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

1
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

• Peter H. Yoon
• U. Maryland, College Park, USA

2
Outline

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

3
TSALLIS ENTROPY

• Part 1.

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 1023 m2 kg s2 K1

• of ways one molecule can be arranged in an
  ideal gas.
• Ref. K. Huang, Statistical Mechanics

9
SB k log 2

• SA k log 2

10
SB k log 2

• SA k log 2

SAB 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
SOLAR WIND ELECTRONS

• Part 2

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
? 8
20
Most probable states

• Helmholtz free energy

21
Boltzmann vs Kappa Distributions as thermodynamic
equilibria
Leubner, 2004 Treumann et al., 2008 Livadiotis
and McComas, 2009
22
Maxwellian (Gaussian) vs Kappa Distribution

• If one defines
• k 1/(1q)
• then Tsallis distribution becomes kappa-like
  distribution (Vasyliunas, 1968),

23
fe(v)
Electron Velocity Distribution
By Linghua Wang, Davin Larsen, Robert Lin
24
BEAM-PLASMA INSTABILITY AND LANGMUIR TURBULENCE

• Part 3

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.
Altshul 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
(No Transcript)
40
(No Transcript)
41
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).
42
Theory
C.-M. Ryu, T. Rhee, T. Umeda, P. H. Yoon, and Y.
Omura, Turbulent acceleration of superthermal
electrons, Phys. Plasmas 14, 100701 (2007).
43
fe(v)
Electron Velocity Distribution
By Linghua Wang, Davin Larsen, Robert Lin
44
TURBULENT EQUILIBRIUM

• Part 4

45
fe(v)
Electron kinetic equation
I(k)
Langmuir wave kinetic equation
46
Steady-State Solution (Quasi-Equilibrium)
Electron kinetic equation
Steady-state solution Hasegawa et al., 1985
47
Langmuir wave kinetic equation
48
Balance of spontaneous emission and induced
emission
Self-consistent kappa distribution but k is
undetermined
49
Langmuir wave kinetic equation
50
To determine k one must also balance
spontaneous and induced scattering (turbulent
equilibrium)
0
51
Steady-state solution (Turbulent
quasi-equilibrium)
52
Theory
Observation
53
(No Transcript)
54
CONCLUSIONS

• Part 5

55
Summary

• Solar wind electrons feature kappa-like
  distribution, implying turbulent
  quasi-equilibrium.
• Alternatively, it implies non-extensive
  equilibrium.
• Turbulent equilibrium non-extensive equilibrium
  (?)