Statistical/Evolutionary Models of Power-laws in Plasmas

Transcription

1 Saisical/Evoluionay Models of owe-laws in lasmas Moden Challenges in Nonlinea lasma hysics Ilan Roh Space Sciences, UC Beeley Halidii GREECE June 009

2 Heavy Tails - Saisics of Enegizaion ocesses - Anomalous Tanspo/Diffusion - Enopy of Supe-diffusive ocesses

3 BASICS: Egodic, wealy ineacing sysem conveges ino Bolzmann-Gibbs saisics/funcion THEREFORE: paicles ha ineac sochasically wih elecomagneic fields and pefom Bownian moion in phase space, chaaceized by sho-ange deviaion and sho em micoscopic memoy will appoach asympoically a Gaussian. Why is hee an ubiquiy of boen powe laws?

4 Example - Sola/Heliospheic Elecons B Elecons popagaing along heliospheic field lines o AU pomp/delayed Elecons popagaing along coonal field lines

5 Relaion beween coonal and I peubaions Acceleaion Sies? 64 MHz NRH images Heliospheic Signaues Injecion of elecons ino heliosphee LASCO/SOHO CME Maia, 00 Type II signaue Enegizaion - occus behind he CME?

6 Impulsive Gadual Disinc delay beween Type III and enegeic elecon injecions os-flae, pos-cme: Gadual elecons

7 Elecon Speca gadual impulsive

8 Wang, 006 os CME econsucion Yellow flae sie Blac aow CME diecion Red closed seched magneic field lines Blue - open, in eclipic plane geen open, non-eclipic

9 Yellow flae sie Blac aow CME diecion Red closed seched magneic field lines Blue - open, in eclipic plane geen open, non-eclipic CME

10 Chaaceisics of elecon ajecoy in he pesence of oblique whisle waves

11 Saisics of Enegizaion ocesses - Anomalous Tanspo/Diffusion - Enopy of Supe-diffusive ocesses

12 Saisical Analysis aicle Disibuion Enegizaion is deemined by an incemen of an independen, idenically disibued iid andom vaiable in he language of saisical mahemaics, wih an assigned pobabiliy disibuion.

13 Cenal Limi Theoem CLT: Sum of andom vaiables wih a finie vaiance will convege wealy o a nomal o Gaussian disibuion. No Heavy Tail. Sable Disibuion: addiion of a lage numbe of andom vaiables peseves he shape of he disibuion infiniely divisible. Nomal Disibuion is an example of a Sable Disibuion aaco disibuion.

14 Nomal Diffusion: Bownian andom wal moion CLT applies, if a disibuion wih finie momens vaiance b no long ange ineacion c sho em memoy - Maovian pocess Anomalous Diffusion: CLT does no hold in is egula fom Resul: Sub-diffusion o exended ails

15 pobabiliy nomal he peseves Convoluion c d c N Y c b a d b a c disibuion in Y d cx bx ax d c b b N bx a a N ax v disibued idenically no independen X X dx x x X N X x ], [ ~ ;, :, ], [ ~ ],, [ ~,.,, ] / exp[ ;, ~ σ σ σ σ πσ σ This sable Disibuion is no unique Nomal Disibuion

16 Genealizaion: he sum of a lage numbe of ime-homogeneous, independen, idenically disibued andom vaiables wih an infinie vaiance will end o a symmeic sew sable Levy disibuion. X < Z < X L N, X X β x dx Gedaneno-Kolgomoov-Levy

17 Lévy sable obabiliy Disibuion ^ l The sable disibuion funcions ae compleely defined by hei Chaaceisic Funcion Fouie Tansfom of a single even pobabiliy λ exp[ c λ iβ an π / sgn λ ] 0 Lévy exponen, c scale, β -sewness Gaussian : ; Loenzian /Cauchy: phase ansiion Nomal law aacs all disibuions wih >

18 Cauchy Holsma Gaussian obabiliy densiy funcion of sable Levy disibuions -exponen; -shif; cσ-widh/scale; β-sewness As deceases, longe ail and naowe coe δ funcion.

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20 Gaussian Cauchy Cumulaive Disibuion Funcion

21 ocedue: Chaaceisic funcion Fouie of one even pobabiliy Muliplicaion in Fouie space insead of convoluion Invese Fouie inegaion a. df wih Finie momens [Gam Chalie Edgewoh 905] ξ~exp-ξ /π / [ λ 3 H 3 ξ/3! n / λ 4 H 4 ξ/4! n ] coe Coecions o Gaussian ξ /σn /, λ i cumulans nomalized o he second momen σ H n Hemie polynomials. The Gaussian coe of he disibuion exends o vey high values

22 b. Lepouoic pobabiliy funcion diveging high > momens ] sin [cos exp : / : ^ 4 / η η η η π p Funcion Chaaceisic p pobabiliy even One π η η η η η ] exp[log ] log[ ^ ^ ^ d e e p p p n i n

23 dy e e D Inegal Dawson D Gaussian n D O D n d i n d n a e e y G G n G i n Φ Φ Φ Φ Φ 0 4 / / 3 3 / 3 / : ;... ] [ 6 ~ } exp[ / {exp 6 ~ ~...] [ π η η π π η η η η Cumulan expansion - log pη aound η / 3 / / log 4 3 ^ η η η η p

24 3 m!! D x ~ [ Σ ] 4 m 3 m m x x 4x x 4 D x ~ / 3x as x > Φ n ξ ~ Φ G ξ / 6π n [ ξ 4 / ξ 6 / 88n] The edge of he cenal coe ~ log n / Due o limied ineacion ime, he obseved disibuion conveges exemely slowly o a Gaussian and exhibis heavy ails.

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26 c. Diveging second pdf momen Chaaceisic funcion wih < ^ l λ exp c λ L, N asympoically exp iλ cn λ csin π / Φ ζ dλ n π ζ / l Γ/ / n / fo ζ And a Gaussian which naows as -->0 a he cenal coe egion m ζ Γm / Φ ζ Σ m / π m m! c Φ ζ ~ Γ/ / πc exp ζ / σ ; σ ~ π / 3/ fo ζ 0

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28 Tuncaed Cauchy Gaussian T-suden

29 - Saisics of Enegizaion ocesses Anomalous Tanspo/Diffusion - Enopy of Supe-diffusive ocesses

30 Random wal: coninuum descipion F i Δ i i ; hase space i± i ± Δ; isoopic Local in space and ime jumps, lim d 0, d o d d, K, Coninuum limi, 4πK / exp / 4K Geen popagao < > ~ ; Cenal limi heoem

31 Coninuous ime Random wal Non-localiy in phase space j d n [A j,n j-n B j,n jn ] Genealizaion Lengh of jump, waiing ime χ, λτ Decoupled empoal/spaial memoy obabiliies λd phase space jump τd waiing ime beween jumps

32 jump ω ν x, v Ω x, x, νn Waiing ime

33 obabiliy, - phase space posiion a ime pobabiliy ζ, of aival a a an ealie ime and saying hee wihou a jump: suvival pobabiliy φ- d τ;, d ζ, φ- Mase equaion ζ, d d ζ, χ-,- o δ Fouie-Laplace Monoll-Weiss τ u 0, u u χ, u

34 A. NORMAL Bownian moion - sho memoy, sho ange τ [exp-/ 0 / 0 ]; oissonian finie chaaceisic ime λ4πσ - exp[-/σ ] finie jump lengh vaiance F-L: τu~ -u 0 and λ~-σ as,u->0,0, u u K Invese Fouie-Laplace diffusion limi; F K x

35 B. SUBDIFFUSIVE ROCESS Long-ailed waiing ime pdf wih finie jump vaiance τ A / 0 --, <; λ4πσ - exp[-/σ ] τu~ -u 0 ; λ~-σ as,u->0,0 σ K u K u u / /, 0

36 ,, K D Tempoal Inegaion emphasizes he non-maovian, long em memoy Soluion conveges o Bownian as --> {D -, } u -,u Invese F-L - Facional F Γ G d x G D 0 ' ' ' 0, Riemann-Liouville

37 Nomal diffusive vs anomalously sub-diffusive 0, K u u 0 /, u K u u, exp, K E K Exponenial elaxaion Seched exponenial Single mode decay 0 ] ~ [ Γ Γ n E n n Miag-Leffle < > [K/Γ], <

38 C. SUERDIFFUSIVE ROCESS Sho memoy waiing ime - long jumps vaiance Maovian ime pdf Levy diveging jumps: < τ u ~ - u 0 ; λ exp-σ ~ -σ, u u 0 K ; K σ / 0

39 ,, D K, } { } { : ' ' ' n n z G z G D Fouie z z z G dz dz d n z G D z G D z z n n n n n z z Γ F F Spaial Inegaion emphasizes he pobabiliy fo a long ange ineacion Soluion conveges o Bownian as --> Riess/Weyl facional opeao

40 Gauss wal Levy fligh

41 Nomal diffusive vs anomalously supe-diffusive 0, K u u K u u, 0 exp, exp, K K Exponenial elaxaion Exended ail Symmeic Levy Single mode decay

42 Soluion: Mellin Tansfom 0 ] Μ[ d F s F F s Γ 0 d e s s Resuling in a se of Γ funcions q q q p p q q D q p p D Γ Γ Γ 0 0 ], [ },, { s F s s p F L Γ M M

43 Gamma funcion holomophic wih simple poles a all he non-posiive ineges; he esidue a -n is - n /n!

44 ds z s a s b s a s b i z H z H s p n i i i q m i i i n i i i m i i i a b mn pq Γ Γ Γ Γ C,, ] ; [ β β π β Alenaing signs slow convegence Seies expansion wih esidue -/l! a he poles b -β s-l, l0,... b l l p n i i i q m i i i n i i i m i i i l m a b mn pq b l z l b a l b b l b a l b b z H β β β β β β β β, 0,,! ] [ ] [ ] [ ] [ ] ; [ Γ Γ Γ Γ j,β j >0; a j,b j complex; oos of a, / b,β lef/igh of C Fox H : genealized Mellin-Banes Meije G Funcions

45 ] / [, / / K L K s C s s s s ds H ξ ξ ] [,,/,/,,/ Γ Γ Γ Γ Roos a s/-m; Γ Γ 0 ~! /, m m m K m m m ξ Explici soluion of he FFE ξ / K poles Slow convegence

46 Geneal: Long waiing imes and lage jumps, D K, < > /

47 - Saisics of Enegizaion ocesses - Anomalous Tanspo/Diffusion Enopy of Supe-diffusive pocesses

48 *Themodynamic popeies - fom micoscopic saes *Sysem in conac wih a lage esevoi *Sho empoal/spaial ineacion andom jumps p Bolzmann-Gibbs-Shannon enopy-exensive ~sysem size S BG S p ln p d Consains: p d ; < > p d σ < Vaiaional Opimizaion; β-lagange muliplie exp β exp β p ; β T /σ β Z β de Gaussian aaco in disibuion space

49 Long ange ineacion maes he enopy non-exensive S q fomaion of ails in he disibuion. q-saisics/enopy Tsallis: q [ p d]/ q S S as q Opimizaion of S q wih q-expecaion consain and β as Lagange muliplie leads o heavy ail disibuions BG < > [ p ] q d p q [ q β ] / q / Z q q β π exp β e q [ q ] /q p q exp q Z β q β exp q d exp β q β

50 <q<3 - Suden disibuion f Γ[ / ] πγ[ / ] / q 3

51 p q q [ q β ] / [ q β < q < 3 ] q d Conveges o Gaussian fo q. Conveges o Cauchy fo q Vaiance finie fo q < 5/3, diveges fo 5/3 <q <3 dxx p x q-vaiance: finie fo q < 3, q p q ~ / q i. e. 3 q / q ; 0 < < p q, N [ β / / N / ] L [ β / / N / ]: 5/ 3 < q < 3 hase Tansiion a q5/3 q-enopy conveges o sable laws disibuions

52 Compaison wih he daa allows o assess a he validiy of he single even pobabiliy densiy funcion b he elaive impoance of he diffeen expansion ems powe law ansiion c he duaion of he ineacion

53 SUMMARY Elecon Speca o o One-even Impulsive pobabiliy O A p E 4 E wih an incomplee asympoic Gaussian evoluion One-even Gadual pobabiliy Lévy O A p E ;.5.0 δ δ E wih a uncaed Levy index a he ange of Holsma-Cauchy

54 ΕΠΙΛΟΓΟΣ Disibuion funcion of elecons and ions in space plasmas due o andom, esonan ineacions wih elecomagneic ubulence displays numeous chaaceisic foms, combining Gaussians wih boen powe laws. The basic single even ineacion pobabiliy deemines he asympoic disibuion wih a phase ansiion a he index of he chaaceisic funcion, while he global ineacion ime deemines he obseved non-asympoic disibuion. The disibuion funcion of he injeced paicles may be consued via geneal agumens of sochasic pocesses, and paameeized via non-exensive enopy.