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1.G. Hu and J. A. Krommes, Phys. Plasmas 1, 863 (1994).
2.In steady state (and only there), it may sometimes be adequate to interpret the ensemble average as a time average. For homogeneous statistics, one may interpret it as a spatial average. For more discussion, see Appendix A of R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975), reprinted by Krieger, Malabar, FL, 1991.
3.J. A. Krommes and G. Hu, Phys. Plasmas 1, 3211 (1994).
4.S. E. Parker and W. W. Lee, Phys. Fluids B 5, 77 (1993).
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6.Y. Chen and R. B. White, Phys. Plasmas 4, 3591 (1997).
7.Strictly speaking, there is a minor distinction between the true PDF (normalized to unity) and the conventional one-particle distribution function [normalized such that ]. This distinction is of little consequence for most of the discussion, and I will frequently refer to f as the PDF.
8.P. C. Martin, E. D. Siggia, and H. A. Rose, Phys. Rev. A 8, 423 (1973).
9.M. Kotschenreuther, W. Dorland, M. A. Beer, and G. W. Hammett, Phys. Plasmas 2, 2381 (1995).
10.D. H. E. Dubin, J. A. Krommes, C. R. Oberman, and W. W. Lee, Phys. Fluids 26, 3524 (1983).
11.It is well known that in a Fourier representation enters multiplied by the Bessel function That coefficient is not made explicit in the subsequent discussion; formally, it can be absorbed into the definition of
12.M. Kotschenreuther, in Proceedings of the 14th International Conference on the Numerical Simulation of Plasmas (Office of Naval Research, Arlington, VA, 1991), paper PT20.
13.Numerical implementation of the term in the algorithm to be described in Sec. II C requires that be evaluated at the positions of the markers. This poses somewhat of a problem; although the particle and thermal fluxes are (or should be) already available as a diagnostic on the Eulerian spatial grid, direct accumulation of the full velocity-dependent G may be noisy and time consuming. It may be adequate to represent in terms of its first few velocity-space moments, in which case G can be simply expressed in terms of possibly the momentum flux, and f. Splines can be used to interpolate the Γs to all x, or it may be adequate to simply use the mapping of the ’s onto the spatial grid; that mapping is known at each time step.
14.W. W. Lee (private communication).
15.For some further remarks about random sources, see, J. A. Krommes, Phys. Plasmas 4, 1342 (1997).
16.Strictly speaking, the sample mean is not a random variable, although it induces one. In this work I will consider the sample mean to be the induced random variable.
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17.reprinted in E. T. Jaynes, Papers on Probability, Statistics, and Statistical Physics, edited by R. D. Rosenkrantz (Kluwer Academic, Dordrecht, 1989), p. 116.
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23.Indeed, Y. Chen was unable to carry out certain small-ν simulations of toroidal Alfvén eigenmodes using the algorithm of Ref. 6 (private communication).
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26.Here “Gaussian” refers to a derivation from Gauss’s law of least constraint (Ref. 25), not to its statistics interpretation.
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30.D. C. Leslie, Developments in the Theory of Turbulence (Clarendon, Oxford, 1973).
31.In computing such parametrizations, the steady state for some W can be used as the initial condition for a subsequent run with a different W.
32.I am grateful to S. Parker and W. Lee for providing the basic code and for answering many detailed questions on gyrokinetic simulation practice.
33.This point has been emphasized in the context of the shear-flow modes by M. Kotschenreuther (private communication).
34.J. A. Krommes and C. Oberman, J. Plasma Phys. 16, 229 (1976).
35.It is not entirely clear that the only possibility is to project out the null space of the collision operator. Even in thermal equilibrium, nonlinear mode coupling generates slow modes that behave hydrodynamically (Ref. 34); this behavior persists in turbulent situations as well. The optimal way of choosing the appropriate subspace on which to apply the W stat is presently not understood.
36.C. E. Leith, J. Atmos. Sci. 28, 145 (1971).
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41.reprinted in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), p. 113.
42.N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
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