Robert Sheldon 1 , Mark Adrian 2 , Shen-Wu Chang 3 , Michael Collier 4

1. Solar Wind Heating as a Non-Markovian Process: L₫vy Flight, Fractional Calculus, and Ú-functions Robert Sheldon1, Mark Adrian2, Shen-Wu Chang3, Michael Collier4 UAH/MSFC, NRC/MSFC, CSPAAR/MSFC, GSFC May 29, 2001

2. The Solar Coronal Issue • (I apologize for repeating the obvious, please bear with me) • Q: Corona is 2MK, photosphere only 5.6kK, so how does heat flow against the gradient? • A: Non-equilibrium heat transport • a) coherence (waves) • b) topology (reconnection) • c) velocity filtration + non-Maxwellians • Each solution has its pros/cons, I want to present some mathematical results that may support (c)

3. Problems, Opportunities • 1) Coherent phenomena (waves), need to randomize to heat. Dissipation at 1-2 Rs turns out to be a problem. [Parker 92] • 2) Topology tangles, like a comb through long hair, collect in clumps or boundaries where they would heat unevenly. In contrast, all observational evidence shows even heating. Nor is there direct observation of nanoflares [Zirker+Cleveland 92]. • 3) Velocity filtration is not itself a heating mechanism, it requires a non-Maxwellian distribution as well. Thus it postpones the problem to one of non-equilibrium thermodynamics in the highly collisional photosphere. [Scudder 92] • 4) Heat conduction too low to smooth out hot spots [Marsden 96] • 5) Non-Kolmogorov spectrum, non-turbulent heating.[Gomez93]

4. Characteristics needed • Robust (nearly independent of magnetic polarity, geometry, location on sun, etc.) • Fine-grained (no evidence of clumps) • Non-turbulent (wrong spectrum) • Non-equilibrium statistical mechanics (it still transports heat the wrong way.) • What we need is a mechanism so fine-grained that it escapes observation, yet so macroscopic that it does not rely on fickle micro/meso-scale physics.

5. Velocity Filtration • A Maxwellian maximizes the entropy for a fixed energy, so switching velocity space via collisions, or removing part of the distribution has no effect on the temperature. • But as Scudder shows graphically, the power-law tail of a kappa-function has a higher temperature than the core, so removing the core leaves hi-T, unlike a Maxwellian which is self-similar.

6. Pin a tail on the distribution? • What sort of stochastic processes generate tails? • For the sake of transparency, we will discuss spatial distributions that depart from Gaussian, and later apply these distributions to velocity space Maxwellians. • If we consider a random stochastic process, such as a drunk staggering around a lamppost, we can plot the resulting spatial distribution of a collection of drunks. • For some simple restrictions, the Central Limit Theorem predicts the distribution will converge to a Gaussian. • Even worse, the distribution that maximizes entropy while conserving energy is a Maxwellian= Gaussian in v-space. • How can we get a tail without violating the physics?

7. Central Limit Theorem • Paul Levy [1927] generalized the C.L.T. • Variance: s 2 = <x2> - <x> 2 = 2Dt • Diffusion: D = (<x2> - <x> 2) / 2T • Probability Distribution Function, P : <xn>=€dx xnP(x) • We just need a different PDF to get a fat tail. • P(x)~x-m • if m < 3, then <x2> = âands 2 ~ t 1<g<2 • well, we lost the 2nd moment, but we have a tail. • What does this do to the physics? What happened to the entropy (or is the energy)?

8. PDF and spatial diffusion P(x) • A slight change in the PDF can change diffusion radically. • Levy-flights • Self-similar m=2.2 m=3.8 x

9. Self-similarity • Paul L₫vy generalized the Central Limit Theorem, by looking for distributions that were "stable", having 3 exclusive properties: • Invariant under addition: X1+X2+...+Xn= cnX + dn • Domain of attraction: cn= n1/Ñ ; 0<Ñ<2 • Canonical characteristic function: H- or Fox-fcns. • In other words, For a given Ñ, there exists a self-similar function, having a canonical form, that all random distributions will converge to. • L₫vy-stable probability distribution functions

10. Stable Distributions Lorentzian/Cauchy Ñ=1 Ñ = 1.5 Gaussian/Normal Ñ= 2

11. L₫vy-stable distributions Linear Log -Self-similar, convergence to these stable distributions which are unimodal, and bell-shaped, but lack 2nd moment -invariant under addition, domain of attraction, char. fn. -Gaussian core, power-law tails (indistinguishable from Ú)

12. Fractional Diffusion • A completely separate mathematical technique has been found to describe L₫vy-stable distributions. • Time-fractional Diffusion Equation • d2Ýf / d2Ýt = D d2f / d2x • where D denotes positive constant with units of L2/T2Ý • Ý=1 wave equation; Ý=1/2 diffusion equation (Gauss) • Anomalous Diffusion • Ý<1/2 = slow diffusion (Cauchy); Ý>1/2 = fast diffusion • Solutions are Mittag-Leffler functions of order 2Ý, they are also Levy-stable pdf. • We can identify Ú-fcn with slow fractional diffusion

13. Fractional Derivatives (19th) Quadratic=>Linear note how neatly it interpolates between the lines Linear=> Constant note how the slope of the fractional deriv exceeds both. It uses global info! (integro-differential)

14. Physical Interpretation • Collier [91] applied Levy-flight to velocity space diffusion and generated Cauchy tails as expected. • Non-Markovian processes have "memory", or correlations in the time-domain. E.g., the lamppost is on a hill. That is, collisions have non-local information=> fractional calculus. • Time-fractional & Space-fractional diffusion equations are equivalent (if there is a velocity somewhere). • Non-local interactions, and/or non-Markovian interactions both produce fractional diffusion. • Many physical systems exhibit super/sub diffusion with fat tails. • Non-adiabatic systems need not conserve energy. E.g. if we maximize entropy holding log(E) constant => power law tails!

15. W.A.G. • 1) Non-local Coulomb collisions in the Earth's ionosphere are thought to put tails on upflowing ions. • 2) Equivalently, large changes in collision frequency are temporally correlated as plasma leaves Sun. • 3) Fluctuations in energy are log-normal distributed, suggesting a pdf already far from Gaussian. • 4) Coulomb cross-section decreases with energy, so that E-field "runaway" modifies the power law of the P, the step size between collisions (and energy gain). • 5) "Sticky vortices" in Poincar₫ plots-Hamiltonian chaos.

16. Nomenclature

17. Abstract • Many space and laboratory plasmas are found to possess non-Maxwellian distribution functions. An empirical function promoted by Stan Olbert, which superposes a Maxwellian core with a power-law tail, has been found to emulate many of the plasma distributions discovered in space. These $\kappa$-functions, with their associatedpower-law tail induced anomalous heat flux, have been used by theorists$^1$\ as the origin of solar coronal heating of solar wind. However, the principle and prerequisite for the robust production of such a non-equilibrium distribution has rarely been explained. We report on recent statistical work$^2$, which shows that the $\kappa$-function is one of a general class of solutions to a time-fractional diffusion equation, known as a L\'evy stable probability distribution. These solutions arise from time-variable probability distribution (or equivalently, a spatially variable probability in a flowing medium), which demonstrate that anomalously high flux, or equivalently, non-equilibrium thermodynamics govern the outflowing solar wind plasma. We will characterize the parameters that control the degree of deviation from a Maxwellian and attempt to draw physical meaning from the mathematical formalism. $^1$Scudder, J. {\it Astrophys. J.}, 1992.\$^2$Mainardi, F. and R. Gorenflo, {\it J. Computational and Appl. Mathematics, Vol. 118}, No 1-2, 283-299 (2000).

18. Fractional Diffusion Examples