lectures in Theoretical Physics (ed. Brittin, W. E. & Dunham, L. G.), vol. I, pp. 120–203. Interscience.Google Scholar
Kubo, R. 1974 Response, relaxation, and fluctuations. In Lecture Notes in Physics: Transport Phenomena (ed. Kirczenow, G. & Marro, J.), vol. 31, pp. 24–124. Springer.Google Scholar
Leslie, D. C. 1973 Developments in the Theory of Turbulence. Clarendon.Google Scholar
Martin, P. C., Siggia, E. D. & Rose, H. A. 1973 Statistical dynamics of classical systems. Phys. Rev. A 8, 423–437.Google Scholar
Mathews, J. & Walker, R. L. 1970 Mathematical Methods of Physics. Benjamin.Google Scholar
McComb, W. D. 1990 The Physics of Fluid Turbulence. Clarendon.Google Scholar
McComb, W. D. 2014 Homogeneous, Isotropic Turbulence. Oxford University Press.Google Scholar
Mikhaǐlovskiǐ, A. B. 1967 Macroscropic description of a collision[al] plasma in a strong magnetic field for stability problems. Zh. Eksp. Teor. Fiz. 52, 943–954; [Sov. Phys. JETP 25 623–630 (1967)].Google Scholar
Mikhaǐlovskiǐ, A. B. & Tsypin, V. S. 1971 Transport equations and gradient instabilities in a high pressure collisional plasma. Plasma Phys. 13, 785–798.Google Scholar
Mikhaǐlovskiǐ, A. B. & Tsypin, V. S. 1984 Transport equations of plasma in a curvilinear magnetic field. Beitr. Plasmaphys. 24, 335–354.Google Scholar
Montgomery, D. C. & Tidman, D. A. 1964 Plasma Kinetic Theory. McGraw-Hill.Google Scholar
Mori, H. 1965 Transport, collective motion, and Brownian motion. Progr. Theoret. Phys. 33, 423–455.Google Scholar
Nemerov, H.1967 Projection. The Atlantic Monthly 219 (5), 87.Google Scholar
Piccirelli, R. A. 1968 Theory of the dynamics of simple fluids for large spatial gradients and long memory. Phys. Rev. 175, 77–98.Google Scholar
Reichl, L. E. 1998 A Modern Course in Statistical Physics, 2nd edn. Wiley (The section on ‘Beyond the Boltzmann Equation’ was radically shortened in the third and fourth editions).Google Scholar
Robertson, H. P. 1940 The invariant theory of isotropic turbulence. Math. Proc. Camb. Phil. Soc. 36, 209–223.Google Scholar
Rose, H. A. 1979 Renormalized kinetic theory of nonequilibrium many-particle classical systems. J. Stat. Phys. 20, 415–447.Google Scholar
Rosenbluth, M. N. & Liu, C. S. 1976 Cross-field energy transport by plasma waves. Phys. Fluids 19, 815–818.Google Scholar
Rostoker, N. & Rosenbluth, M. N. 1960 Test particles in a completely ionized plasma. Phys. Fluids 3, 1–14.Google Scholar
Rudnick, J. & Gaspari, G. 2004 Elements of the Random Walk: An Introduction for Advanced Students and Researchers. Cambridge University Press.Google Scholar
Struchtrup, H. 2005 Failures of the Burnett and super-Burnett equations in steady state processes. Contin. Mech. Thermodyn. 17, 43–50.Google Scholar
Struchtrup, H. & Torrolhon, M. 2003 Regularization of Grad’s 13 moment equations: derivation and linear analysis. Phys. Fluids 15, 2668–2680.Google Scholar
Swanson, D. G. 2008 Plasma Kinetic Theory. CRC Press.Google Scholar
Taylor, G. I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. (2) 20, 196–212; reprinted in Turbulence: Classic Papers on Statistical Theory (ed. S. K. Friedlander and L. Topper) (Interscience, 1961), pp. 1–17.Google Scholar
Wang, M. C. & Uhlenbeck, G. E. 1945 On the theory of the Brownian motion II. Rev. Mod. Phys. 17, 323–342; reprinted in Selected Papers on Noise and Stochastic Processes (ed. N. Wax) (Dover, 1954), pp. 113–132.Google Scholar
Williams, E. A.1973 On the theory of fluctuations in plasma. PhD thesis, Princeton University.Google Scholar
Wong, C. K., MacLennan, J. A., Lindenfeld, M. & Dufty, J. W. 1978 Theory of nonlinear transport in Burnett order. J. Chem. Phys. 68, 1563–1753.Google Scholar
Zwanzig, R. 2001 Nonequilibrium Statistical Mechanics. Oxford University Press.Google Scholar
Supplementary material: File

Krommes supplementary material

Krommes supplementary material 1

Download Krommes supplementary material(File)
File 226.4 KB