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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
The Solar Coronal Issue
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• (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)
Problems, Opportunities
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• 1) Coherent phenomena (waves), need to randomize to heat.
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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]
Characteristics needed
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• 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 finegrained that it escapes observation, yet so
macroscopic that it does not rely on fickle
micro/meso-scale physics.
Velocity Filtration
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• 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 kappafunction has a higher temperature than the core, so removing the
core leaves hi-T, unlike a Maxwellian which is self-similar.
Pin a tail on the distribution?
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• 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.
Central Limit Theorem
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• 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> = â and s 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)?
PDF and spatial diffusion
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P(x)
m=2.2
m=3.8
x
• A slight change
in the PDF can
change diffusion
radically.
• Levy-flights
• Self-similar
Self-similarity
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• 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 selfsimilar function, having a canonical form, that all
random distributions will converge to.
• L₫vy-stable probability distribution functions
Stable Distributions
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Lorentzian/Cauchy Ñ=1
Ñ = 1.5
Gaussian/Normal Ñ= 2
L₫vy-stable distributions
Linear
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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)
Fractional Diffusion
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• A completely separate mathematical technique has
been found to describe L₫vy-stable distributions.
• Time-fractional Diffusion Equation
2Ý
2Ý
2
2
– d f/d t = Dd f/d x
d
da
–
–
–
–
–
–
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
Fractional Derivatives
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th
(19 )
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)
Physical Interpretation
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• Collier [91] applied Levy-flight to velocity space diffusion and
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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!
W.A.G.
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• 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
Nomenclature
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Abstract
•
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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 nonequilibrium 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, nonequilibrium 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, 283299 (2000).
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Fractional Diffusion Examples
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