Abstract
In a recent paper, Maasjost and Elsässer (ME) concluded, from the results of numerical experiments and heuristic arguments, that the Bourret and the direct-interaction approximation (DIA) are “of no use in connection with the stochastic acceleration problem” because (1) their predictions were equivalent to that of the simpler Fokker-Planck (FP) theory, and (2) either all or none of the closures were in good agreement with the data. Here some analytically tractable cases are studied and used to test the accuracy of these closures. The cause of the discrepancy (2) is found to be the highly non-Gaussian nature of the force used by ME, a point not stressed by them. For the case where the force is a position-independent Ornstein-Uhlenbeck (i.e., Gaussian) process, an effective Kubo numberK can be defined. ForK≪1 an FP description is adequate, and conclusion (1) of ME follows; however, forK>1 the DIA behaves much better qualitatively than the other two closures. For the non-Gaussian stochastic force used by ME, all common approximations fail, in agreement with (2).
Similar content being viewed by others
References
P. A. Sturrock,Phys. Rev. 114:186 (1966).
D. E. Hall and P. A. Sturrock,Phys. Fluids 10:2620 (1967).
S. A. Orszag and R. H. Kraichnan,Phys. Fluids 10:1720 (1967).
S. A. Orszag, inProceedings of the Symposium on Turbulence of Fluids and Plasmas (Polytechnic Press, Brooklyn, New York), p. 17.
W. B. Thompson and J. Hubbard,Rev. Mod. Phys. 32:714 (1960).
J. Hubbard,Proc. Roy. Soc. A 260:25 (1961).
R. Kubo,J. Math. Phys. 4:174 (1963).
R. C. Bourret,Can. J. Phys. 40:782 (1962).
N. G. van Kampen,Stochastic Processes in Physics and Chemistry, Chap. XIV (North-Holland, Amsterdam, 1981);Physica 74:215 (1974).
J. B. Keller,Proc. Symp. Appl. Math. 16 (Amer. Math. Soc., Providence, 164), p. 894.
R. Kubo, inFluctuation, Relaxation, and Resonance in Magnetic Systems, D. ter Haar, ed. (Oliver and Boyd, Edinburg, 1962), p. 23.
W. Maasjost and K. Elsässer,J. Stat. Phys. 28:783 (1982).
R. H. Kraichnan,J. Math. Phys. 2:124 (1961).
J. A. Krommes, inHandbook of Plasma Physics, vol. 2, A. A. Galeev and R. N. Sudan, eds. (North-Holland, Amsterdam, 1984), p. 183 and references therein.
Author information
Authors and Affiliations
Additional information
Work supported partly by U.S. D.o.E. Contract No. DE-ACO2-76-CHO-3073 and partly by the National Science Foundation under Grant No. NSF PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration.
Rights and permissions
About this article
Cite this article
Dimits, A.M., Krommes, J.A. Stochastic particle acceleration and statistical closures. J Stat Phys 44, 879–906 (1986). https://doi.org/10.1007/BF01011912
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01011912