Statistical/Evolutionary Models of
Power-laws in Plasmas
Modern Challenges in Nonlinear Plasma Physics
Ilan Roth
Space Sciences, UC Berkeley
Halkidiki
GREECE
June 2009
Heavy Tails
- Statistics of Energization Processes
- Anomalous Transport/Diffusion
- Entropy of Super-diffusive Processes
BASICS: Ergodic, weakly interacting system
converges into Boltzmann-Gibbs statistics/function
THEREFORE: particles that interact stochastically with
electromagnetic fields and perform Brownian motion in
phase space, characterized by short-range deviation and
short term microscopic memory will approach
asymptotically a Gaussian.
Why is there an ubiquity of (broken) power laws?
Example - Solar/Heliospheric Electrons
B
Electrons propagating along
heliospheric field lines to 1 AU
prompt/delayed
Electrons propagating
along coronal field lines
Heliospheric
Signatures
Relation between
coronal and IP
perturbations
Acceleration Sites?
164 MHz
NRH images
Injection of
electrons into
heliosphere
LASCO/SOHO
CME
Type II signature
Energization - occurs behind the CME?
Maia, 2001
Impulsive
Gradual
Distinct delay between Type III
and energetic electron injections
Post-flare, post-CME: Gradual electrons
Electron Spectra
gradual
impulsive
Wang,
2006
Post CME reconstruction
Yellow – flare site
Black arrow – CME direction
Red – closed (stretched) magnetic field lines
Blue - open, in ecliptic plane
green – open, non-ecliptic
Yellow – flare site
Black arrow – CME direction
Red – closed (stretched) magnetic field lines
Blue - open, in ecliptic plane
green – open, non-ecliptic
CME
Characteristics of electron trajectory in the
presence of (oblique) whistler waves
Statistics of Energization Processes
- Anomalous Transport/Diffusion
- Entropy of Super-diffusive Processes
Statistical Analysis – Particle Distribution
Energization is determined by an increment of an
independent, identically distributed (iid) random
variable (in the language of statistical mathematics),
with an assigned probability distribution.
Central Limit Theorem (CLT):
Sum of random variables with a finite
variance will converge (weakly) to a normal
or Gaussian distribution.
No Heavy Tail.
Stable Distribution: addition of a large
number of random variables preserves the
shape of the distribution (infinitely divisible).
Normal Distribution is an example of a
Stable Distribution (“attractor distribution”).
Normal Diffusion:
Brownian (random walk) motion – CLT applies, if
a) distribution with finite moments (variance)
b) no long range interaction
c) short term memory - Markovian process
Anomalous Diffusion:
CLT does not hold in its regular form
Result: Sub-diffusion or extended tails
Normal Distribution
X ~ N ( μ , σ ); P ( X ≤ x) =
2
1
2πσ 2
x
2
2
−
−
μ
σ
exp[
(
x
)
/
2
]dx
∫
−∞
X 1 , X 2 − independent , (not )identically − distributed r.v,
aX 1 ~ N [aμ , (aσ ) 2 ], bX 2 ~ N [bμ , (bσ ) 2 ]
∃c, d : aX 1 + bX 2 = cX + d = Y
(in distribution)
c 2 = a 2 + b 2 , d = (a + b − c) μ ; Y ~ N [cμ + d , (cσ ) 2 ]
Convolution
preserves the (normal )
probability
This (stable) Distribution is not unique
Generalization: the sum of a large
number of time-homogeneous,
independent, identically distributed
random variables with an infinite
variance will tend to a symmetric
(skew) stable Levy distribution.
X2
P ( X 1 < Z N < X 2 ) 6 ∫ Lα ,β ( x)dx
X1
(Gedanenko-Kolgomorov-Levy)
Lévy (α stable) Probability Distribution
The stable distribution functions are completely
defined by their Characteristic Function
(Fourier Transform of a single event probability)
^
α
l α (λ ) = exp[−c λ (1 − iβ tan(πα / 2) sgn(λ ))]
0 ≤ α ≤ 2 – Lévy exponent, c – scale, β - skewness
( Gaussian : α = 2; Lorentzian /Cauchy: α = 1)
α=2 – “phase transition”
Normal law attracts all distributions with α > 2
Cauchy
Holtsmark
Gaussian
Probability density function of stable Levy distributions
α-exponent; μ-shift; c=σ-width/scale; β-skewness
As α decreases, longer tail and narrower coreÆδ function.
Gaussian
Cauchy
Cumulative Distribution Function
Procedure:
Characteristic function – Fourier of one event probability
Multiplication in Fourier space (instead of convolution)
Inverse Fourier integration
a. Pdf with Finite moments
[Gram (1883) - Charlier (1905) - Edgeworth (1905)]
P(ξ)~exp(-ξ 2)/(2π)1/2[1+ λ3H3(ξ)/(3! n1/2) + λ4H4(ξ)/(4! n)+…]
core
Corrections to Gaussian
ξ =(r/σn1/2), λi cumulants normalized to the second moment σ
Hn Hermite polynomials.
The Gaussian core of the distribution extends to very high values
b. Leptokurtotic probability function
(diverging high (>2) moments)
2 / π 1/ 2
probability : p(r ) =
(1 + r 4 )
One event
^
Characteristic Function : p(η ) = exp(
^
−η
^
p(η ) = exp[log( p (η )]
Pn (r ) = ∫ e
irη
^
e
n log [ p (η )]
2
)[cos
dη
2π
η
2
+ sin
η
2
]
Cumulant expansion - log p(η) around η =0
^
3
log p (η ) = −η / 2 + η / 3 2 + 11η 4 / 48 + ....
Φ n (r ) = ∫ eirη e −η
2
2
/2
[1 +
aη
3
n
+ ...]
dη
~
2π
∂3
1
2
2
η
−
r
−
−
ir
dη}
{exp(
/
2
)
exp[
(
)
~ Φ G ( r ) + 1/ 2
3
∫
2 6π n) ∂r
r
D ( 4)
(−1)
Φ n ( r ) ~ Φ G ( r ) + 1/ 2
D′′′( 1/ 2 ) + O[
] + ...
2
n
2 6π n
Φ G (r ) − Gaussian;
D − Dawson Integral : D (r ) = e
−r
2
r
∫e
0
y2
dy
2
1
3
(2m − 1)!!
∞
D( x) ~ [1 + 2 + 4 + Σ m = 3 m 2 m ]
2x
2x
4x
2 x
D′′′( x) ~ −1 / 3x 4
as x − > ∞
1
Φ n (ξ ) ~ Φ G (ξ ) + 1/ 2
[ξ − 4 / 12 + ξ −6 / 288n]
2 6π n)
The edge of the central core ~ (log n)1/2
Due to limited interaction time, the observed
distribution converges extremely slowly to a
Gaussian and exhibits heavy tails.
c. Diverging second pdf moment
Characteristic
function with α<2
^
α
l α (λ ) = exp(−c λ )
∞
α
Lα , N (r ) = ∫ exp(iλr − cn λ )dλ = n −1/ α lα (r / n1/ α )
∞
asymptotically
Φα (ζ ) =
c sin(απ / 2)Γ(1 / α )α
πζ
1+α
for ζ → ∞
And a Gaussian which narrows as α -->0 at the central core region
1
(−ζ 2 ) m Γ(2m + 1 / α )
Φα (ζ ) =
Σ
πα
(2m)!
c ( 2 m +1/ α )
m
Γ(1 / α )
2
2
3/ 2
Φα (ζ ) ~
exp(
−
ζ
/
σ
)
σ
~
π
/
2
α
;
παc1/ α
for ζ → 0
Gaussian
Truncated
Cauchy
T-student
- Statistics of Energization Processes
Anomalous Transport/Diffusion
- Entropy of Super-diffusive Processes
Random walk: continuum description Æ FP
1
1
Pi (t + Δt ) = Pi +1 (t ) + Pi −1 (t );
2
2
ri ±1 = ri ± Δr ;
isotropic
Phase space
jumps
Local in space and time
∂P (r , t )
∂2
(dr ) 2 ∂ 2
= lim
P(r , t ) ≡ K 2 P(r , t )
2
∂t
∂r
∂r
dr →0 , dt →o dt
P(r , t ) = (4πKt ) −1 / 2 exp(−r 2 / 4 Kt )
<r2> ~ tα ; α=1
Continuum limit
Green propagator
Central limit theorem
Continuous time Random walk
Non-locality in phase space
Pj(t+dt)=∑n=1 [Aj,nPj-n(t)+ Bj,nPj+n (t)]
Generalization
Length of jump, waiting time
χ(r, t) = λ(r)τ(t)
Decoupled temporal/spatial memory
Probabilities
λ(r)dr phase space jump
τ(t)dt waiting time between jumps
jump
ν=
ω − k ( x , t ) v ( x, t )
Ω ( x, t )
ν=n
Waiting time
Probability P(r,t) - phase space position r at time t
probability ζ(r,t’) of arrival at r at an earlier time t’ and
staying there without a jump:
“survival” probability φ(t)=1-∫dt τ(t);
P(r,t) = ∫dt’ ζ(r,t’) φ(t-t’)
Master equation
ζ (r,t)= ∫dr’ ∫dt’ ζ(r’,t’) χ(r-r’,t-t’) + Po(r) δ(t)
Fourier-Laplace
(Montroll-Weiss)
P(k , u ) =
P0
1 − τ (u )
u 1 − χ (k , u )
A. NORMAL Brownian motion - short memory, short range
τ(t) = [exp(-t/t0)/ t0]; Poissonian (finite) characteristic time
λ(r)=(4πσ )-1exp[-(r/2σ)2]
finite jump length variance
F-L: τ(u)~ 1-ut0 and λ(k)~1-σ2 k2 as (k,u)->(0,0)
1
P(k , u ) =
u + Kk 2
Inverse Fourier-Laplace Ædiffusion limit; FP
∂P
∂2
=K 2 P
∂t
∂x
B. SUBDIFFUSIVE PROCESS
Long-tailed waiting time pdf with finite jump variance
τ(t) = A (t/t0)-1-α , α<1;
λ(r)=(4πσ )-1exp[-(r/2σ)2]
τ(u)~ 1-(ut0)α ;
λ(k)~1-σ2k2 as (k,u)->(0,0)
P0 (k ) / u
P(k , u ) =
1 + Kα u −α k 2
Kα = σ 2 / t α
t
Riemann-Liouville
'
1
G
(
t
)
'
−α
dt
0 Dt G ( x, t ) =
∫
Γ(α ) 0 (t − t ' )1−α
£{D-α P(r,t) } = u-α P(r,u)
Inverse F-L -Æ Fractional FP
∂P(r , t )
∂
1− α
= Dt Kα 2 P(r , t )
∂t
∂r
2
Temporal Integration emphasizes the
non-Markovian, long term memory
Solution converges to Brownian as α-->1
Normal diffusive
vs
P0 (k )
P(k , u ) =
u + Kk 2
anomalously sub-diffusive
P0 (k ) / u
P(k , u ) =
1 + Kα u −α k 2
Single mode decay
P(k , t ) = exp(− Kk 2t )
Exponential relaxation
P(k , t ) = Eα (− Kα k 2t α )
Stretched exponential
Mittag-Leffler
∞
Eα (− t ) = ∑
α
n =0
(−t α ) n
~ [t α Γ(1 − α )]−1
Γ(1 + αn)
<r2> =[K/Γ(α+1)] tα , α<1
C. SUPERDIFFUSIVE PROCESS
Short memory waiting time - long jumps variance
Markovian time pdf
μ
τ (u) ~ 1 - ut 0 ;
μ
μ
λ (k) = exp(-σ k ) ~ 1 - σ k
P(k , u ) =
μ
Levy diverging jumps: μ<2
μ
P0 (k )
μ
u+K k
K = σ / t0
μ
;
μ
Riess/Weyl fractional operator
μ
D
−∞
z G ( z ) = −∞
n
1
d
Dzn −( n − μ )G ( z ) =
Γ(n − μ ) dz n
μ
μ
Fourier : F{−∞ Dz G ( z )} = − k F{G ( z )}
z
'
G
(
z
)
'
∫−∞dz ( z − z ' )1−( n−μ )
μ ∈ (n − 1, n)
∂P(r , t )
μ
μ
= K −∞ Dr P(r , t )
∂t
Spatial Integration emphasizes the
probability for a long range interaction
Solution converges to Brownian as μ-->2
Gauss walk
Levy flight
Normal diffusive
vs
P0 (k )
P(k , u ) =
u + Kk 2
anomalously super-diffusive
P0 (k )
P(k , u ) =
u + K μk μ
Single mode decay
P(k , t ) = exp(− Kk t )
2
Exponential relaxation
μ
μ
P(k , t ) = exp(− K k t )
Extended tail
Symmetric Levy
Solution: Mellin Transform
∞
Μ[ F (t )] = F ( s) = ∫ F (t )t s −1dt
0
M{L( F (t ), p), s} = Γ(s)M[ F (t ),1 − s]
Resulting in a set of Γ functions
∞
Γ( s ) = ∫ e −t t s −1dt
0
Γ(1 + p ) p − q
1
q
−q
⇒⇒
=
D
t
t
D
t
=
1
r
0
0
Γ(1 + p − q )
Γ(1 − q )
q p
r
Gamma function – holomorphic with simple poles at all
the non-positive integers; the residue at -n is (-1)n/n!
Fox (H) : generalized Mellin-Barnes (Meijer G) Functions
m
1
( a ,α )
mn
H ( z ) = H pq [ z;( b , β ) ] =
2πi ∫C
n
∏ Γ(b − β s)∏ Γ(1 − a − α s)
i
i
i =1
q
i
i
i =1
p
∏ Γ(1 − b + β s) ∏ Γ(a − α s)
i
i = m +1
i
i
i
i = n +1
αj,βj >0; aj,bj –complex; roots of (a, α)/ (b,β ) left/right of C
Series expansion with residue (-1)/l! at the poles bk-βks=-l, l=0,...
β i (bk + l ) n
α i (bk + l )
(b +l )
b
a
Γ
−
Γ
−
+
[
]
[
1
]
∏
∏
i
i
l
βk
βk
(−1) z β
i =1, ≠ k
i =1
q
β i (bk + l ) p
α i (bk + l ) l!bk
b
a
Γ
−
+
Γ
−
[
1
]
[
]
∏
∏
i
i
β
β
i = m +1
i = n +1
k
k
m
m
mn
H pq
[ z;((ba,,αβ )) ] = ∑
k =1
∞
∑
l =0
Alternating signs – slow convergence
k
k
z s ds
poles
Explicit solution of the FFPE
s
Γ(1 − s )Γ( )
1
1
μ s
11
(1,1 / μ )(1,1 / 2 )
P(r , t ) =
H 22 [ξ |(1,1)(1,1/ 2 ) ] =
ds
ξ
∫
μr
μ r C Γ( s )Γ(1 − s )
μ
2
r
ξ=
Roots at s/μ=-m;
( K μ t )1/ μ
1
P(r , t ) =
μr
Γ(mμ ) (−1) m − mμ
Kt
ξ
~
∑
m!
r 1+ μ
m = 0 Γ ( − mμ / 2)
∞
Slow convergence
1/ μ
P (r , t ) = ( Kt ) −1/ μ Lμ [ r /( Kt ) ]
General: Long waiting times and large jumps
μ
∂P(r , t )
1−α
μ ∂
= Dt Kα μ P(r , t )
∂t
∂r
< r >= t
2
α /μ
- Statistics of Energization Processes
- Anomalous Transport/Diffusion
Entropy of Super-diffusive processes
*Thermodynamic properties - from microscopic states
*System in contact with a large reservoir
*Short temporal/spatial interaction – random jumps p(r)
Boltzmann-Gibbs-Shannon entropy-extensive ~system size
S BG = S1 = − ∫ p (r ) ln p (r )dr
Constrains:
∫ p(r )dr = 1;
< r 2 >= ∫ r 2 p (r )dr = σ 2 < ∞
Variational Optimization; β-Lagrange multiplier
exp(− β r 2 ) exp(− βr 2 )
p1 (r ) =
=
2 ;
−β r
Z1 ( β )
∫ dre
β = T −1 = 1 /(2σ 2 )
Gaussian – attractor in distribution space
Long range interaction makes the entropy non-extensive
Æformation of tails in the distribution.
q-statistics/entropy (Tsallis):
q
S = [1 − ∫ p (r )dr ] /( q − 1)
q
→ S BG = S1 as q → 1
< r 2 >= ∫ r 2 [ p (r )]q dr
Optimization of Sq with q-expectation constraint
and β as Lagrange multiplier leads to heavy tail distributions
p q ( r ) = [1 − (1 − q ) β r ]
2
1 /( 1 − q )
/Zq
→
qÆ1
eqr = [1 + (1 − q )r ]1/(1− q ) =⇒⇒ pq (r ) =
exp q (− β r 2 )
Z q (β )
β
exp( − β r 2 )
π
=
exp q (− β r 2 )
2
dr
exp
(
−
β
r
)
q
∫
1<q<3 - Student t distribution
r 2 −( k +1) / 2
Γ[(k + 1) / 2]
f (r ) =
(1 + )
k
πk Γ[k / 2]
q=
3+ k
1+ k
p q ( r ) = [1 − (1 − q ) β r 2 ]
1
( 1− q )
/
∞<q<3
∫
[1 − (1 − q ) β r 2 ]
Converges to Gaussian for q=1.
Converges to Cauchy for q=2
Variance – finite for q < 5/3 , diverges for 5/3 <q <3
q-variance:
∫
pq (r ) ~ r − 2 /( q −1)
dxx 2 p ( x) q finite for q < 3,
i.e. α = (3 − q ) /(q − 1); 0 < α < 2
pq (r , N ) = [ β 1/ 2 / N 1/ α ]Lα [( β 1/ 2 / N 1/ α )r ] : 5 / 3 < q < 3
“Phase Transition” at q=5/3 (α=2)
q-entropy converges to stable laws (distributions)
1
( 1− q )
dr
Comparison with the data allows to assess
a) the validity of the single event probability
density function
b) the relative importance of the different
expansion terms – power law transition
c) the “duration” of the interaction
One-event Impulsive probability
SUMMARY
O
Electron Spectra
A
p( E ) =
(1 + E 4 )
with an incomplete asymptotic
Gaussian evolution
o
o
------------------------------------------------One-event Gradual probability
α=0.5-1.0 (Lévy)
O
A
; δ = 1.5 − 2.0
p( E ) =
δ
(1 + E )
with a truncated Levy index α
at the range of Holtsmark-Cauchy
ΕΠΙΛΟΓΟΣ
Distribution function of electrons and ions in space
plasmas due to random, resonant interactions with
electromagnetic turbulence displays numerous
characteristic forms, combining Gaussians with (broken)
power laws. The basic single event interaction probability
determines the asymptotic distribution with a phase
transition at the index of the characteristic function α=2,
while the global interaction time determines the observed
non-asymptotic distribution. The distribution function of
the injected particles may be construed via general
arguments of stochastic processes, and parameterized via
non-extensive entropy.