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Particle acceleration and entropy considerations ∗ JohnC Brown†, GregoryBeekman†, NormanGray† AlexanderLMacKinnon‡ 5 9 12 January1995 9 1 n Abstract a Maxwellian distribution might place a theoretical thermody- J namic constraint on the fraction of magnetic energy release 3 Possibleentropyconstraintsonparticleaccelerationspectraare likely to go into such acceleration. Brown (1993) has raised 2 discussed. thisissueagainandillustratedthepointwithasimpleexample ofthethermal/nonthermalentropydifference. 1 Solarflaremodelsinvokeavarietyofinitialdistributionsof v theprimaryenergyreleaseovertheparticlesoftheflareplasma Inthispaperweputthisquestiononamorequantitativebasis 3 –ie.,thepartitionoftheenergybetweenthermalandnonther- byexaminingtheentropyofmoregeneralparticledistribution 8 mal components. It is suggested that, while this partition can functions. Essentially the idea is that, solely on energy con- 0 1 take any value as far as energyis concerned, the entropyof a servationgrounds(firstlawofthermodynamics),depositionof 0 particledistributionmayprovideausefulmeasureofthelikeli- energyE amongN coldparticlescouldequallywellresultin 5 hoodofitsbeingproducedforaprescribedtotalenergy. (for example) a Maxwellian of temperature 2E/3Nk, a sin- 9 The Gibbs’ entropy is calculated for several nonthermal gleparticleofenergyE withtherestcold,ora deltafunction / h isotropicdistributionfunctionsf,forasingleparticlespecies, distribution of all the particles at particle energy E/N. In a p andcomparedwiththatofaMaxwellian,alldistributionshav- statistical sense, however, it is obvious that the last two are - o ing the same total number and energy of particles. Specula- much less probable than the first – ie., they have much lower r tions are made on the relevance of some of the results to the entropy. (Evolution of a plasma distribution function f from t s cosmic ray power-law spectrum, on their relation to the ob- a cold Maxwellian to a nonthermal form with high mean en- a served frequency distribution of nonthermal flare hard X-ray ergydoesofcourseinvolvea particleentropyincreaseand so : v spectrum parameters and on the additional energy release re- isnotprecludedbythesecondlawofthermodynamics. How- i X quiredtoachievelowerentropyfs. ever,whilenotimpossiblesuchanoutcomeisstatisticallymuch r Keywords:entropy;acceleration;flares;cosmicrays. less probable than production of a hot Maxwellian f since it a haslowerentropy). Quantifyinghowmuchlower the entropy (probability)isshouldhelpconstrain,inastatisticalsense,the 1 Introduction likely forms of particle distributions produced by prescribed energy release conditions or conversely constrain the energy releaseconditionsrequiredtoyieldaprescribedparticledistri- Models in which energetic particles are invoked as a major bution. component of primary energy release in the impulsive phase of solar flares have been both popular and controversial for We recognise at the outset that the present calculations are exploratory and rest on simplifying assumptions which need decades(see eg, reviews(Brown, 1991; Simnett, 1991)). De- furtherstudy. Inparticularweassume thatentropyconsidera- spite everimprovingdata andtheoreticalmodelling,however, itremainsunresolvedhowimportantsuchparticlesareinflare tionsarerelevant–thisraisesissuesregardingequilibriumand closureofthesystem,forexample,whichhavebeenaddressed energy transport. Brown and Smith (1980) suggested that inbroadtermsbyTolman(1959,pp.560-564)–butwefeelthat the lower entropy of accelerated particles compared to a pure this will be so provided we deal only with newly-accelerated ∗ToappearinA&A.Preprintno.Astro-95/02,andastro-ph/9501083. particles outside the (presumably small) accelerating volume. †Dept. ofPhysics and Astronomy, University ofGlasgow, Glasgow G12 Secondly, we compare here only the entropies of alternative 8QQ,U.K. ‡Dept.ofAdultandContinuingEducation,UniversityofGlasgow,Glasgow formsoftheparticledistributionfunctionwhichcouldbegen- G128QQ,UK eratedbydepositionofenergyE amongN particlesin a pre- 1 scribed volumeV. We neglect, forthe moment, entropycon- = kN0 g(u)lng(u)d3u tributionsfromothersystemdegreesoffreedomand,inpartic- − Zu ular,fromplasmawaves,andfromthechangingmagneticfield 1 presumedresponsibleforgeneratingtheflareparticledistribu- +kN0(cid:26)lnv03− N0 Zrn(r)lnn(r)d3r(cid:27) (4) tion. The possible importance of including plasma waves as additionaldegreesoffreedominconsideringtheevaluationof where plasmaentropyduringparticleaccelerationhasbeendiscussed N0 = n(r)d3r (5) in (Grognard, 1983). Our neglect of magnetic field entropy Zr is based on the heuristic argument that the field is associated isthetotalnumberofparticles. Ifweconsiderdifferentdistri- solelywithanorderedzeroentropyflowcomponentofthepar- butionfunctionsf withthesamespatialdistributionn(r)and ticles–theGibbsentropy flnfd3vofanyparticledis- totalvolume(ie., same N0) and use the same velocityunitv0 tribution function f(v) is∼unRcRhRanged by adding a systematic throughoutthenwecanmeasuretheentropydifferencesbythe driftv0 (ie.,v v+v0). ‘scaledentropy’ → 1 1 2 Definitions Σ= S kN0 lnv03 n(r)lnn(r)d3r kN0 (cid:18) − (cid:26) − N0 Zr (cid:27)(cid:19) (6) InthispaperwewilldealsolelywiththeGibbsentropySofthe sothatby(5),Σcanbeexpressedsolelyintermsofg(u),via 3-D (velocity space) distribution function f(r,v) for a single speciesofnonrelativisticparticleofmassm,definedby Σ g = g(u)lng(u)d3u (7) { } −Zu S = k flnfd3rd3v (1) andwenote,intermsofg,normalisation(3)is − ZvZr where k is Boltzmann’sconstant. For simplicity, we consider g(u)d3u=1. (8) only distributions in which the velocity v distribution is sep- Zu arable from the configuration space r dependence. With this We donotconsiderfurtherherethespatialstructurecontri- restriction,wemaywrite butiontotheentropy(lasttermin(6))–i.e.,weonlyconsider f(r,v)=n(r)G(v) (2) entropiesofdifferentg(u)butthesamen(r).Also,throughout the rest of this paper we consider only isotropic f(v) so that wheren(r)isthetotalparticlespacedensityatrand ud3u→ 0∞4πu2du. Wewillbecomparingtherelativeen- tRropiesofvRariousdistributionsgwiththatofapureMaxwellian G(v)d3v=1 (3) ofthesameN0andofthesametotalenergyE(ie.,wearecom- Zv paringidenticalplasmas with the same total energydeposited in them). Itis thereforeconvenientto use as velocityunitv0, Because (1) is not linear in f, the absolute value of S de- thethermalspeedintheMaxwellianplasma pendsonthevelocityandcoordinateunitsused.Here,however, we will only be concerned with entropy differences between v0 =(2kT0/m)1/2 (9) differentG(v). Ifwetakesomecharacteristicspeedv0 asve- locityunitandwrite v = v0uandG(v) g(u)/v03 then(1) whereT0 = 2E/3N0k isthetemperatureoftheplasmawhen and(3)yield → all of E is deposited as heat in a pure Maxwellian. With this choice of v0 and with g isotropic the normalisation con- S = k n(r)g(u)ln[n(r)g(u)/v3]d3rd3u dition (8) ensuring the same number of particles is that g(u) 0 − ZuZr shouldsatisfy ∞ 1 = k n(r)lnn(r)d3r g(u)d3u g(u)u2du= (10) − Zr Zu Z0 4π k n(r)d3r g(u)lng(u)d3u whiletheconstraintensuringthesametotalenergybecomes − Zr Zu ∞ 3 4 +klnv03Zrn(r)d3rZug(u)d3u Z0 g(u)u du= 8π. (11) 2 3 Scaled entropies of specific distribu- 3.3 Pure powerlaw gPL(u) tion functions Acommonlyoccurringformofacceleratedparticlespectrumin solarflaresandelsewhereinastrophysicsisthepowerlawv−α. 3.1 PureMaxwelliangM(u) Suchf(v)divergesunlessflattenedortruncatedatsmallv, so wefirstconsideratruncatedpurepowerlawcomprisingallthe Expressed in terms of g, a pure Maxwellian satisfying con- plasma particles and again with the same total energy as the straints(10)and(11)is pureMaxwellian3.1.Withnumberconstraint(10),thishasthe e−u2 form gM(u)= π3/2 (12) 0 −α u<u1 g =α 3 u (17) which, in (7), integrates straightforwardly to give the scaled PL  4π−u31 (cid:18)u1(cid:19) u≥u1 entropy 3 wherethedimensionlesslowenergycut-offspeedu1hastobe Σ = (1+lnπ). (13) M 2 relatedtoαtosatisfytotalenergyconstraint(11),viz. 3.2 Maxwellianwithbump intailgBIT(u) 3α 5 1/2 u1 = − (18) (cid:18)2α 3(cid:19) One of the forms of distribution function with an accelerated − component is a Maxwellian with a ‘bump in tail’. Here we and we note that α > 5 is required for finite total energy. consideronlyasimplecasetoallowanalytictreatment,namely We note that the relation between α and the spectral index δ abumpintailcentredonspeedu1,andofnarrowwidth∆u normally used (in solar flare physics) to describe the non- ≪ 1 over which the bump contribution to g is a constant added relativisticparticlefluxspectrumdifferentialinkineticenergy to the Maxwellian componentwhich describes the rest of the (Brown,1971)isα=2δ+1. plasma(cfBrown(1993)). Becauseofenergyconstraint(11), Substitutionof(17)and(18)in(7)leadstothescaledentropy thetemperatureofthisMaxwelliancomponentisreducedbya factorτ <1relativetothepureMaxwellianofcase3.1,bythe δ+ 1 27π2(δ 2)3 energyresidentinthebump. Ifthefractionoftheparticlesby ΣPL(δ)= δ 12 + 12ln(cid:20) 2 (δ−1)5(cid:21) (19) numberinthebumpandin theMaxwellianarerespectivelyφ − − and(1 φ)(tosatisfythenumberconstraint(10))then − 3.4 Maxwellianwitha powerlaw tail φ g (u)= 1−φ e−u2/τ + 4πu2∆u uin∆u The conventionalwisdom regardingthe behaviour of the real BIT (πτ)3/2  01 otherwise distributionfuntionforthebulkoftheelectronsinasolarflare (14) isthatitisroughlyMaxwellianatlowvelocities(wherecolli-  with sionsdominate)butofpowerlaw(orothernonthermalform)at 1 2φu2 high velocities (where mean free paths become long and run- τ = − 3 1 (15) 1 φ awaymayoccur). Sucha particlemodelspectrumfindssome − and we note that the condition u2 3/2φ has to be met supportinrecentinversionsofhighresolutionbremsstrahlung 1 ≤ hard x-ray spectra (Johns and Lin, 1992; Thompson et al., since τ 0, equality holding when the Maxwellian compo- ≥ 1992;Piana,1994)thoughtheseinferencesareambiguousbe- nentof(1 φ)N0 remainscold. − cause of the effects of noise (Craig and Brown, 1986) and of Substituting (14) and (15) in (7) and approximatingthe in- tegrals on the assumptionthat ∆u 1 and φ/(4πu2∆u) averaging over an inhomogeneous source (Brown, 1971). It 1 1 φe−u21/τ/(πτ)3/2 (ie, the bump≪is locally ‘large’)we o≫b- isthereforeinstructivetoconsidersuchadistributionfromthe − viewpointofentropy. Aconvenientfunctionalformwhichex- tainforthescaledentropy hibitstheseasymptoticbehavioursatlowandhighv,andwhich ΣBIT(φ,u1,∆u)= circumvents most of the analytic messiness associated with a piecewisedescriptionisf(v) (1+v2/v2)−α/2 (Robinson, φ 3 1 ∼ −φln(cid:18)4πu2∆u(cid:19)+ 2(1−φ) 1993). The high velocity spectral index is α, as in 3.3, and 1 the temperature T defining the shape at small v is fixed by α 1+lnπ 5ln(1 φ)+ln 1 2φu21 (.16) andv1 suchthat 21mv12 =αkT. Similarlytocases3.2and3.3, ×(cid:20) − 3 − (cid:18) − 3 (cid:19)(cid:21) α and v1 have to be interrelated to satisfy constraints (10) 3 and (11), namely u1 = v1/v0 = (α 5)/2. Taking these − constraintsintoaccount,theresultinpgg(u)is 2 3/2 Γ α 2 −α/2 2 2 g (u)= 1+ u MPL (cid:20)π(α 5)(cid:21) Γ (cid:0)α−3(cid:1) (cid:18) α 5 (cid:19) − 2 − (cid:0) (cid:1) (20) whereΓ(x)isthegammafunction. Substitutionof(20)in(7)gives,forthiscase(withα=2δ+ 1asinSect.3.3) Figure1isinfilefig1.ps Γ(δ+ 1) Σ (δ) = ln 2 MPL − (cid:20)π3/2(δ 2)3/2Γ(δ 1)(cid:21) − − (δ+ 1)[ψ(δ 1) ψ(δ+ 1)](21) − 2 − − 2 wherethepsifunctionis d ψ(x) [lnΓ(x)]. (22) ≡ dx 4 Entropy comparison of nonthermal Figure 1: Scaled entropy ∆Σ Σ Σ, relative to the M ≡ − distributions with a pure Maxwellian Maxwellian for (a) a pure truncated power law and (b) a Maxwellianwithapower-lawtail. We now compare the entropies of nonthermal distributions (16),(19),and(21)withthatofthepureMaxwellian(13). 4.2 Pure powerlaw Infigure1a,weshowthequantity∆Σ (δ)=Σ Σ as 4.1 Bump-in-tail PL M − PL a functionof δ forthe purepower-lawcase. Thekey features ofthiscurveare This case was already briefly discussed by Brown (1993). Its mostimportantfeaturesare: 1. Thereisaminimumin∆ΣPL whichoccursatδ = δ0 ≡ (3 + √7)/2 2.82. This is the value of δ at which a 1. Asφ 0,thepureMaxwellianentropy(13)isrecovered. ≈ → sharplytruncatedpowerlawmostcloselyapproachesthe 2. Foranyφ,u1(ie,nomatterhowfewparticlesareacceler- entropyoftheMaxwellian. atedortohowlowaspeed),therelativeentropy → −∞ 2. As δ , Σ and ∆Σ + . This is as∆u 0–ie,asafinitenumberofparticlesinf(u)is → ∞ PL → −∞ PL → ∞ → because as δ , more and more of the particles are crowdedintoanarbitrarilynarrowvelocityrange.Consid- → ∞ eringtherelativeentropyasthenegativeofthelogofthe concentrated at a single velocity u1 = 3/2 (by (18)) and the distribution is more and more lipke a delta func- probabilityP,thisquantifiesthequalitativeimprobability tion,withassociatedlowprobability(cf.,4.1). Thesame statementinSect.1concerningtherelativeimprobability limitationapplieshereasinpoint2ofSect.4.1. of nonthermal distributions. We note, however, that the divergence of Σ to is an artefact of taking the ap- 3. As δ 2, ∆Σ + logarithmically. This is be- proximatecontinuum−f∞unctionexpression(1)forentropy cause→thedistribuPtLion→func∞tionthenbecomesveryflatand toapointwherethesmoothnessassumptionmadebreaks widewithadecreasingprobabilityoftheincreasingnum- down(seeAppendix). berof particlesneededatu to satisfy thetotalen- → ∞ ergyconstraint(11). 3. Nomatterhowsmallthefractionφofparticlesaccelerated is,asu21 → 32φ,ΣBIT →−∞becausethefractionofthe AnotherwaytoexpresstheentropycomparisonbetweengM total energyin fast particles 1 atthatpoint, which is and g is to consider the temperature T (or total energy) → PL δ arbitrarilyimprobable.Thesameremarkasin2applies. whichapureMaxwellianwouldneedtohaveinordertohave 4 as low an entropy as gPL with the same total number N0 of in ∆ΣMPL which tends to + as δ 2 and falls as δ in- ∞ → particles. The generalisation of (12) to a Maxwellian of tem- creases. This is essentially because, for a prescribedtotal N0 peratureτ = T/T0 is gM(u,τ) = exp( u2/τ)/(πτ)3/2 and andE,thesteeppowerlawtailmoreandmorecloselyresem- − of(13)is bles the Maxwellian for large δ. On the other hand as δ de- Σ (τ)= 3[1+ln(πτ)]. (23) creases,theentropybecomessmallerandsmallerasdeviation M 2 fromtheMaxwellianincreases. Ifthisresultisinterpretedsta- Equating(23)with(19)wecanfindthedimensionlesstem- tisticallyin termsoftherelativelikelihoodofatailofgivenδ perature τ(δ)=Tδ/T0requiredfortheMaxwellianentropyto occurring, as comparedto pure heating, it means that smaller beaslowasthepowerlaw,viz. lnτ =2[Σ Σ ]/3,or PL PL− M andsmallerδshouldoccurlessandlessfrequently. 4 δ 3 δ 2 Observational inference of δ in solar flares is achieved lnτPL(δ)= 3(δ− 1) +ln(cid:20)(2π)1/3(δ −1)5/3(cid:21) (24) from the spectral index γ of bremsstrahlung hard X-rays (eg, − − Brown(1971)). Intermsoftheelectronaccelerationspectrum whichalsohasamaximumatthevalueofδ =δ0. theeventintegratedγisrelatedtoδbyδ =γ+1(forcollision Theaboveresultsallowsomequantitativeinterpretation,al- dominatedthicktargetenergylosses)andby δ =γ 1forcol- − beit somewhat speculative. First, if we consider transient en- lisionally thin targets. Kane (1974)has reported the statistics ergyreleaseevents,thentheentropiesofdistributionfunctions of observed occurrence of different γ values. The qualitative ofdifferentδshouldbemeasuresoftherelativeprobabilitiesof trend of these data is similar to that predictedfrom the above theseδsbeingrealisedif alltheenergywentintoapurepower- entropyargument–ie,asharpdeclineofnumbersofeventsof law. Ina largeset ofsuch eventsthese shouldreflectin some smallspectralindexandafrequencydistributionflatteningoff waytherelativefrequencyofoccurrenceofdifferentδs. Inpar- atlargerindices(thoughthedataareinstrumentallylimitedto ticular,valuesofδnearδ0,whereΣPLhasamaximum,should indicesγ <6). be commonest (This is certainly not the case in the observed Inview∼ofthesimplicationsmadeinthepresenttheoretical frequencydistribution of flare δ values inferred from hard X- analysis(homogeneoussourcevolume,neglectofwavesandof ray bursts, but nor are flare spectra those of pure power laws anisotropy)the trend of the theoreticaland observed distribu- withnoMaxwelliancomponent!–cf.Sect.4.3). tionsisremarkablysimilar,andworthyoffurtherinvestigation. Second, we note thatthe abovepower-lawentropyanalysis alsoappliestotherelativisticregimewhenvisreplacedbythe 5 Conclusions momemtump andthemaximumentropypower-lawspectrum againhasindexδ0. Thisδ0 isveryclosetotheobservedmean spectralindexofcosmicrayparticlesoveraverywideenergy The lower entropy of a hot plasma with an accelerated com- range(Longair,1981)–afactpointedouttousbyBell(1994). ponent, as compared to a pure Maxwellian of the same total Thatis,ifanunspecifiedmechanismoperatestoforceapower- particlenumberandenergy,quantifiestheextentto whichthe lawformonthedistributionfunction,underconditionsofcon- hardest non-thermalcomponentsare further from equilibrium stanttotalparticlenumberandenergy,thenthehighestentropy andsoshouldoccurlessfrequently.Calculationoftheentropy (most likely) state is that with a power-law index close to the difference in the case of a Maxwellian with a power-law tail observedcosmic-rayone. Thisfactshouldbeexploredfurther. of indexδ leadsto a predictedfrequencydistributionofspec- tral indices in qualitative agreementwith the general trend of the distributioninferredfromthe statistics of flare hard X-ray 4.3 Maxwellianwithpower-law tail bursts. In figure 1b we show the variation with δ of ∆Σ (δ) AlldistributionsexcepttheMaxwellianarenon-equilibrium MPL ≡ Σ Σ (equivalenttoτ )forthiscase,definedsimilarly ones, which means that we cannot define a temperature for M MPL δ − tothosein4.2,basedonequations(13)and(21),viz. themandso,inturn,thatwecannotuseourexpressionforthe entropies of these distributions in any non-trivial purely ther- 3 Γ(δ+ 1) ∆Σ (δ) = +ln 2 modynamical argument. We can, however, make a cautious MPL 2 (cid:20)(δ 2)3/2Γ(δ 1)(cid:21) statistical argument,aswe havedonein Sects. 4.2and 4.3, to − − +(δ+ 1)[ψ(δ 1) ψ(δ+ 1)](25) theextentthat, givena processwhichproducesa certainnon- 2 − − 2 (26) thermalparticledistribution,weexpectsomecharacteristicsof that distribution to be more likely than others. We have no Inthiscase,whichisexpectedtobemuchclosertothesolar doubtthattherearemore, andmoredetailed,argumentstobe flare situation than the pure power law, there is no extremum madeusingtheseresults. 5 The close coincidence of the observed cosmic-ray spectral G(u )δ3u particles in the volume centred on u . Conse- i i indexwiththatofamaximumentropypurepower-lawislike- quently,thetotalnumberofwaysofdistributingtheN particles wisetantalising. amongstthecellswillbethemultinomialΩ = N!/ N !. If i i we take δ3u to be large enough that each of the NQi is large, thenwecanapproximatelnΩusingStirling’sformulalnN! Acknowledgements ≈ NlnN N,toobtain − GB gratefully acknowledges the support of a Royal Society lnΩ N g(u )lng(u )δ3u. (28) i i ≈− of Edinburgh Cormack Vacation Research Scholarship. JCB, Xi NG,andALMgratefullyacknowledgethesupportofaPPARC wherewehaveusedthenormalisation grant, an EEC Contract, and of the Research Centre for The- oreticalAstrophysics,UniversityofSydney,wherepartofthis N = G(u )δ3u=N, (29) i i workwascarriedout.Thepaperhasbenefitedfromdiscussions Xi Xi oftheentropyproblemwithnumerouscolleaguesandinpartic- ularwithDBMelrose,PRobinson,A.M.Thompson,andAR andthenusedthelargenessofG(u )δ3utoignorelnδ3u,com- i Bell. pared with lnG(u ), before substituting G(u) Ng(u). If, i finally,thescaleδ3uissmallenoughthatweca≡ntakethedis- tributionfunctiong(u)tobeconstantoverit, thenwe canap- Appendix: the limitations of Eq. (7) proximatethis final estimate for the distributionfunction(28) byanintegral,andwritetheentropy(27)as As pointed out in Sect. 4.1, the divergence of the ‘scaled entropy’ Σ to −∞ is an artefact of the approximationsmade in Σ[g] S[g]/Nk= g(u)lng(u)d3u, (30) deriving(1).Tomakethisclear,wepresenthereabriefderiva- ≡ −Z tionof(7)whichconcentratesonthestepsbeforeEq.(1)rather wherewehavereplacedtheapproximationsignbyanequality. than,asinSect.2,onthephysicsbehindthedefinitionofΣ. Compare(7). Crucially,thevalidityoftheapproximation(30) Asabove,wecanwrite(2) dependsonthedistributionfunctionG(u)beingsuchthatthere f(r,v)=n(r)G(v), canbeinfactascaleδ3uwhichsimultaneouslyhassufficiently large N and sufficiently constant g(u). If this is not so, we i where n(r) and G(v) are position and velocity distribution generatetheartefactsdescribedinSects.4.1and4.2. functions,respectively. Sincetheseareindependent,thestatis- tical weightfor the distributionf(r,v) – the numberof ways References of realising it – is simply Ω[f] = Ω[n(r)] Ω[G(v)], so that the entropyof the distributionf(n,v) wil×l be the sum of Bell,A.R.,1994, personalcommunication the entropies in position and velocity spaces separately; this means that we can immediately ignore the constant contribu- Brown,J.C.,1971, Sol.Phys.18,489 tionofn(r)totheentropy,andinsteadconcentrateontheen- tropyofthevelocitydistributionalone.Intheseterms,thisis Brown, J.C., 1991, in(CulhaneandJordan,1991),pp413– 424 S[G]=klnΩ[G], (27) Brown,J.C.,1993, in(Uchidaetal.,1993) where k is Boltzmann’s constant, and Ω[G] is the statistical Brown,J.C.andSmith,D.F.,1980, Rep.Prog.Phys.43,125 weightofthedistribution,orthenumberofwaysinwhichthe distributioncanberealised. Notethat,sinceΩ[G]isapositive Craig, I. J. D. and Brown, J. C., 1986, Inverse Problems in integer,theentropyS[G]hasaminimumofzero(whichoccurs Astronomy, AdamHilger(Bristol) whenG(v)isadeltafunction–thereisonlyonewayofgiving Culhane, J. L. and Jordan, C. (eds.), 1991, The Physics of alltheparticlesthesamevelocity). First definea dimensionless‘velocity’u v/v0, forsome SolarFlares,Vol.336AofPhil.Trans.Roy.Soc.London ≡ arbitraryspeedv0. Imaginedividingtheaccessible volumeof Grognard,R.J.-M.,1983, Phys.Lett.95A(8),432 velocityspaceintoafinitenumberν ofsmall,butnotinfinites- imal, volumes, size δ3u, at u , so that there will be N Johns,C.M.andLin,R.P.,1992, Sol.Phys.137(1),121 i i ≡ 6 Kane, S. R., 1974, IAU Symposiumno.57, pp105–141,D. Reidel Longair,M.S.,1981, HighEnergyAstrophysics, Cambridge UniversityPress Piana,M.,1994, A&A288(3),949 Robinson,P.,1993, personalcommunication Simnett,G.M.,1991,in(CulhaneandJordan,1991),pp439– 450 Thompson, A. M., Brown, J. C., Craig, I. J. D., and Fulber, C.,1992, A&A265(1),278 Tolman,R.C.,1959, PrinciplesofStatisticalMechanics, In- ternationalSeriesofMonographsonPhysics,OxfordUniver- sityPress Uchida, Y., Watanabe, T., Shibata, K., and Hudson, H. S. (eds.), 1993, X-Ray Solar Physics from Yohkoh, No. 10 in FrontiersScience,UniversalAcademicPress,Inc 7 This figure "fig1-1.png" is available in "png"(cid:10) format from: http://arXiv.org/ps/astro-ph/9501083v1 (a) 3 2 L P Σ - M Σ 1 0 2 4 6 8 10 δ (b) 4 3 PL 2 M Σ - M 1 Σ 0 −1 2 4 6 8 10 δ