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Turbulent equilibrium, and superhalo solar wind electron distribution, and nonextensive entropy

INTERNATIONAL ASTROPHYSICS FORUM 2011. Frontiers in Space Environment Research, Alpbach/June 20-24, 2011. Turbulent equilibrium, and superhalo solar wind electron distribution, and nonextensive entropy. Peter H. Yoon U. Maryland, College Park, USA. Outline. Tsallis entropy

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Turbulent equilibrium, and superhalo solar wind electron distribution, and nonextensive entropy

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  1. INTERNATIONAL ASTROPHYSICS FORUM 2011 Frontiers in Space Environment Research, Alpbach/June 20-24, 2011 Turbulent equilibrium, and superhalo solar wind electron distribution, and nonextensive entropy Peter H. Yoon U. Maryland, College Park, USA

  2. Outline • Tsallis entropy • Solar wind electrons • Beam-plasma instability • Turbulent equilibrium • Conclusions

  3. Part 1. TSALLIS ENTROPY

  4. Boltzmann entropy • S = k log W • W: # of ways a single molecule in an ideal gas can be arranged. • What does this mean?

  5. A Toy Example • Two colored balls:

  6. A Toy Example • Two colored balls: • In how many ways can the balls be arranged?

  7. A Toy Example • Two colored balls: • In how many ways can the balls be arranged? • Answer: 2 • S = k log 2

  8. S(W) = k log W kB = 1.3806503 x 10–23 m2 kg s–2 K–1 • : # of ways one molecule can be arranged in an ideal gas. • [Ref. K. Huang, Statistical Mechanics]

  9. SA = k log 2 SB = k log 2

  10. SA = k log 2 SB = k log 2 SA+B = k log 4= k log 2 + k log 2 = SA + SB

  11. Extensivity of Boltzmann Entropy S(W) = k log W S(W’) = k log W’ S(WW’) = k log WW’ = S(W) + S(W’)

  12. Non-Extensive Entropy S(W) = k log W S(W’) = k log W’ S(WW’) ≠ S(W) + S(W’)

  13. Non-Extensive Entropy q defines the degree of non-extensivity. q = 1 (Boltzmann limit)

  14. Boltzmann vs Tsallis Entropy Continuum limit Discrete versions

  15. Part 2 SOLAR WIND ELECTRONS

  16. 2007 January 9 Linghua Wang, Robert P. Lin, Chadi Salem

  17. fe(v) Electron Velocity Distribution By Linghua Wang, Davin Larsen, Robert Lin

  18. Gaussian vs Kappa Distribution [Kappa distribution: Olbert, Vasyliunas]

  19. Kappa Model Energetic (superthermal) tail  = ∞

  20. Most probable states • Helmholtz free energy

  21. Boltzmann vsKappaDistributions as thermodynamic equilibria [Leubner, 2004; Treumann et al., 2008; Livadiotis and McComas, 2009]

  22. Maxwellian (Gaussian) vs Kappa Distribution • If one defines • k= 1/(1–q) then Tsallis distribution becomes kappa-like distribution (Vasyliunas, 1968),

  23. fe(v) Electron Velocity Distribution By Linghua Wang, Davin Larsen, Robert Lin

  24. Part 3 BEAM-PLASMA INSTABILITY AND LANGMUIR TURBULENCE

  25. Exospheric model [Scudder & Olbert, 1979; Pierrard et al., 2009, …] Turbulence model [this talk]

  26. Bump-on-tail instability • A. Vedenov, E. P. Velikhov, R. Z. Sagdeev, Nucl. Fusion 1, 82 (1961). • W. E. Drummond and D. Pines, Nucl. Fusion Suppl. 3, 1049 (1962).

  27. Quasi-linear beam-plasma interaction Spontaneous drag (discrete particle effect) Velocity space diffusion Spontaneous emission (fluctuation-dissipation theorem) Induced emission (Landau damping/ Quasi-linear growth/damping rate)

  28. Quasi-linear beam-plasma interaction

  29. Weak turbulence theory L. M. Gorbunov, V. V. Pustovalov, and V. P. Silin, Sov. Phys. JETP 20, 967 (1965) L. M. Al’tshul’ and V. I. Karpman, Sov Phys. JETP 20, 1043 (1965) L. M. Kovrizhnykh, Sov. Phys. JETP 21, 744 (1965) B. B. Kadomtsev, Plasma Turbulence (Academic Press, 1965) V. N. Tsytovich, Sov. Phys. USPEKHI 9, 805 (1967) V. N. Tsytovich, Nonlinear Effects in Plasma (Plenum Press, 1970) V. N. Tsytovich, Theory of Turbulent Plasma (Consultants Bureau, 1977) A. G. Sitenko, Fluctuations and Non-Linear Wave Interactions in Plasmas (Pergamon, 1982)

  30. Equation for fe(v) Spontaneous drag (discrete particle effect) Velocity space diffusion

  31. Equation for I(k) Spontaneous emission (fluctuation-dissipation theorem) Induced emission (Landau damping/ Quasi-linear growth/damping rate)

  32. Equation for I(k) Linear wave-particle resonance

  33. Spontaneous decay Induced decay

  34. Nonlinear wave-wave resonance

  35. Spontaneous scattering Induced scattering (scattering off thermal ions)

  36. Nonlinear wave-particle resonance

  37. Discrete-particle (collisional) effect ~ g = 1/(nlD3)

  38. Weak turbulence theory [Muschietti & Dum, 1991; Ziebell et al., 2001; Kontar & Pecseli, 2002]

  39. Long-time behavior of bump-on-tail Langmuir instability P. H. Yoon, T. Rhee, and C.-M. Ryu, Self-consistent generation of superthermal electrons by beam-plasma interaction, PRL 95, 215003 (2005).

  40. Theory C.-M. Ryu, T. Rhee, T. Umeda, P. H. Yoon, and Y. Omura, Turbulent acceleration of superthermal electrons, Phys. Plasmas 14, 100701 (2007).

  41. fe(v) Electron Velocity Distribution By Linghua Wang, Davin Larsen, Robert Lin

  42. Part 4 TURBULENT EQUILIBRIUM

  43. fe(v) Electron kinetic equation I(k) Langmuir wave kinetic equation

  44. Steady-State Solution (Quasi-Equilibrium) Electron kinetic equation Steady-state solution [Hasegawa et al., 1985]

  45. Langmuir wave kinetic equation

  46. • Balance of spontaneous emission and induced emission: • Self-consistent kappa distribution but k is undetermined:

  47. Langmuir wave kinetic equation

  48. • To determine k one must also balance spontaneous and induced scattering (turbulent equilibrium): =0

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