ManuscriptpreparedforAnnalesGeophysicæ withversion3.2oftheLATEXclasscopernicus.cls. Date: 12May2014 Superdiffusion revisited in view of collisionless reconnection 4 R.A.Treumann∗1,2andW.Baumjohann3 1 0 1DepartmentofGeophysicsandEnvironmentalSciences,MunichUniversity,Munich,Germany 2 2DepartmentofPhysicsandAstronomy,DartmouthCollege,HanoverNH03755,USA y 3SpaceResearchInstitute,AustrianAcademyofSciences,Graz,Austria a M ∗VisitingtheInternationalSpaceScienceInstitute,Bern,Switzerland 9 Abstract. The concept of diffusion in collisionless space whereclassicaldiffusionispracticallyinhibitedonallphys- ] h plasmas like those near the magnetopause and in the geo- icallyinterestingprocesses. (Thelessinterestingcaseγ<1 p magnetictailduringreconnectionisreexaminedmakinguse would refer to subdiffusion.) One of those processes is re- 35 - ofthedivisionofparticleorbitsintowaitingorbitsandbreak- connection, the dominant mechanism for plasma and mag- e c5 outsintoballisticmotionlyingatthebottom,forinstance,of neticfieldtransportacrossmagneticboundariesrepresented a Lévy flights. The rms average displacement in this case in- bythincurrentsheets/layers. p creases with time, describing superdiffusion, though faster Reconnectionhastheenormousadvantageoverglobaldif- s . than classical, is still a weak process, being however strong 40 fusionofbeinglocalized,withthemainphysicsofmagnetic s enough for supporting fast reconnection. Referring to two mergingandplasmamixingtakingplaceinanextraordinar- c i kindsofnumericalparticle-in-cellsimulationswedetermine ily small spatial region the linear size shorter than the elec- s10 y theanomalousdiffusioncoefficient,theanomalouscollision troninertiallength<λe=c/ωe. Inthisnote,basedonavail- h frequencyonwhichthediffusionprocessisbased,andcon- able numerical simulations, we demonstrate by estimating p structarelationbetweenthediffusioncoefficientsandthere- theanomalouscollisionfrequencyν thatmagneticmerging 45 a [ sistivescale. Theanomalouscollisionfrequencyfromelec- during reconnection can well be understood as a localized 2 tron pseudo-viscosity in reconnection turns out of being of anomalous diffusion process. This result satisfactorily uni- 15 v the order of the lower-hybrid frequency with the latter pro- fies the two originally different views on plasma transport 9 vidingalowerlimit,thusmakingsimilarassumptionsphys- acrossanapparentlyimpermeableboundarylikethemagne- 1 icallymeaningful. Tentativethoughnotcompletelyjustified topause. 5 50 2 useoftheκdistributionyieldsκ≈6inthereconnectiondif- Anomalous diffusion is also of interest in cosmic ray . fusionregion,andtheanomalousdiffusioncoefficienttheor- physics, where it is frequently described as quasilinear dif- 120 derofseveraltimesBohmdiffusivity. fusion resulting from wave-particle interactions, formulated 0 4 in the Fokker-Planck phase space-diffusion formalism. Un- Keywords. Diffusion, Lévy flights, κ-distribution, Recon- 1 fortunately, most of the observed diffusive particle spectra nection 55 : (cf., e.g. Christon et al., 1989, 1991, for the most elaborate v i observations in near-Earth space) barely exhibit the shapes X resulting from quasilinear diffusion. They turn out power ar25 1 Introduction law both in energy and momentum space, most frequently beingdescribedbestbyso-calledκ-distributions 60 Anomalousdiffusionisthesummaryheadingofallprocesses where the ensemble averaged mean-square displacement (cid:18) x2 (cid:19)−(κ+1+d/2) p(κ|x)=A 1+ (1) (cid:104)x2(cid:105)∝tγ deviates from linear time dependence γ=1 with κ κ(cid:96)2 classical (Einstein) diffusion coefficient D =2Tν /m, cl c with normalization factor A, d dimensionality, and (cid:96) corre- with T temperature, and ν classical binary collision fre- 30 c lation length (cf., e.g., Livadiotis & McComas, 2010, 2011, quency. For γ>1 one speaks of superdiffusion, which is 2013, for an almost complete compilation of the properties of particular importance in the collisionless space plasma ofκ-distributions)withhigh-energy/high-momentumslopes 65 Correspondenceto: R.A.Treumann to which the parameters κ are related. Estimated κ values (rudolf.treumann@geophysik.uni-muenchen.de) from the magnetospheric observations range in the interval 2 R.A.TreumannandW.Baumjohann: Superdiffusion 5<κ<10 (Christon et al., 1991). Such distributions were connection of the above probability spectrum to real space introducedbyVasyliunas(1968), followingasuggestionby distributions,inparticulartoκdistributions,isnon-trivial. S.Olbert,asbestfits.1 Inthetime-asymptoticlimit,κdistri- Thediffusionprocesscanbeenvisagedasconsistingofa 70 butionswereexplicitlyderivedbyHasegawaetal.(1985)and sequenceofnsteps(cf.,e.g.,Treumann,1997)bridgingthe 110 Yoon et al. (2012). Their q-equivalent relation to superdif- timefromt=0tot=t withtheparticlejumpingfromfirst n fusion has also been suggested (Tsallis et al, 1995; Prato waitingtonthwaitingposition,theexpectationvalueofthe & Tsallis, 1999; Bologna et al, 2000; Gell-Mann & Tsallis, latterbecomes 75 2004,andreferencestherein). (cid:90) n (cid:89) Forthepresentpuroposeswemakenodirectuseofthese (cid:104)x2(n)(cid:105)= x2p(n|x)ddx, p(n)= p(i). (3) distributionsasthey,apparently,playnoroleinreconnection. 1 Rather,aswedemonstrate,anomalousdiffusioninreconnec- The nth expectation value is proportional to the random 115 tion results from processes leading to waiting statistics and mean square of the displacement x2 and a power of the causinggyro-viscosity. 80 elapsedtimesequence. WorkinginFourier(ormomentum) spacek,multiplicationoftheprobabilitiesyields 2 Diffusionprocess p(n|k)=pn(k)∝exp(−ankα)∼ p(k(cid:48)) (4) withp(k(cid:48))theprobabilityofthenthtimestep. Hencek(cid:48)= Collisionlessdissipationandrelateddiffusionismediatedin120 kn1/α. Any real space coordinate therefore scales as x→ awidersensebycollisionlessturbulence(cf.,e.g.,Allegrini xn−1/α. Forthereal-spaceprobabilitythisimpliesthat etal.,1996). Herebinarycollisiontimesτ (cid:29)τ byfarex- c a 85 ceed anomalous interaction times. Any real non-collisional p(n|x)ddx −→ p(cid:0)x/n1/α(cid:1)ddx/nd/α (5) diffusion proceeds on times much shorter than classical (in comparison infinite) diffusion times with absolute values of yielding from Eq. (3) for the nth displacement expectation anomalousdiffusioncoefficientsbeingsmall. value 125 The superdiffusion process can be considered as a se- (cid:10)x2(n)(cid:11)=n2/α(cid:104)x2(cid:105) (6) quence of «waiting times» when the particle is in a quasi- 90 stationarytrappedstatefollowedby«breakouts»intoballis- withα<2notpreciselyknownbuttobedeterminedbelow tic motion until the next trapping and waiting period starts fromnumericalsimulations. Themean-squaredisplacement (Shlesinger et al., 1987; Klafter et al., 1990). Such particle shouldbeobtainedfromthesecondmomentoftheunderly- motions are typical, for instance, for Lévy flights (cf., e.g., ingreal-spacedistributionfunction,forinstancetheκdistri- 130 95 Shlesingeretal.,1993). bution,yielding Working in d-dimensions, the probability of a particle to occupy a particular volume element during a process, as- (cid:104)x2(cid:105)=21dκ(κ+1)(cid:96)2 (7) sumed to be caused by some unspecified (nonlinear) inter- anexpressionwewillmaketentative(notfullyjustifiedand action between particles and plasma waves, is most conve- forthepresentpurposesmarginal)useofonlyattheveryend nientlyformulatedinwavenumberspacekwithprobability 100 inapplicationtoreconnection. 135 spectrum p(k)∝exp(−akα), (2) 3 Diffusioncoefficient where a is some positive constant, and 0<α∈R a real In using probability steps n, time has been discretized into number. α≥2 reproduces the classical Gaussian probabil- piecesoffreeflight,waitingandsomekindofinteraction. In 105 ity spectra (Tsallis et al, 1995). Non-Gaussian spectra have theaveragetheinteractioniscoveredbyafictitiousanoma- flatter tails implying α<2, indicating superdiffusion. The lous collision frequency ν . Ordinary binary collision fre- 140 a quencies ν are very small, suggesting a scaling ν (cid:29)ν c a c 1Theoretical attempts of justifying solar wind κ-distributions with the anomalous timescale ν−1=τ (cid:28)τ =ν−1 much a a c c followed, invoking wave-particle interactions with inclusion of less than the collision timescale τ . The diffusion process c residualbinarycollisions(Scudder&Olbert,1979).Statisticalme- takesplaceinatimet<τ .Replacingthetimestepsn→ν t c a chanicalargumentswerebasedonnonextensivestatisticalmechan- themeansquarenthdisplacementbecomes ics(Tsallis,1988;Gell-Mann&Tsallis,2004). Fromkineticthe-145 ory they were identified as collisionless turbulent quasi-stationary (cid:104)x2(t)(cid:105)=(cid:104)x2(cid:105)(ν t)2/α (8) a statesfarfromthermalequilibriumresultingfromanomalouswave- particle interactions (Treumann, 1999a,b). There the role of the With γ=2/α it defines the anomalous diffusion coefficient temperatureT asthermodynamicderivativewasclarified(seealso D whenmultiplyingbyτ−1=ν a a a Livadiotis&McComas,2010).Therelationbetweenthenonexten- siveqandtheκparameterswasgivenfirstinTreumann(1997). Da(d,t)=(cid:104)x2(cid:105)(νat)2/ανa≡Dca(νat)2/α (9) R.A.TreumannandW.Baumjohann: Superdiffusion 3 as a function of time tν . Since ν (cid:29)ν , it is much less which 525000 had high energies and contribute most to the 150 a a c than the classical diffusion coefficient which in this case mean displacement as well as for the high time resolution, wouldcorrespondtofreeflight. Underanomalouscollisions the statistical error of the measurement is smaller than the the free flight is abruptly interrupted and reduced to non- widthofthelineinthisfigure!) 205 stochastic diffusion by the finite anomalous collision fre- Adoptingtheprobabilityspectrumbasedtheorytheexper- quencyν . imentallydeterminedslopeof2/α≈1.17oftheaveragedis- 155 a placementinFigure1tellsthatinthesesimulationsonehad 4 Evolution α≈1.71 (experimental) (10) Estimates of diffusion coefficients respectively γ based on210 avaluesubstantiallyfarawayfromtheGaussianlimitspec- tral slope α=2 and being less than it, thus indicating quite observations in space plasma are not only rare but unreli- strongsuperdiffusion. able. Theysufferfromthepracticalimpossibilityofanysuf- ficiently precise determination of particle displacements as 160 functionoftimeandthesubsequenttransitiontotheasymp- 5 Transitiontocollisionalstate totic state. In addition they are mostly based on quasilinear theories of particular instabilities (Sagdeev, 1966; Anomalousdiffusionproceedsonafasterthanclassicaltime Liewer & Krall, 1973; Huba et al., 1977; Davidson, 1978; scale with time dependent diffusion coefficient which justi- Sagdeev, 1979; Huba et al., 1981; LaBelle & Treumann,215 165 fies the term superdiffusion. In spite of this, the coefficient 1988; Treumann et al., 1991; Yoon et al., 2002; Matthaeus D =(cid:104)x2(cid:105)ν infrontofthetimefactordeterminingtheab- ca a etal.,2003;Daughtonetal.,2004;Riccietal.,2005;Royter- solutemagnitudeofthediffusionisgenerallysmall. Itdoes shteynetal.,2012;Izutsuetal.,2013)whichdonotproperly not compensate for the absolute smallness of the diffusion accountforanynonlinearinteractions. coefficient. When, after a long time has elapsed the order 220 We therefore refer to high-resolution particle-in-cell sim- 170 of the classical collision time τ , classical diffusion takes c ulations (Scholer et al., 2000) performed in order to deter- overscatteringsomeparticlestolarger,someothersbackto mine the cross-magnetic field diffusion of ions near quasi- smaller displacements and setting the collisionless process perpendicularshocks. TheresultsarecompiledinFigure1. temporarilyoutofwork.Theaveragedisplacementofthevi- Theright-handsideofthefigureshowsonemacro-particle olentlyscatteredparticleswhosedisplacementlinehasbeen 225 orbit arbitrarily selected out of the large number of parti- 175 smeared out suddenly over a large spatial domain may now cles used in the simulation to determine their instantaneous followtheclassicallineartemporaryincrease. displacements from the origins of their trajectories in the One single elapsed binary collision time may not suffice simulationasfunctionofsimulationtimemeasuredinunits to stop the nonlinear collisionless interaction process. The of their identical (energy-independent) gyration frequency widelyscatteredparticlepopulationmaystillhavesufficient 230 ω =eB/m in the total magnetic field, which is the sum 180 ci i freedom to organize again into a softened collisionless dif- of the ambient and the self-consistently generated turbulent fusion which lasts until the next binary collision time has wavemagneticfield. Theparticleshiftsitspositionperpen- passed. During this second collisionless period the slope dicular to the magnetic field from its start point to the end should be flatter than the initial collisionless slope, and af- point in the simulation. It is found in a slowly changing ter statistically sufficiently many periods of elapsed classi- 235 waiting position, performs jumps to new waiting positions, 185 calcollisiontimesnocollisionlessmechanismsrevivesany- andendsupduringafinaljump. Suchanorbititneitheradi- more. Diffusion has by then become completely classical. abaticnorstochastic. ThesesequencesareschematicallyshowninFigure2. TheleftpartoftheFigureshowstheaveragedisplacement, ensemble averaged over the entire particle population, as 190 functionofsimulationtime.Afterperforminganinitialoscil- 6 Discussion lationtheaveragedisplacementssettleintoanaboutsmooth continuously increasing curve of constant slope (cid:104)(∆x)2(cid:105)∝ Waiting statistics offers an approach to anomalous diffu- 240 t1.17. sion in various regions of space plasmas where classical The slope of the final evolution of the average displace- (andneo-classical)diffusionprocessesareinappropriate,vi- mentisclosetobutbynomeansidenticalwithclassicaldif- olently failing to explain the transport of plasma and mag- 195 fusionwhichisshownbytheslopeofthetwostraightlines netic field. Application to numerical simulations near col- in the figure. Though the deviation in the slope is small, it lisionless shocks determined the value of α≈1.71 which 245 is nevertheless substantial and statistically significant, indi- turnsouttobeclosetobutsufficientlyfarbelowitsclassical catingasuperdiffusiveprocesswhichdeviatesfromclassical (Gaussian) limit α=2 for identifying superdiffusion. Su- diffusion. (We should note that, because of the large num- perdiffusioncoefficientsobtainedaresmallbutincreasewith 200 berof∼6.3×106 macro-particlesusedinthesimulationof time. 4 R.A.TreumannandW.Baumjohann: Superdiffusion Ion trajectory 20 Mean Perpendicular Displacement 3 10 start 0 ‘super-diffusion’ 2 10 7_ waiting 〉 ∝ t1 . 1 7 ≈ t6 2x(Δ) ∝ t1 x / λi-20 jumping 〈 shock 1 10 simulation data classical diffusion -40 0 10 end 1 2 3 10 10 10 -60 Time (tω ) -20 0 20 ci y / λ i Fig.1.Two-dimensionalnumericalsimulationresultsofthemeandownstreamperpendiculardisplacementofionsnearaquasi-perpendicular supercriticalshock(shocknormalangleθ=87◦,AlfvénicMachnumberM =4asfunctionofsimulationtime(simulationdatatakenfrom A Scholeretal.,2000,courtesyAmericanGeophysicalUnion). Distancesaremeasuredinioninertiallengthsλ =c/ω withω ionplasma i i i frequency.Left:Theparticledisplacementperformsaninitialdampedoscillationbeforesettlingintoacontinuousdiffusiveincreaseattime about ω t∼40 [in units of the ion gyro frequency ω ]. The further time-evolution deviates apparently only slightly from the classical ci ci (linear) increase of the mean displacement, following a (cid:104)(∆x)2(cid:105)∝(ω t)1.17 power law. (Note that simulation-time limitations did not ci allowmonitoringthelong-timeevolutionoftheensemble-averagedsquaredisplacement,thusinhibitingdeterminationofthefinalstateof thediffusionprocess.) Right:Latetimetrajectoryofanarbitraryionofthesampleused. Theorbitisprojectedintotheplaneperpendicular tothemeanmagneticfieldwhichconsistsofasuperpositionoftheambientandwavemagneticfields.Theionorbitisneitheranundisturbed gyro-oscillationnorasmoothstochastictrajectory. Itconsistsofwaiting(trappedgyrating)partsandpartswhentheionsuddenlyjumps aheadalongdistancecausebysomebriefbutintenseinteractionbetweentheparticleandwavespectrum. Thisbreakoutofgyrationis typicalforrareextremeeventslikethoseinLévyflightsreferredtointhepresentpaper. The present theory is based on constant α for the en- maining constant. Under such circumstances the diffusion 250 tire diffusion process. This might be unrealistic. Real coefficient in Figure 2 never approaches the classical limit 270 powers α[W (t)] will turn out functionals of the time- butsettlesinsteadonitsmuchloweranomalouscollisionless w dependent turbulent wave levels W (t) which are gener- level (cid:10)Dfin(cid:11). The related processes lie outside the present w a ated self-consistently in the underlying turbulent collision- investigation. We may, however, estimate a lower bound lesswave-particleinteraction(foraderivationofthephase- on the average final diffusion coefficient (cid:10)Dfin(cid:11) assuming 255 a spacedistributioninparticularwave-particleinteractionscf., τ ≈ν−1,whichyields 275 f a e.g.,Hasegawaetal.,1985;Yoonetal.,2012,yieldingtime- asymptoticvaluesofthephase-spacepower-lawindexκde- D (cid:46)(cid:10)Dfin(cid:11) (12) pendingonwavepowerW ). ca a w Itmaybeexpectedthat,withincreasingwavelevelW (t), 260 w In the following we list a few practical consequences of anewcollisionlessequilibriumwillbereachedwherethedif- ourtheorywhichfocusononeofthemostinterestingprob- fusionprocess,infinitetimet∼τ ,approachesanothernew f lemsincollisionlessplasmaphysics,themechanismofcol- andapproximatelyconstantdiffusivity lisionlessreconnectionofmagneticfields. 280 lim D (t)−→Dfin(t(cid:38)τ )0 yields the trivial require- tp thespectrumiscompletelydescribedbytheBesselfunction. ment α<5/2 valid for all interesting cases including the720 Physically it indicates that in the long range the spectrum Gaussian. 675 corresponding to the κ distribution is indeed, as expected, Another condition is obtained from the requirement that far away from Gaussian behavior. It, however, has no sim- p(α|x) must be a real probability distribution. Setting ilarity whatsoever to the model probability spectrum in Eq. (cid:61)(k )=0 one concludes that α =1+3/2n > 1, with tp d=1 (2). Theκdistributiondescribesstrongcorrelationsbetween n=1,2,.... Hence, α = 5,7,3,11,... can assume dis-725 d=1 2 4 2 8 the particles with, if at all, weak dependence of α on κ (at cretevaluesonlywhich,forlargen,convergeto1. 680 thebesttohigherorder). Forarbitrarydandα=2−α(cid:48)≈2,i.e. α(cid:48)(cid:28)1,thetwoap- In order to get a feeling of such a weak dependence one proximatesolutionsfortheturningpointbecomebothpurely may manipulate the exponential contributed by the Bessel imaginarykt(p1,2)=−i[2(d−1)/x; x(1−α(cid:48)/2)/2a−4(d− function and its frontal k¯ dependent factor in the above ex- 730 1)/x]. The reality condition for the real space probability pressionforlargek¯tobecome requires treating the complex turning point integral. Even 685 (cid:104) (cid:16) (cid:17)(cid:105) closetotheGaussianlimitα=2calculationoftherealspace k¯ζexp−(k¯/2)=exp −1k¯ 1−lnk¯2ζ/k¯ 2 probability distribution from the hypothetical non-Gaussian (cid:104) (cid:16) (cid:17)(cid:105) spectrumEq. (2)isnontrivial,causingseriousdoubtsinthe ≈exp −1k¯exp −lnk¯2ζ/k¯ (A6) 2 assumed generality of the kα model spectrum. No further (cid:104) (cid:105) usefulinformationisobtained. ≈exp −1k¯(1−2ζ/k¯) 690 2 The x-dependence of the solution of the d=1 integral is contained in the factor (aα/x)(α−2)/2(α−1) which may735 Again,thisexponentialreproduceswhencarryingoutthedif- ferentiations. Hence one finds approximately for the func- be interpreted as the large-x limit of the κ distribution for tional d=1. It can, by comparison, be reconciled only for val- 695 ues 0<αd=1=(1−κ)/(34−κ)< 52, yielding 0<κ< 34, a α[κ]≈1−2ζ/k¯, ζ=κ+1−d/2 (A7) narrow and extreme range only at this dimensionality. The lower limit on κ gives 4 (cid:46)α . Imposing the Gaussian depending on the dimensionality d and the spectral scale k¯. 3 d=1 limit α=2 yields finally 0<κ≤ 1. This is rather differ- For d=1 it is independent of the differential operation. It 2 740 entfromtheGaussianlimitκ→∞oftheκ-distribution, in isobviousthatα=2,theGaussianlimit,isnotcontainedin 10 R.A.TreumannandW.Baumjohann: Superdiffusion thisconditionwhichshowsthatαissmallandforlargek¯is Thisisfinallyseenwhenstraightforwardlytakingthelarge oforderα∼O(1),asalreadyinferredabove. Moreover,for order expansion of the Bessel function for κ(cid:29)1. Then the √ largeκonehasfromthisexpressionα≈1− κ/kexplicitly abovespectrumasymptoticallybehavesas 775 excludingthelimitκ→∞unlessuniformlyalsok→∞. 745 exp−κ¯η The last condition is satisfied for k¯>2ζ only providing p (k)∼D(d−1)√ (A10) κ κ¯(1+k¯2/4κ¯2)1/4 anotherconditionholds, 0<κ< k2 (cid:34)1±(cid:115)1−4(cid:96)(cid:18)1−d(cid:19)(cid:35)2≈(cid:18) k (cid:19)2 (A8) lwni(cid:104)th(k¯/κ¯2κ¯=)(cid:16)121(+κ(cid:112)+11++k¯2d//42κ¯)2,(cid:17)(cid:105).anIdt isηno=t o(cid:112)bv1io+usk¯2h/o4wκ¯2th+is 16(cid:96)2 k 2 2(cid:96) spectrummatchesanexp(−ak2)functionforκ→∞. which,forinstance,yields0<κ∼1forthemodestrequire- Takingκ(cid:29)kyieldsadependence∼exp(−κ¯−k¯2/8κ¯)→ 780 ment k/(cid:96)∼2, thus identifying a sufficiently large range of 0 for κ→∞. On the other hand, in the limit k(cid:29)κ for 750 validity of κ for all three spatial dimensions in treating the all fixed κ, one has η∼k¯/2κ¯ and thus a dependence ∼ averagedisplacementproblem. Noindependentrelationbe- exp−k¯/2. tweenαandκhasyetbeenobtained. This α=1 spectrum had been elucidated already above TheonlywayofincludingtheGaussianlimitα=2isvia and also in Treumann (1997). It represents the asymptotic 785 application of renormalization group methods to the func- spectral behavior of κ distributions, indicating that asymp- 755 tionalEq. (A7). Thisprocedureyields totically these are practically independent of κ and thus, at the best, are extreme versions α=1 of spectra of the form α[κ]≈2+exp[−(1+2ζ/k¯)] (A9) of Eq. (2), indicating the presence of extraordinarily strong correlations. which for fixed k and large κ exhibits a dependence α∼790 √ exp(− κ/k). Thisformindeedincludesthelimit lim α= κ→∞ 2.However,itisnotofanyinterestasthislimitisapproached 760 fromabove,yieldingα(cid:38)2forallκ. One may notice that the exponential power spectrum is notonlyapproximatehere,ithasnotevenbeenimposednor used in the calculation of the power spectrum belonging to theκdistribution. Hence,iftheκdistributionisbelievedto 765 describe power spectra of the type of Eq. (2) invluding the underlyingphysicalprocesses,thenthiscanonlybetakenas veryapproximate. Generallyspeaking,thoughtheκ-distributioncontainsthe Gaussianprobabilityforκ→∞,thereconstructionofasim- 770 pleGaussianprobabilityspectrumfromtheκdistributionis notsuccessful.

Superdiffusion revisited in view of collisionless reconnection

Similar Superdiffusion revisited in view of collisionless reconnection