Abstract
The investigation of the dynamics of individual (noninteracting) charged particles in a magnetic trap is probably the simplest of the problems of prolonged plasma containment for controlled thermonuclear fusion. Nonetheless, even this “simple” problem is quite rich in content and is far from completely solved, notwithstanding many years’ efforts in this direction (see, e.g., [2–4]). In addition, the dynamics of an individual particle is an integral part of the more complicated problem of collective processes in a plasma. Finally, the problem of containing a single particle in a magnetic trap must be faced every time when a new scheme or a substantial modification of an old method of magnetic confinement of a plasma appears.* An example is Dimov’s ambipolar (tandem mirror) trap [7]. It is one of the so-called open systems of plasma confinement, or traps with “magnetic mirrors,” which will be discussed below. We shall refer to them for brevity simply as traps.
Keywords
Magnetic Trap Stochastic Component Unperturbed System Magnetic Line Nonlinear ResonancePreview
Unable to display preview. Download preview PDF.
References
- 1.G. I. Budker, “Thermonuclear reactions in a system with magnetic mirrors. Contribution to the problem of direct conversion of nuclear energy into electricity,” in: Collected Works [in Russian], G. I. Budker (ed.), Nauka, Moscow (1982), p. 72.Google Scholar
- 2.B. V. Chirikov, “The problem of stability of motion of a charged magnetic trap,” Fiz. Plazmy, 4, No. 3, 521 (1978).Google Scholar
- 3.R. H. Cohen, “Orbital resonances in nonaxisymmetric mirror machines,” Comments Plasma Phys. Controlled Fusion, 4, No. 6, 157 (1979).ADSGoogle Scholar
- 4.B. V. Chirikov, “Adiabatic invariants and stochasticity in magnetic confinement systems,” Proceedings Intern. Conf. on Plasma Physics, Nagoya (1980), Vol. II, p. 176.Google Scholar
- 5.R. H. Cohen, G. Rowlands, and J. H. Foote, “Non-adiabaticity in mirror machines,” Phys. Fluids, 21, No. 4, 627 (1978).ADSCrossRefGoogle Scholar
- 6.B. V. Chirikov, “Homogeneous model of resonant diffu sion of particles in an open magnetic trap,” Fiz. Plazmy, 5, No. 4, 880 (1979).Google Scholar
- 7.G. I. Dimov, V. V. Zakaidakov, and M. E. Kishinevskii, “Thermonuclear trap with tandem mirrors,” Fiz. Plazmy, 2, No. 4, 597 (1976).Google Scholar
- 7a.T. K. Fowler and B. G. Logan, “The tandem mirror reactor,” Comments Plasma Phys. Controlled Fusion, 2, No. 6, 167 (1977).Google Scholar
- 8.B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,” Phys. Rep., 52, No. 5, 263 (1979).MathSciNetADSCrossRefGoogle Scholar
- 9.A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer, Berlin (1983).MATHGoogle Scholar
- 10.V. I. Arnol’d, “Small denominators and the problem of stable motion in classical and celestial mechanics,” Usp. Mat. Nauk, 18, No. 6, 91 (1963).Google Scholar
- 11.G. I. Budker, V. V. Mirnov, and D. D. Ryutov, “Influence of magnetic-field corrugation on expansion and cooling of a dense plasma,” Pis’ma Zh. Eksp. Teor. Fiz., 14, No. 5, 320 (1971).Google Scholar
- 12.N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).MATHGoogle Scholar
- 13.R. W. B. Best, “On the motion of charged particles in a slightly damped sinusoidal potential wave,” Physica, 40, No. 2, 182 (1968).MathSciNetADSCrossRefGoogle Scholar
- 14.A. S. Bakai and Yu. P. Stepanovskii, Adiabatic Invariants [in Russian], Naukova Dumka, Kiev (1981).Google Scholar
- 15.J. E. Howard, “Nonadiabatic particle motion in cusped magnetic fields,” Phys. Fluids, 14, No. 11, 2378 (1971).ADSCrossRefGoogle Scholar
- 16.A. M. Dykhne and A. V. Chaplik, “Change of adiabatic invariant of a particle in a magnetic field,” Zh. Eksp. Teor. Fiz., 40, No. 2, 666 (1961).Google Scholar
- 17.M. Kruskal, Adiabatic Invariants [Russian translation], Inostr. Lit., Moscow (1962).Google Scholar
- 18.C. S. Gardner, “Magnetic moment in second order for axisymmetric static field,” Phys. Fluids, 9, No. 10, 1997 (1966).ADSCrossRefGoogle Scholar
- 19.O. B. Firsov, “Repulsion of charged particle from regions with strong magnetic field. (Accuracy of adiabatic invariant),” in: Plasma Physics and the Problem of Controlled Thermonuclear Reactions [in Russian], Izd. Akad. Nauk SSSR, Moscow (1958), Vol. III, p. 259.Google Scholar
- 20.R. J. Hastie, G. D. Hobbs, and J. B. Taylor, “Non-adiabatic behavior of inhomogeneous magnetic fields,” Plasma Physics and Controlled Thermonuclear Fusion Research, IAEA (1969), Vol. I, p. 389.Google Scholar
- 21.E. M. Krushkal’, “Nonadiabatic motion of particles in nonuniform magnetic fields,” Zh. Tekh. Fiz., 42, No. 11, 2288 (1972).Google Scholar
- 22.I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Academic Press, New York (1965).Google Scholar
- 23.J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys., 20, No. 6, 1183 (1979).ADSCrossRefGoogle Scholar
- 24.F. M. Izrailev, B. V. Chirikov, and D. L. Shepelyanskii, “Dynamic stochasticity in classical mechanics,” Preprint Inst. Nucl Phys., Siberian Branch, Acad. Sei. USSR, No. 80–209, Novosibirsk (1980).Google Scholar
- 25.M. N. Rosenbluth, “Superadiabaticity in mirror machines,” Phys. Rev. Lett., 29, No. 7, 408 (1972).ADSCrossRefGoogle Scholar
- 26.I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).MATHGoogle Scholar
- 27.V. M. Alekseev and M. V. Yakobson, “Symbolic dynamics and hyperbolic dynamic systems,” Suppl. to Methods of Symbolic Dynamics, R. Bowen (ed.) [Russian translation], Mir, Moscow (1979).Google Scholar
- 27a.A. A. Brudno, “Entropy and algorithmic complexity of trajectories of a dynamic system,” Preprint of All-Union Inst. for System Research, Moscow (1980).Google Scholar
- 28.B. V. Chirikov, “Nature of stochastic laws of classical mechanics,” in: Methodological and Philosophical Problems of Physics [in Russian], Nauka, Novosibirsk (1982), p. 181.Google Scholar
- 29.N. N. Bogolyubov, “Problems of dynamic theory in statistical physics,” in: Selected Works [in Russian], Naukova Dumka, Kiev (1970), Vol. 2, p. 99.Google Scholar
- 30.J. L. Lebowitz, “Hamiltonian flows and rigorous results in nonequilibrium statistical mechanics,” in: Statistical Mechanics. New Concepts, New Problems, New Applications, Univ. of Chicago Press, Chicago (1972), p. 41.Google Scholar
- 31.R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York (1975).MATHGoogle Scholar
- 32.G. E. Norman and L. S. Polak, “Irreversibility in classical statistical mechanics,” Dokl. Akad. Nauk SSSR, 263, No. 2, 337 (1982).Google Scholar
- 33.E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Pergamon Press, Oxford (1981).Google Scholar
- 34.A. B. Rechester and R. B. White, “Calculation of turbulent diffusion for the Chirikov-Taylor model,” Phys. Rev. Lett., 44, No. 24, 1586 (1980).MathSciNetADSCrossRefGoogle Scholar
- 34a.A. B. Rechester, M. N. Rosenbluth, and R. B. White, “Fourier-space paths applied to the calculation of diffusion for the Chirikov-Taylor model,” Phys. Rev., A23, No. 5, 2264 (1981).MathSciNetADSGoogle Scholar
- 35.J. R. Cary, J. D. Meiss, and A. Bhattacharjee, “Statistical characterization of periodic measure-preserving mappings,” Phys. Rev., A23, No. 5, 2744 (1981).MathSciNetADSGoogle Scholar
- 36.L. A. Bunimovich and Ya. G. Sinai, “Statistical properties of Lorentz gas with periodic configuration of scatterers,” Commun. Math. Phys., 78, 479 (1981).MathSciNetADSMATHCrossRefGoogle Scholar
- 37.C. Grebogi and A. N. Kaufman, “Decay of statistical dependence in chaotic orbits of deterministic mappings,” Phys. Rev., A24, No. 5, 2829 (1981).MathSciNetADSGoogle Scholar
- 38.G. V. Gadiyak and F. M. Izrailev, “Structure of transition zone of nonlinear resonance,” Dokl. Akad. Nauk SSSR, 218, No. 6, 1302 (1974).Google Scholar
- 39.B. V. Chirikov and D. L. Shepelyanskii, “Statistics of Poincaré returns and structure of stochastic layer of nonlinear resonance,” Preprint, Inst. Nucl. Phys, Siberian Branch, Acad. Sci. USSR, No. 81–69, Novosibirsk (1981).Google Scholar
- 40.D. Bora, P. I. John, Y. C. Saxena, and R. K. Varma, “Multiple lifetimes in the nonadiabatic leakage of particles from magnetic mirror traps,” Plasma Physics, 22, No. 7, 653 (1980).ADSCrossRefGoogle Scholar
- 41.D. F. Escande and F. Doveil, “Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems,” J. Stat. Phys., 26, No. 2, 257 (1981).MathSciNetADSCrossRefGoogle Scholar
- 42.L. P. Kadanoff, “Scaling for a critical Kolmogorov Arnold-Moser trajectory,” Phys. Rev. Lett., 47, No. 23, 1641 (1981).MathSciNetADSCrossRefGoogle Scholar
- 42a.S. J. Shenker and L. P. Kadanoff, “Critical behavior of a KAM surface: empirical results,” J. Stat. Phys., 27, No. 4, 631 (1982).MathSciNetADSCrossRefGoogle Scholar
- 43.L. D. Landau, “Kinetic equation for rarefied gases in strong fields,” Zh. Eksp. Teor. Fiz., 7, No. 2, 203 (1937).Google Scholar
- 44.S. T. Belyaev, “Kinetic equation for rarefied gases in strong fields,” in: Plasma Physics and Problem of Controlled Thermonuclear Reactions [inRussian], Izd. Akad. Nauk SSSR, Moscow (1958), Vol. III, p. 50.Google Scholar
- 45.B. V. Chirikov and D. L. Shepelyanskii, “Diffusion in multiple passage through nonlinear resonance,” Preprint Inst. Nucl. Phys., Siberian Branch, Acad. Sci. USSR, No. 80–211, Novosibirsk (1980).Google Scholar
- 46.M. A. Lieberman and A. J. Lichtenberg, “Stochastic and adiabatic behavior of particles accelerated by periodic forces,” Phys. Rev., A5, No. 4, 1852 (1972).ADSGoogle Scholar
- 47.A. Brahic, “Numerical study of a simple dynamical system,” Astron. Astrophys., 12, 98 (1971).ADSMATHGoogle Scholar
- 48.T. A. Zhdanova and F. M. Izrailev, “On Fermi statistical acceleration,” Preprint Inst. Nucl. Phys., Siberian Branch, Acad. Sci. USSR, No. 121–74, Novosibirsk (1974).Google Scholar
- 49.R. H. Cohen, “Stochastic motion of particles in mirror machines,” in: Intrinsic Stochasticity in Plasma, G. Laval and D. Gresillon (eds.), Edition de Physique, Orsay (1979).Google Scholar
- 50.D. F. Escande, “Primary resonances do not overlap,” in: Intrinsic Stochasticity in Plasma, G. Laval and D. Gresillon (eds.), Edition de Physique, Orsay (1979).Google Scholar
- 51.R. J. Goldston, R. B. White, and A. H. Boozer, “Confinement of high-energy particles in tokamaks,” Phys. Rev. Lett., 47, No. 9, 647 (1981).ADSCrossRefGoogle Scholar
- 52.L. I. Mandel’shtam, Complete Works [in Russian], Izd. Akad. Nauk SSSR, Moscow (1948), Vol. I, p. 297.Google Scholar
- 53.V. I. Arnol’d, “Instability of dynamic systems with many degrees of freedom,” Dokl. Akad. Nauk SSSR, 156, No. 1, 9 (1964).MathSciNetGoogle Scholar
- 54.N. N. Nekhoroshev, “Exponential estimate of the stability time of nearly integrable Hamiltonian systems,” Usp. Mat. Nauk, 32, No. 6, 5 (1977).MATHGoogle Scholar