Non-Extensive Approach to Quark Matter
aa r X i v : . [ h e p - ph ] D ec EPJ manuscript No. (will be inserted by the editor)
Non-Extensive Approach to Quark Matter
Tam´as S. Bir´o , G´abor Purcsel and K´aroly ¨Urm¨ossy KFKI Research Institute for Particle and Nuclear PhysicsReceived: date / Revised version: date
Abstract.
We review the idea of generating non-extensive stationary distributions based on abstract com-position rules of the subsystem energies, in particular the parton cascade method, using a Boltzmannequation with relativistic kinematics and modified two-body energy composition rules. The thermodynam-ical behavior of such model systems is investigated. As an application hadronic spectra with power-lawtails are analyzed in the framework of a quark coalescence model.
PACS.
Power-law tailed distributions are abundant in Nature andin human technology ranging from high energy particlespectra to fluctuations in stock markets or connectivitystatistics in the Internet. It would be natural to explainthis abundance by a universal, statistical limiting distri-bution since different causes result in similar outcomes.A more prestigious attempt is to set such phenomenainto a united framework of non-extensive thermodynam-ics, based on certain generalizations of familiar basic for-mulas. In particular generalizations of the Boltzmann –Gibbs – Shannon entropy formula were seeked as fundingstones for such a general treatment [1,2,3,4,5,6,7,8].Several basic questions arise during this enterprise:among those the uniqueness of equilibrium state and theentropy function describing irreversibility, the connectionbetween composition rules for basic thermodynamical quan-tities between two large subsystems and the extensivitylimit for a system with a large number of degrees of free-dom, and the very question that which microscopical mech-anisms lead to such a distribution. Is this a sign of non-equilibrium, of incomplete equilibrium or just of a new,generalized kind of equilibrium? Applying and justifyinga statistical, least thermodynamical approach to high en-ergy heavy ion collisions, as it is being central in the ex-perimental quest for quark matter, in particular requiresclarification of the above questions. Any inference to athermal state and a physical temperature of the quarkmatter from single particle spectra must connect the fitparameter measuring the spectral slope to basic princi-ples of thermodynamics.In recent years we have been succeeding towards an-swers to the above problems. After facing the fact thattransverse momentum spectra fit well to a cut power-law distribution towards much higher values than just the simple Gibbs-exponential, a particular parton cascade ap-proach was suggested in Ref.[9] for generating these dis-tributions. It has been observed that the stationary dis-tribution generated and maintained by a Boltzmann typeequation is intimately related to the energy compositionrule used in two-particle encounters. A simple modifica-tion of the kinetic energy addition rule among two part-ners, which in high energy collisions is probably relatedto the relativistic kinematics, leads to the observed result.A general treatment of abstract composition rules is pre-sented in Ref.[10], where the non-extensivity property isrelated to the deformation of addition rule and hence tothe deformation of the classical Gibbs exponential.The physics’ question to begin with is the source ofnon-extensivity, especially for the two most relevant quan-tities, energy and entropy. It is relatively easy to con-struct examples with non-extensive energy, whenever theinteraction retails a fractal structure in the phase spaceand therefore cannot be neglected in the large volume –large particle number limit, as it is traditional in classicalthermodynamics. It is much harder to understand non-extensive entropy, however. In order to shed some lightto possible mechanisms by which non-extensivity in one-particle variables, like entropy and energy, can occur inphysical systems, let us investigate a very particular case.We assume that in an N-particle system there are two-particle correlations left and seek for their relative con-tributions to total energy and entropy. For the sake ofdemonstration we regard the following special form of thetwo-particle density: ρ = f ( p ) f ( p ) g ( r ) , (1)which is factorizing in the momentum space via one-particledistribution functions, but is connected in the coordinatespace via the pair-distribution function, g ( r ), of the rela-tive coordinates. Tam´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter
The trace over states is determined via phase spaceintegrals, normalized to satisfy the following conditions in d spatial and momentum dimensions: R d d r V, R d d r g ( r ) = V eff , R d d p (2 π ¯ h ) d f ( p ) = n, (2)with V being the total volume, V eff the available volumefor a partner of a given particle, and n the average (mean)density in the system. We normalize the integrals so that nV = N and nV eff = N − ρ = N ( N − ρ = Tr ( ρ ) = ( N − f ( p ) . (3)The total entropy of a correlated pair in matter, − Tr ( ρ ln ρ ) / Tr ( ρ ), is expressed by S = − Z d d p (2 π ¯ h ) d Z d d p (2 π ¯ h ) d Z d d r Z d d r ρ ln ρ N ( N − . (4)In calculating this quantity two further individual inte-grals occur: s = − Z d d p (2 π ¯ h ) d f ( p ) ln f ( p ) ,V info = − Z d d r g ( r ) ln g ( r ) . (5)Using these notations one arrives at: S = 2 sn + V info V eff . (6)Generalizing the above expression valid for the two-particledensity, ρ , to an N -particle density, ρ ...N , factorizedinto N ( N − / S N N = sn + n V info . (7)The entropy of such a system is considered to be extensive,as long as the specific ratio remains finite in the largeparticle number limit: lim N →∞ S N N < ∞ . (8)In this sense dangerous pair distributions are those, forwhich V info increases with N at fixed mean density n . Ina familiar piece of matter the pair distribution function g ( r ) approaches one at large distances, in these cases V info is finite and hence the entropy is extensive. In case ofa quark gluon plasma, however, some infrared magneticmodes remain non-perturbative and hence long range cor-relations remain. As a consequence − g ln g may not tend Fig. 1.
Schematic plots of the pair distribution function andthe corresponding energy and entropy contributions in short(top) and long range (bottom) correlated matter assuming a1 /r and a σr type pair potential, respectively. to zero fast enough and therefore the integral V info ineq.(5) may increase as a function of N . For example con-sidering power-law type pair distribution functions, like g ( r ) = r a / (1 + r b ), the corresponding integrals up to alarge radius, R scale like V eff ∼ R d + a − b and like V info ∼ R d + a − b ln R . In this case the specific entropy for large N becomes S N N −→ sn + const . ( a − b ) N ln N (9)with some unspecified constant. For a = b this would leadto a non-extensive entropy. For other possible sources ofnon-extensive entropy see Ref.[11].The total energy can be calculated in a similar way.Assuming a v ( r ) pair-potential depending on the relativecoordinate only and individual kinetic energies, K ( p i ), wearrive at E N N = en + n V pair (10)with e = Z d d p (2 π ¯ h ) d f ( p ) K ( p ) ,V pair = Z d d r g ( r ) v ( r ) . (11)Some typical g ( r ) functions are shown in Fig.1. On thetop figure a pair distribution function tending to one atlarge distances and a Coulomb-like pair potential, whileon the bottom figure a linear confining potential, v ( r ) ∼ am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 3 r and a power-law tailed pair distribution function areassumed.In most physical systems studied traditionally in ther-modynamics, like gases, liquids, plasmas, etc. the function g ( r ) approaches the value one at large distances. Thereforethere are no non-extensive contributions to the entropyper particle. For the energy the situation is different sofar, since v ( r ) ∼ r − b might not approach zero for largedistances fast enough, producing this way a contributionto the energy per particle, E N /N which may even divergein the large N limit. Such a case is an unscreened 1 /r -likepotential in three dimensions. Non-extensive quantities, whose amount per particle is notfinite in the thermodynamic limit, are also not additive,because the repeated composition by simple addition rulesalways leads to a result proportional to the number ofsteps. It is possible, however, that one is able to find an-other quantity, a certain function of the non-additive one,which is additive. This way the non-extensive thermody-namics can be treated by mathematical algorithms whichwere designed for additive composition rules. In this sec-tion we analyze the mathematical background of compos-ing energy and/or entropy of subsystems and then repeat-ing this composition. The thermodynamical limit is ap-pointed to the infinite repetition of the composition withan infinitesimal amount [10].
Let us denote an abstract pairwise composition rule by themapping ( x, y ) → h ( x, y ). The important question arises,that what happens if we repeat such a composition rulearbitrarily long, each time applying to an infinitesimalamount: This way one deals with the thermodynamicallimit of composition rules corresponding to the energy orthe entropy. The effective rule in this limit, which appliesto results of repeated rules themselves, has special prop-erties then.From the starting rule we demand only a trivial prop-erty: that the composition with zero should be the identity h ( x,
0) = x. (12)We do not assume in general symmetry (commutativity),such as h ( y, x ) = h ( x, y ) nor we demand associativity h ( h ( x, y ) , z ) = h ( x, h ( y, z )) . (13)Her we note that the general solution of the associativityequation (13) is given by h ( x, y ) = X − ( X ( x ) + X ( y )) (14)with X ( x ) being a strict monotonic function [12]. We shallrefer to this mapping function as the ”formal logarithm”, because it maps the arbitrary composition rule h ( x, y ) tothe addition by taking the X -function of eq.(14): X ( h ( x, y )) = X ( x ) + X ( y ) . (15)Due to this construction the generalized analogs to classi-cal extensive (and additive) quantities are their formal log-arithms, whenever the composition rule is associative. Asa consequence stationary distributions, in particular thoseobtained by solving generalized Boltzmann equations [9],are the Gibbs exponentials of the formal logarithm, f ( x ) = 1 Z e − βX ( x ) . (16)Let us now regard a large number of iterations, N , ofa general composition rule. We apply it to a small amount y/N and repeat this ( N −
1) times, constructing this waythe quantity x N ( y ) := h ◦ . . . ◦ h | {z } N − (cid:16) yN , . . . , yN (cid:17) . (17)We consider the large- N limit,lim N →∞ x N ( y ) < ∞ , (18)if this is finite for a finite y , we can apply all formulasof classical thermodynamics usually applied to extensivequantities. Such a limiting quantity is extensive, but notnecessarily additive. Our goal is to obtain the asymptoticcomposition rule, x N + N = ϕ ( x N , x N ) (19)in the limit N , N → ∞ . The recursion for the n -th stepof this repetitive composition is given by x n = h (cid:16) x n − , yN (cid:17) , (20)starting with x = 0. Subtracting x n − = h ( x n − ,
0) fromboth sides we arrive at x n − x n − = h ( x n − , yN ) − h ( x n − , . (21)Denoting by t = ( n − /N the extensivity share alreadyachieved, one step takes ∆t = 1 /N , and the above recur-sion can be Taylor-expanded for a small y/N = y∆t : x ( t + ∆t ) − x ( t ) = y ∆t ∂∂y h ( x ( t ) , y ) (cid:12)(cid:12)(cid:12)(cid:12) y =0 + + O ( ∆t ) . (22)In the large N ( ∆t →
0) limit this becomes equivalentto a differential equation similar to a renormalization flowequation: dxdt = y h ′ ( x, + ) . (23)In this expression h ′ ( x, + ) denotes the partial derivativeof the rule h ( x, y ) with respect to its second argumenttaken when this value approaches zero from above. Note Tam´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter that the uniformity of subdivisions to y/N is not neces-sary; all infinitesimal divisions summing up to y by t = 1lead to the same differential flow equation.The solution of eq.(23), L ( x ) = x Z dzh ′ ( z, + ) = y t, (24)defines the additive mapping of x , i.e. the formal loga-rithm L ( x ). By the help of this the following asymptoticcomposition rule arises: x := ϕ ( x , x ) = L − ( L ( x ) + L ( x )) ; (25)it is already associative and commutative. Commutativityis trivial and associativity is also easily proved: ϕ ( ϕ ( x , x ) , x ) = L − ( L ( ϕ ( x , x )) + L ( x ))= L − ( L ( x ) + L ( x ) + L ( x ))= L − ( L ( x ) + L ( ϕ ( x , x )))= ϕ ( x , ϕ ( x , x )) . (26)It is interesting to check that all associative rules aremapped to themselves in the above limit . Given an asso-ciative composition rule, h ( x, y ), it possesses a formal log-arithm, X ( x ), which is additive: X ( h ( x, y )) = X ( x ) + X ( y ) . (27)Now taking the derivative of this equality with respect tothe second argument we obtain X ′ ( h ) ∂h/∂y = X ′ ( y ) (28)which taken at y = 0 becomes h ′ ( x, + ) = X ′ (0) X ′ ( h ( x, . (29)Due to the property h ( x,
0) = x (equivalently X (0) = 0)the formal logarithm of the asymptotic composition ruleis given by L ( x ) = x Z X ′ ( z ) X ′ (0) dz = X ( x ) X ′ (0) ; (30)it is proportional to the formal logarithm of the startingrule. Therefore the asymptotic rule is exactly the same aswe begun with: ϕ ( x, y ) = h ( x, y ). The freedom in a factorof the formal logarithm is used to set X ′ (0) = 1. This wayany associative composition rule describes a limiting ruleof a class of non-associative rules. The stationary distribution eq.(16) in the large- N limitcontains the formal logarithm, L ( x ). In fact the composedfunction, e a = exp ◦ L is the one, which is frequently called a ’deformed exponential’ in the literature. Its inverse, ln a = L − ◦ ln is then the corresponding ’deformed logarithm’ .These functions are inverse to each other. Further proper-ties of the traditional exponential and logarithm functionsare, however, not automatically inherited. In particularreciprocals and negatives follow different rules as we areused to.In the particular case of scaling formal logarithms, L a ( x ) = 1 a L ( ax ) , (31)several interesting identities hold, among others the fol-lowings: L ( x ) = x,L − a ( x ) = 1 a L − ( ax ) , ln a (1 /x ) = − ln − a ( x ) , /e a ( x ) = e − a ( − x ) (32)Since a = q −
1, the a ∗ = − a duality corresponds to the q ∗ = 2 − q Tsallis-duality. This can be important for theparticle-hole relation for fermions:1 − e a ( − x ) + 1 = 1 e − a ( x ) + 1 . (33)Let us now list some important particular rules andtheir asymptotic pendants considered in applications ofnon-extensive statistics to physical systems.The trivial (and classical) addition is the simplest com-position rule: h ( x, y ) = x + y . In this case h ′ ( x, + ) = 1and one obtains L ( x ) = x Z dz = x. (34)The original Gibbs exponentials, e − βE /Z , result as sta-tionary distributions from any Monte Carlo type algo-rithm using the additive composition rule. The asymptoticrule is also the addition ϕ ( x, y ) = x + y .Another rule leading to the so-called q -exponential dis-tribution [13] is given by h ( x, y ) = x + y + axy with theparameter a proportional to q − h ′ ( x, + ) = 1 + ax and L ( x ) = x Z dz az = 1 a ln(1 + ax ) . (35)This formal logarithm leads to a stationary distributionwith power-law tail as the function composition exp ◦ L on the power − β : f ( E ) = 1 Z e − βa ln(1+ aE ) = 1 Z (1 + aE ) − β/a . (36)On the other hand, assuming such a non-additive compo-sition rule for the generalized entropy, a special formula am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 5 can be constructed as the expectation value of the inverseof this function, of the deformed logarithm, L − ◦ ln. Oneobtains S = Z f e − a ln( f ) − a = 1 a Z ( f − a − f ) . (37)The asymptotic composition rule again coincides with theoriginal one: ϕ ( x, y ) = x + y + axy . We note here thatthe formal logarithm of the integrated expression is the(additive) R´enyi entropy: L ( S ) = 11 − q ln Z f q , (38)with a = 1 − q and R f = 1.A further rule has been suggested by Kaniadakis[14],based on the sinh function. The formal logarithm is givenas L ( x ) = 1 κ Arsh( κx ) , (39)and its inverse becomes L − ( t ) = sinh( κt ) /κ . The station-ary distribution, composed by exp ◦ L , is f eq ( p ) = 1 Z (cid:16) κp + p κ p (cid:17) − β/κ . (40)For large arguments it gives a power-law in the momen-tum p and hence also in the relativistic energy. The corre-sponding entropy formula is the average of L − ◦ ln overthe allowed phase space: S K = − Z fκ sinh( κ ln f ) = Z f − κ − f κ κ . (41)The composition formula can be reduced to h ( x, y ) = x p κ y + y p κ x . (42)For low arguments it is additive, h ( x, y ) ≈ x + y , for highones it is multiplicative, h ( x, y ) ≈ κxy . It has been mo-tivated by the relativistic kinematics of massive particles.Interpreting the parameter as κ = 1 /mc , one deals with κp = sinh η , so the formal logarithm becomes proportionalto the rapidity, L ( p ) = mcη . This implies a stationary dis-tribution like exp ( − βmcη ), which has not yet ever beenobserved in particle spectra stemming from relativisticheavy ion collisions. For such a purpose it is tempting toconsider some further scenarios based on other quantitiesthan suggested above (see next section).The rule leading to a stretched exponential station-ary distribution, often considered in problems related toanomalous diffusion and Levy-flights, is given by h ( x, y ) = (cid:0) x b + y b (cid:1) /b . Here the partial derivative is evaluated at asmall positive argument, ǫ = y/ N . One obtains h ′ ( x, ǫ ) = c ( ǫ ) x − b with a factor depending on ǫ and for given val-ues of b diverging in the ǫ = 0 limit. However, this canbe accommodated by our procedure; we obtain the formallogarithm L ( x ) = c ( ǫ ) x b /b , and therefore the asymptoticrule ϕ ( x, y ) = (cid:0) x b + y b (cid:1) /b . Again, constant factors in the formal logarithm can be eliminated without loss of anyinformation.Now let us investigate a non-associative rule; its asymp-totic limit cannot be itself. We regard a linear combinationof arithmetic and harmonic means: h ( x, y ) = x + y + a xyx + y (43)The rescaling flow derivative is given by h ′ ( x, + ) = 1 + a and – being a constant – it leads to L ( x ) = x/ (1 + a ) andwith that to the addition as the asymptotic rule: ϕ ( x, y ) = x + y .As an interesting rule we discuss the relativistic for-mula for collinear velocity composition, h ( x, y ) = x + y xy/c . (44)This rule is associative, and it also preserves its form inthe thermodynamic limit. The fiducial derivative is givenby h ′ ( x, + ) = 1 − x /c and the formal logarithm, L ( x ) = c atanh ( x/c ) turns out to be the rapidity. The asymptoticcomposition rule recovers the original one.There are also general types of composition rules, whichmutate into a simpler asymptotic form. For our discussionparticularly important are rules of the form h ( x, y ) = x + y + G ( xy ) (45)with a general function G ( z ), restricted by the property G (0) = 0 only. In this case h ′ ( x,
0) = 1 + G ′ (0) x asymp-totically leads to a Tsallis-Pareto distribution with theparameter q − G ′ (0). In this section we review a particular type of pair inter-action, which can be expressed as a function of the ki-netic energies of the individual particles. The relation torelativistic kinematics is established by the fact, that weconsider such dependence through the Lorentz-invariantrelative four-momentum square variable: E = E + E + U ( Q ) . (46)We study whether relativistic speeds alone can cause ”non-extensivity”, i.e. a power-law tailed kinetic energy distri-bution. The relativistic formula for Q is given by: Q = ( p − p ) − ( E − E ) (47)with p i , E i being relativistic momenta and full energiesof interacting bodies. Expressed by the energies and theangle Θ between the two momenta this becomes a linearexpression of cos Θ : Q = 2 ( E E − p p cos Θ ) − ( m + m ) (48)with p i = p E i − m i for i = 1 ,
2. Here we use relativisticunits ( c = 1) and assume the masses m and m , respec-tively, for the interacting partners. It is useful to note thatwriting eq.(48) as Q = 2( A − B cos Θ ) we have A ± B = E E −
12 ( m + m ) ± p p . (49) Tam´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter
For the sake of simplification we average over the rel-ative directions of the respective momenta and obtain h U ( Q ) i = 12 π Z U (2 A − B cos Θ ) sin Θ dΘ = F (2 A + 2 B ) − F (2 A − B )4 B , (50)with U ( w ) = dF/dw . It is easy to derive by the substitu-tion w = 2( A − B cos Θ ). The rule for the kinetic energy, K i = E i − m i , composition is given by K = K + K + F (2 A + 2 B ) − F (2 A − B )4 B (51)The quantities A and B can be expressed by the respec-tive kinetic energies and masses: A = K K + ( m K + m K ) −
12 ( m − m ) ,B = K K ( K + 2 m )( K + 2 m ) . (52)One observes that the product of kinetic energies occursdue to kinematic reasons.Taylor expanding the integral of the unknown function U ( w ) around w = 2 A and ensuring the h ( x,
0) = x , aswell as the h (0 , y ) = y property, we obtain the followingcomposition rule for the relativistic kinetic energies: h ( x, y ) = x + y − U (2 m x + m ) − U (2 m y + m )+ U ( m ) + ∞ X j =0 U (2 j ) (2 A ) (4 B ) j (2 j + 1)! (53)with m = − ( m − m ) , A = xy + ( m x + m y ) + m / B = 4 xy ( x + 2 m )( y + 2 m ). For unequal masses, m = m this composition rule is not symmetric. Since ∂A∂y ( x,
0) = m + x,∂B ∂y ( x,
0) = 2 m x ( x + 2 m ) , (54)the derivative leading to the formal logarithm of the asymp-totic rule becomes an expression with a finite number ofterms h ′ ( x,
0) = 1 − m U ′ ( m ) + 2( m + x ) U ′ ( z )+ 43 m x (2 m + x ) U ′′ ( z ) , (55)with z = 2 A ( x,
0) = 2 m x + m . In all traditional ap-proaches the interaction energy U is independent of Q .In such cases h ′ ( x,
0) = 1 and the simple addition is theasymptotic composition rule. Therefore the stationary en-ergy distribution is of Boltzmann-Gibbs type. For Q de-pendent interactions on the other hand it is important toconsider the extreme relativistic kinematics. In this casethe replacement m = m = 0 leads directly to h ′ ( x,
0) = 1 + 2 x U ′ (0) . (56) As discussed in the previous subsection this generates aTsallis-Pareto distribution in the relativistic kinetic en-ergy. This result includes for U ′ = 0 the traditional mo-mentum independent interaction case leading to the ad-dition as asymptotic rule for non-relativistic kinetic en-ergies, and hence to the Boltzmann-Gibbs distribution.We note that in the relativistic kinematics the linear as-sumption, U ′ = α = const . also leads to a Tsallis-Paretodistribution due to h ′ ( x,
0) = 1 + 2 αx . There are two possible approaches in constructing a gener-alized entropy formula: i) either to use a non-additive en-tropy for independent events with factorizing probability,or ii) to search for an additive entropy while the commonprobability is not factorizing in the individual probabili-ties. In both cases the entropy density function, σ ( p ) to aprobability p can be obtained from the composition rule h ( x, y ).First we consider a non-additive entropy formula forfactorizing probabilities, i.e. X i,j w ij σ ( w ij ) = h X i p i σ ( p i ) , X j q j σ ( q j ) (57)with w ij = p i q j . We would like to construct the function σ ( p ) by knowing h ( x, y ). Let us inspect the equipartitioncase, p i = 1 /N , q j = 1 /N . In this case w ij = 1 / ( N N ).Eq.(57) leads to σ ( ab ) = h ( σ ( a ) , σ ( b )) (58)with a = 1 /N and b = 1 /N . This requires the samecomposition rule for micro- and macro-entropy: X ij p i q j h ( σ ( p i ) , σ ( q j )) = h X i p i σ ( p i ) , X j q j σ ( q j ) . (59)This h -extensivity can so far only be satisfied by the Tsal-lis rule h ( x, y ) = x + y + axy . On the other hand if q = 1and all other q j = 0 for j = 0, we obtain two constraints: σ ( p i ) = h ( σ ( p i ) , σ (1)) σ (0) = h ( σ ( p i ) , σ (0)) (60)from which it follows h ( x,
0) = x with σ (1) = 0 (the un-expectedness of a sure event is zero) and σ (0) = ∞ , too.Based on the properties of the known h ( x, y ), in thethermodynamical limit it is associative and hence possessa formal logarithm, L ( x ). Therefore L ( σ ( ab )) = L ( σ ( a )) + L ( σ ( b )) , (61)whose general solution is given by L ◦ σ = β ln. Accordingto the tradition β = − k B = 1, and therefore the entropy density functionis expressed by the deformed logarithm: σ ( p ) = L − ( − ln p ) = ln a (cid:18) p (cid:19) . (62) am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 7 It is possible to ask another question: if the construc-tion rule for the common probability is not the simpleproduct, but it is known, what should the entropy densityfunction be in order to lead to the addition rule for thetotal entropy. So given the formula w ij = e h (ln p i , ln q j ) (63)how to construct σ ( p ) such that X ij w ij σ ( w ij ) = X i p i σ ( p i ) + X j q j σ ( q j ) (64)is fulfilled. We have c = w ij = e h (ln a, ln b ) (65)as the known composite probability and seek for the en-tropy density function, σ ( a ) satisfying c σ ( c ) = ab ( σ ( a ) + σ ( b )) . (66)We solve this functional equation by deriving with respectto b and take the result at b = 1. Since ∂c∂b = e h (ln a, ln b ) h ′ (ln a, ln b ) 1 b (67)we arrive at ah ′ (ln a,
0) ( σ ( a ) + aσ ′ ( a )) = aσ ( a ) + aσ (1) + aσ ′ (1) . (68)Using now that σ (1) = 0 and h ′ ( x,
0) = 1 /L ′ ( x ) with theformal logarithm L associated to the composition rule, h ,we obtain – using the variable x = ln a – dσdx + σ = L ′ ( x ) ( σ + β ) , (69)with β = σ ′ (1) constant. The final solution is expressedby the formal logarithm of the asymptotic rule as σ ( a ) = βe L (ln a ) − ln a Z ln a L ′ ( u ) e u − L ( u ) du. (70)It is interesting to note, that using L − a of the deformedlogarithm as the function L belonging to the product com-position rule (65), one assumes ln a ( w ij ) = ln a ( p i ) + ln a ( q j )and arrives at p σ ( p ) = − e ln a p Z ln a p e − u e a ( u ) du. (71) It is a false belief that only the exponential distributioncan be the stationary solution to the Boltzmann equa-tion: this statement is true only i) if the two-particle dis-tributions factorize, ii) the two-particle energies are addi-tively composed from the single-particle energies ( E = h ( E , E ) = E + E ) and iii) the collision rate is multi-linear in the two-particle (and two-hole) densities. A gen-eralization of the original Boltzmann equation has beenpioneered by Kaniadakis[15] investigating nonlinear den-sity dependence of the collision rates. An ′′ H ′′ q theorem forthe particular Tsallis form of the collision rate has beenderived by Lima, Silva and Plastino[16].A possible generalization of the Boltzmann equationuses an altered form of the ’Stosszahlansatz’ and allows foran evolution equation of a function of the original phasespace occupation factor, F ( f ): DF ( f ) = Z w ( G − G ) (72)with DF = p µ p ∂ µ F (73)total (Vlasov-) derivative, with a 1234-symmetric colli-sion rate including Dirac-delta distributions for momen-tum and energy composition rules in two-to-two collisions(which also may be of generalized type by using corre-sponding formal logarithms), and finally the generalizedproduct for the two-particle density factor, G = e a (ln a ( f ) + ln a ( f )) (74)using the deformed exponential and logarithm functions.Based on this, a particular expression for the entropy cur-rent density can be defined: S µ = − Z p µ p σ ( F ( f )) . (75)The entropy density form, σ ( F ) always can be constructedin a way, that the second theorem of thermodynamics isfulfilled. The local source for the entropy is namely givenby ∂ µ S µ = − Z σ ′ ( F ( f )) DF ( f ) . (76)Utilizing the generalized Boltzmann equation (72) and ex-changing the index 1 with 2, 3 and 4 while w staysinvariant and obviously G ij = G ji , one arrives at ∂ µ S µ = 14 Z w ( σ ′ + σ ′ − σ ′ − σ ′ ) ( G − G )(77)with σ ′ i = σ ′ ( F ( f i )) for i = 1 , , ,
4. This quantity isalways non-negative, i.e.( Φ ( G ) − Φ ( G )) ( G − G ) ≥ , (78)if and only if Φ ( G ) = σ ′ ( F ( f )) + σ ′ ( F ( f )) (79)is a monotonic rising function. Inspecting the generalizedStosszahlansatz eq.(74) one finds that this splitting to the Tam´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter sum of respective functions of f and f is only possible,if Φ ( t ) ∝ ln a ( t ). Therefore we conclude that σ ′ ( F ( f )) = α ln a ( f ) + β (80)with α ≥ β undetermined constants. (This deriva-tion followed the spirit of Ref.[15].)The generalized entropy density as a function of theone-particle phase space occupation density is hence givenby σ ( f ) = Z F ′ ( f ) ( α ln a ( f ) + β ) df. (81)The traditional Boltzmann formula arises for F ( f ) = f and ln a ( f ) = ln( f ) (i.e. a = 0). Lavagno et.al. [17] consid-ered F ( f ) = f q and ln a ( f ) = ( f q − − / ( q −
1) (i.e. a =( q −
1) and Tsallis composition rule for ln f ). In the caseof h ( x, y ) = x + y + axy one considers ln a ( x ) = ( x a − /a , e a ( t ) = (1 + at ) /a and G = ( f a + f a − /a . For a small a parameter it is G ≈ f f (1 − a ln( f ) ln( f ) + . . . ).We note that the detailed balance distribution is givenby the condition G = G , while the corresponding en-ergy composition rule applies L ( E ) + L ( E ) = L ( E ) + L ( E ). This is possible only if ln a ( f i ) = − ( L ( E i ) − µ ) /T ,so f eq ( E ) = e a (cid:18) µ − L ( E ) T (cid:19) . (82)The parameters T and µ are arbitrary constants for beinga stationary solution of the generalized Boltzmann equa-tion, but they can be related to the total energy and par-ticle number in a given application. Our fist numerical approach [9] was restricted to the useof abstract composition rules in the energy balance part:we equated the energy of the reacting parts before andafter the collision via an abstract energy composition rule h ( E , E ) = h ( E , E ) . (83)Although this rule cannot be specified without furtherknowledge, according to our results presented in the previ-ous section, in the thermodynamical limit an asymptoticrule can be considered, with a formal logarithm. The par-ton cascade simulation based on a Boltzmann equation ishence modified by considering L ( E ) + L ( E ) = L ( E ) + L ( E ) . (84)At the same time we applied F ( f ) = f and a = 0. Ap-plying such a general energy composition rule consideredin the thermodynamical limit, the rate of change of theone-particle distribution is given by˙ f = Z w [ f f − f f ] . (85) -2-1 0 1 2-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 p y p x Tsallis kinematics E = 4 GeV, P = 3.6 GeV, m = 0.14 GeV a = +0.0 a = +0.1 a = - 0.1 a = - 0.2-2-1.5-1-0.5 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 p y p x Tsallis kinematics E = 4 GeV, P = 2 GeV, m = 0.94 GeVp p p p p a = +0.0 a = +0.1 a = - 0.1 Fig. 2.
Momentum vectors for pions (upper) and protons(lower) at E = 4 GeV pair energy according to eq.(86) withthe rule h ( x, y ) = x + y + axy for the energies. with the symmetric transition probability w includingthe constraint ∆ = δ ( p + p − p − p ) δ ( h ( E , E ) − h ( E , E )) . (86)In the figure 2 the possible pairs of momentum vectorsare shown for the h ( x, y ) = x + y + axy energy compo-sition rule for pions ( m = 0 .
14 GeV mass) and protons( m = 0 .
94 GeV mass), respectively. The two-dimensionalcuts for the endpoints of the respective vectors form anellipsoid in the traditional a = 0 case, while this surfaceis deformed for nonzero extensivity parameters, as seen inthe figure.In a stationary state the f ( E i ) distributions depend onthe phase space points through the energy variables only(this is to be checked on experimentally observed hadrontransverse momentum spectra at mid-rapidity by the socalled m T -scaling) and the detailed balance principle re-quires f ( E ) f ( E ) = f ( E ) f ( E ) . (87)With the generalized constraint (86) this relation is satis-fied by f ( E ) = f (0) exp( − L ( E ) /T ) . (88)For the Tsallis-type energy addition rule[18,19], one ob-tains cut power-law stationary distribution, f ( E ) = f (0) (1 + bE ) − /bT . (89) am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 9 Connecting this to the Tsallis parametrization one uses q = 1 + bT . Since the energy addition rule conserves in atwo by two collision the quantity h ( E , E ), the new en-ergies after such an event lie on the h ( E , E )=constantsurface. Due to the additivity of the formal logarithm ofthe single particle kinetic energies, L ( E i ), the total sum L tot = P i L ( E i ), is a conserved quantity. This rule wasapplied in numerical simulations [9,22]. During the numer-ical searches for stationary distributions only the tacit as-sumption of constant transition probability rates has beenapplied; the evolution results are obtained in terms of thenumber of pairwise momentum exchange events, not interms of real time. Parton cascade simulations usually consider pairwise colli-sions with energy and momentum conservation inside thetwo-particle system. The pairs to collide are chosen ran-domly from an ensemble of particles and the new mo-menta are generated randomly according to the aboveconstraints. This way the probability is uniform in thetwo-particle phase space, provided the conditions for mo-mentum and energy sums (in our more general case forthe energy composition) are satisfied: d w = w δ ( p + p − P ) δ ( h ( E , E ) − H ) d p d p . (90)The constant w is fixed by the normalization of the in-tegral of this probability density to one (or to the actualcollision rate in real-time simulations). Since there are sixdegrees of freedom and four constraints, two free quan-tities have to be chosen randomly. It is, however, a del-icate procedure to ensure the random uniformity in thetwo-particle phase space for a general energy compositionrule.It is customary to introduce the sum and difference ofthe momentum vectors by p , = 12 P ± q . (91)Using this notation the momentum sum constraint can beintegrated out trivially and - since the Jacobean of thetransformation (91) is one - we arrive at d w = w δ ( h ( E , E ) − H ) d q. (92)For the addition rule, h ( E , E ) = E + E , it is enoughto obtain the direction of the vector q accordingly whileits magnitude is constrained by the energy sum. It is astraightforward task to do it in the center of mass sys-tem, where the momentum sum vector, P , vanishes: thedirection of the difference vector q in this system is uni-form on a spherical surface. A Lorentz-transformation intothis system, a random azimuthal angle and a random co-sine, and finally a back transformation provide the newmomenta after a collision.Since we are dealing with a constraint more generalin the energy variables, first we transform the problem of randomly choosing the difference vector q into a problemof choosing proper energies after the collision. The energiesare expressed by the free dispersion relations E , − m , = 14 P + q ± P q cos θ, (93)where P and q denote the lengths of the correspondingvectors and θ the angle between them. From this two equa-tions one easily derives the following energy differentials:2 E dE = 2 qdq + P cos θdq − P q sin θdθ, E dE = 2 qdq − P cos θdq + P q sin θdθ. (94)The phase space volume element can be expressed easilyby using the wedge product form: d q = dq ∧ q sin θdθ ∧ qdφ (95)which upon using eq.(94) can be written as d q = E E P dE ∧ dE ∧ dφ. (96)Now using the energy composition constraint we arrive ata probability density which is not uniform in the energy: d w = w δ ( h ( E , E ) − H ) E E P dE dE dφ. (97)One uses the constraint to eliminate say E from the aboveformula and considers d w = w E E P h ′ ( E , E ) dE dφ. (98)In the general case the differential probability density, dw/dE , is a complicated function of the energy. Its inte-gral, w ( E ) has to be uniformly distributed.In the case of a Tsallis composition rule one obtains E = ( H − E ) / (1 + aE ) and we arrive at d w = w P E ( H − E )(1 + aE ) dE dφ. (99)This expression can be integrated giving d w = 12 π dρ dφ, (100)with ρ ( E , a ) = (2 + aH ) ln(1 + aE ) − aE aE (2 + aH + aE )(2 + aH ) ln(1 + aH ) − aH (101)when properly normalized. The only problem is that ρ ( E )cannot be inverted analytically. Even in the traditionalcase with a = 0, the inversion requires the solution of athird order equation: ρ ( E, a = 0) = 3(
E/H ) − E/H ) (102)is distributed uniformly between zero and one. It meansthat E is between zero and H , the total composed energy. d w / d x xRandom Energy Density Distribution aH = 1.0 aH = 0.0 aH = -0.5 Fig. 3.
The scaled differential probability density, dw/dx isshown as a function of the random energy share of one of thecollided particles x = E /H (cf. eq.(99)) for different values of aH . Full circles and boxes denote the random energy deviatesobtained numerically using the rejection method. After having E and E the momenta can be recon-structed with the help of a vector triad describing thedirection of the momentum sum, P . They are given as n i = P i /P and e i = ( − n ⊥ , n n /n ⊥ , n n /n ⊥ ) wherethe notation n ⊥ = p n + n stands for the componentperpendicular to the first axis. The third orthogonal unitvector is f i = (0 , − n /n ⊥ , n /n ⊥ ). The momentum differ-ence vector is hence reconstructed as q = q || n + q ⊥ (cos φ e + sin φ f ) (103)with q || = E − E P ,q = E + E − P ,q ⊥ = q q − q || . (104) First we show some snapshots of the colliding partons inthe p x − p y phase space cut at different stages of the evo-lution marked by the average number of collisions perparticle, t (cf. Fig.4). At the beginning t = 0 we pre-pared two distributions at a given energy per particle andthen Lorentz boosted each with y B = 2 units of rapid-ity in opposite ways in the p x -direction. The dark dotsrepresent particle momenta stemming from the respec-tively boosted original sets. The evolution towards a zerocentered and isotropic distribution of momenta signals al-ready that thermal equilibration happens.Fig.5 presents results of a simple test particle simula-tion with the rule h ( x, y ) = x + y + axy with a = 0 (left)and a = 2 (right), respectively. We started with a uniformenergy-shell distribution between zero and E = 1 with Fig. 4.
Snapshots of phase space cuts in the p x − p y plane forcolliding partons with the deformed energy composition rule h ( x, y ) = x + y − . xy at t = 0 , . , , and 3 (from top tobottom).am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 11 f( E ) EEvolution to Boltzmann-Gibbs (a=0)0.00.10.51.03.010. 0.0001 0.001 0.01 0.1 1 10 0 1 2 3 4 5 6 f( E ) EEvolution to Tsallis-Pareto (a=2)0.00.10.51.03.010.
Fig. 5.
Evolution of single particle energy kinetic energydistributions for massless particles towards the Boltzmann-Gibbs distribution for h ( E , E ) = E + E (left part) andtowards the Tsallis-Pareto one (right part) for h ( E , E ) = E + E + 2 E E . The curves are normalized to the same in-tegral R E f ( E ) dE . a fixed number of particles N = 10 . The one-particleenergy distribution evolves towards the well-known expo-nential curve for a = 0, shown in the left part of Fig.5.These snapshots were taken initially and after 0 . , . , , a = 2, the energy distribution approachesa Tsallis-Pareto distribution.It is in order to make some remark on the energy con-servation. For h ( x, y ) = x + y we simulate a closed systemwith elastic collisions: The sum, E tot = P Ni =1 E i , doesnot change in any of the binary collisions. This is differ-ent by using a non-extensive formula for h ( x, y ). Witha constant positive (negative) a , the bare energy sum isdecreasing (increasing) while approaching the stationarydistribution, while the sum of the formal logarithms of theenergy remains constant. Open systems may gain or looseenergy during their evolution towards a stationary state. In order to investigate the equilibration of non-extensivesystems we start with two subsystems, equilibrated sepa-rately. In order to prepare these systems the non-extensiveBoltzmann equation can be solved numerically in a partoncascade simulation as described in the previous section. Asan alternative way we use initial momentum distributionsprepared by Monte Carlo rejection techniques in the formof eq.(88), with different energy per particle but a commonparameter a for the one an the other half of the particles.Then random binary collisions between randomly chosenpairs of particles are evaluated. By doing so we apply therules X ( E ) + X ( E ) = X ( E ) + X ( E ) , (105) p + p = p + p . (106)In each step of the simulation we select two particles tocollide. Then we find the value for the new momentumof the first particle ( p ) satisfying the above constraintsbut otherwise random. Then applying eq. 106 we calcu-late the momentum of the second outgoing particle ( p ).In these particular simulations we use the free dispersionrelation for massless particles ( E i ( p i ) = | p i | ), since weare interested in the extreme relativistic kinematics case.A typical simulation includes 10 − collisions among10 − particles. After 3 − X ( E tot ) = N X i =1 X ( E i ) , P = N X i =1 p i , N = N X i =1 . (107)We use the rule h ( x, y ) = x + y + axy for the energy compo-sition, here a ∼ ( q − /T is the non-extensivity parameter.Our model reconstructs the traditional Boltzmann-Gibbsthermodynamics in the limit of a = 0.As a preparation for the study of non-extensive ther-mal equilibration, we perform simulations on two largesubsystems with particle numbers N = N/ N = N/
2, total (quasi-)energies X ( E ) and X ( E ) and non-extensivity parameter a . The unified system is taken asan initial state with N = N + N particles. Our results show that the subsystems do equilibrate, theytend towards having a common stationary distribution.We present examples with different initial conditions.We fix the particle numbers for each subsystem, N = N = 100 000. The number of collisions in a typical sim-ulation is N coll = 1 000 000, so that N coll / ( N + N ) = 5collisions happen per particle. This quantity we use as anevolution parameter instead of the real time. This way wedo not have to know differential cross sections; from the f( E ) EThermalization of Boltzmann-Gibbs spectrainit1 T = 0.1init2 T = 0.5final1 T = 0.3final2 T = 0.3 0.001 0.01 0.1 1 10 100 0.1 1 10 f( E ) EThermalization of a=2 Tsallis-Pareto spectrainit1 T = 0.1init2 T = 0.5final1 T = 0.3final2 T = 0.3
Fig. 6.
Equilibration of two Boltzmann-Gibbs systems ( a =0, upper figure) plotted on a linear - logarithmic scale andequilibration of two Tsallis-type non-extensive systems ( a =2, lower figure) plotted on a double logarithmic scale. Theseare results for three-dimensional systems with 10 collisions perparticle on the average. Each subsystem consists of 100 . viewpoint of the fact of equilibration its rapidity does notmatter.In the figure 6 energy distribution curves are shown:the initial and the final ones and the ones after 5 collisionsper particle, respectively. The upper part plots a Boltz-mann system (simulation with a = 0) the lower one a Tsal-lis system with the energy composition rule using a = 2.In the upper half a logarithmic – linear plot is shown whilein the lower half a double logarithmic plot. These choicesare selected by the respective high energy asymptotics;exponential for a Boltzmann-Gibbs, while power-law fora Tsallis-Pareto distribution. It is hard to distinguish theenergy distributions in the subsystems in the final state,the simulation curves are very close to each other. There-fore we conclude, that within numerical uncertainties acommon stationary energy distribution is achieved. Seeking for a canonical equilibrium state we have to max-imize the total entropy given by a general compositionrule, S ( E , E ), at the same time satisfying a constraintwhich is in the general case also non-additive: h ( E , E )is constant. For the moment we neglect the dependenceon further thermodynamical variables; usually the parti-cle number N and the volume V is regarded to be propor-tional and extensive.In the traditional case both the entropy and the en-ergy are combined additively: S ( E , E ) = S ( E ) + S ( E )and h ( E , E ) = E + E . In the general case by usingcorresponding formal logarithms the quantities Y ( S ) and X ( E ) have to be considered as additive. Since for asso-ciative rules the formal logarithm is strict monotonic, themaximum of the total entropy is achieved where Y ( S ) hasits extreme. The general canonical principle is thereforegiven by Y ( S ) − βX ( E ) = max . (108)The parameter β at this point is a Lagrange multiplier.Applying this for the equilibration of two large subsys-tems, and assuming that the entropy of each systems de-pends only on its own energy, one arrives at the equilib-rium condition Y ′ ( S ( E )) X ′ ( E ) S ′ ( E ) = Y ′ ( S ( E )) X ′ ( E ) S ′ ( E ) = 1 T . (109)Comparing this with the general canonical form eq.(108)we obtain that β = 1 /T , and T is an absolute tempera-ture in the classical thermodynamical sense. Its relationto the entropy, however, has been generalized. In par-ticular for an additive entropy, but non-additive energycomposition rule, one arrives at 1 /T = S ′ ( E ) /X ′ ( E ).The relation of this quantity to the logarithmic spectralslope, 1 /T slope = − d ln f /dE = S ′ ( E ) leads to a prac-tical tool for the analysis of particle spectra in experi-ments. For the Pareto-Tsallis distribution it is given by T slope = T /X ′ ( E ) = T (1 + aE ) = T + ( q − E . The naiveeffort to extract a temperature from energy spectra of par-ticles, as it is a widespread usage in relativistic heavy ionstudies, only works if q = 1, i.e. for spectra exponential inthe particle energy. Otherwise an energy dependent slope,and a curved spectrum in the logarithmic plot has to beinterpreted.The inverse logarithmic slopes of single-particle kineticenergy spectra in the generalized case are functions of theenergy: T slope = − ∂∂E ln f ( E ) . (110)For the Tsallis-Pareto distribution they are linear func-tions, T slope = T + ( q − E . Such slope parameters areplotted in Ref.[22] for the respective subsystems beforeand after equilibration (10 collisions per particle on theaverage). Within numerical uncertainties it is clear thatcommon- a systems do equilibrate at a common tempera-ture also in the a = 0 case. am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 13 S/NTS S/Ncomposed S/N S/NTS S/Ncomposed S/N
Fig. 7.
The evolution of the Boltzmann entropy per particleduring collisions with non-additive energy composition rules:the hotter body cools, the cooler body warms up, while thetotal entropy also increases. In the insertion a magnification ofthe curves is shown.
The (in our case Boltzmann) entropy also evolves dueto the collisions. In Fig.7 the evolution of the entropy perparticle is plotted for the hot and cool subsystems, andfor the total system respectively. Since the composite sys-tem is combined from equal numbers of particles in eachsubsystem, the total entropy per particle starts with thearithmetic mean of the respective specific entropies. Thisvalue, however, rises somewhat, featuring a trend accord-ing to the second law of thermodynamics.
As an application of the above reviewed treatment of non-extensivity, in this section we demonstrate that hadronictransverse momentum spectra stemming from relativisticheavy ion collisions can be well described by cut power-law spectra in statistical models. In order to do so, onehas to disentangle effects of a possible transverse flow onthese spectra and then test whether the result complieswith the thermal assumption; i.e. that the dependence onmomenta is through a dependence on the kinetic energy, E − µ = E − m only. The µ = m assumption correspondsto a vanishing Fermi momentum for fermions, so this isthe natural assumption at zero net baryon density. There-fore transverse momentum spectra at mid-rapidity are ex-pected to follow such statistical model assumptions thebest. It has been long discussed, how a temperature can be con-jectured from observations on particle spectra producedin relativistic heavy ion collisions. One intriguing way is to look at the transverse momentum, p T , spectra aroundmid-rapidity. The different identified hadrons, mostly pi-ons, kaons, protons and antiprotons, have to demonstratethat their abundance in the momentum space depends ontheir kinetic energy; this phenomenon at zero rapidity isthe so-called m T − m -scaling. The transverse mass is givenas m T = p m + p T , at strictly zero rapidity this is thetotal relativistic energy.The analysis is made a little more involved by the factthat the source emitting the detected hadrons is flowingin all directions. The most prominent effects are due to arelativistic transverse flow with velocity v T (and a corre-sponding Lorentz factor γ T = 1 / p − v T in units where c = 1). The relativistic energy of a particle in the frame ofthe emitting source cell is given by the J¨uttner variable: E = u µ p µ = γ T m T cosh( y − η ) − γ T v T p T cos( ϕ − Φ ) . (111)Here the four-velocity of the source and the actual four-momentum of the particle are parametrized by rapidityand angle variables: u µ = ( γ T cosh η, γ T sinh η, γ T v T cos Φ, γ T v T sin Φ ) ,p µ = ( m T cosh y, m T sinh y, p T cos ϕ, p T sin ϕ ) . (112)We consider a thermal model for the particle spectra; thenthe yield is supposed to depend on the J¨uttner variable E given by eq.(111). Assuming a general distribution f ( E ) ∼ exp ( − ( X ( E ) − m ) /T ), which is monotonic decreasing, onefinds its maximum at the minimum of E . This variable isminimal at the rapidity y min = η , and angle ϕ min = Φ ,giving E min = γ T m T − γ T v T p T . (113)This Lorentz-boosted transverse energy reaches its mini-mum at the transverse momentum value p T, min = mγ T v T ,leading to m T, min = mγ T and E min = m . The expansionaround this minimum in the p T -distribution is an effectiveGaussian: e − ( E − m ) /T ≈ exp (cid:18) − ( p T − mγ T v T ) mγ T T γ T (cid:19) . (114)In fact, according to experimental findings at RHICthe observed particle spectra have to be corrected for atransverse flow in order to reach m T -scaling. We conjecture that the power-law tails observed in hadronicspectra may stem from non-extensivity of the suddenlyhadronizing quark matter. We look for a connection be-tween quark and hadron spectra in the framework of thequark coalescence model. A coalescence of a quark and anantiquark into a meson produces a yield proportional tothe quantity: F ( p ) = Z f ( E ( P / q )) f ( E ( P / − q )) C ( q ) d q. (115) pt[GeV]0 2 4 6 8 10 12 14 d N / ( p t * dp t) / f i tt i ng Fig. 8.
General shape of p T spectra for pions, kaons and an-tiprotons in relativistic heavy ion experiments (upper figure).A fit is done by using for X ( E ) the Tsallis-Pareto form withparameters T and a , corresponding to a common temperatureof T ( m i ) = 0 .
160 MeV for the different particles. and a trans-verse flow velocity v T = 0 .
52. In the lower part the ratio ofthe Tsallis fit to the experimental values can be inspected in alinear plot.
Here we integrate over the relative momentum of the quarkswith a coalescence factor, C ( q ), for which a simple modelhas been utilized [23]. For common momenta much largerthan the relative one | P | ≫ | q | on obtains F ( P ) ≈ f ( E ( P / Z C ( q ) d q. (116)In particular light hadrons made from massless quarks fol-low the quark-scaling rule: f hadron ( E ) ∝ f n ( E/n ) . (117)As a consequence particular properties of the non-extensivethermal model between quark and hadron matter alsoscale: T mesons = T baryons = T quarks for the temperature,while q mesons − q quarks − / q baryons − q quarks − / T slope = T + ( q − E − m ) , (118)the rise of these slopes reflect the non-extensivity param-eters. In Fig.9 we show the test of the coalescence modelprediction for the meson to baryon ratio. E[GeV]0 1 2 3 4 5 6 7 − d E / d l n f( E ) Pion
Kaon
Anti−proton
Fig. 9.
Inverse logarithmic slopes, T slope ( E ) = − dE/d ln f ( E )extracted by numerical derivation from the experimentalhadronic spectra (after subtracting a common flow effect). Thefull lines correspond to a common meson and baryon fit theirsteepness keeping the ratio 2:3 predicted by the quark coales-cence picture. q qu a r k m (GeV)Extracted values for the quark q-parameter 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 T s l op e = T + ( q qu a r k - ) m m (GeV)Quark coelasecence slopes at zero p t Fig. 10.
The q parameter of quark matter extracted fromhadronic spectra assuming quark coalescence at a suddenhadron formation (upper picture). The spectral inverse slopeas a function of the minimal energy E min = m agree with thelinear prediction from the coalescence scaling. Furthermore these predictions of the non-extensive phe-nomenology meet the curves from pQCD calculations, withthe following surmised properties of quark matter at RHIC: T = 140 . . .
180 MeV, q = 1 . v T = 0 .
6. [24]. Deviationsfrom the product rule suggested by the simplest quark co-alescence idea occur at p T values lower than 1 GeV. Weattribute these to a further constituent in real hadrons,namely non-perturbative gluons simulated by a string en-ergy contribution[26].We note that a stringy interaction remainder abovethe color deconfinement temperature T c ≈
170 MeV in am´as S. Bir´o, G´abor Purcsel, K´aroly ¨Urm¨ossy: Non-Extensive Approach to Quark Matter 15 quark gluon plasma also describes the main effects on thequark matter equation of state seen in lattice QCD calcu-lations successfully[27]. Both the presence of a string likepair potential for a however minor percentage of pairs, as amicroscopic model, and the assumption of non-extensivity, q >
1, as a descriptive phenomenology are able to explainthe value of the energy per particle,
E/N = 6 T = 1 GeV,which has been found by fitting statistical hadronic reso-nance gas models with Boltzmann distributions.Finally some remarks are in order to the Tsallis-Paretofits to energy spectra. In several cases naively a fit is donein the original form: f ( E ) ∼ (cid:18) q − ET (cid:19) − q − (119)to which the following inverse logarithmic slope depen-dence belongs: T slope = T + ( q − E. (120)A more sophisticated approach (as one suggested in [17]),however, uses the original Tsallis-Pareto form for the num-ber density distributions of particles and for the generat-ing thermodynamical potential, for the logarithm of thecanonical partition function. This way in this second ap-proach the energy distribution is described by the q -thpower of the naive factor: f ( E ) ∼ (cid:18) q − E ˜ T (cid:19) − ˜ q ˜ q − (121)and the corresponding inverse slope T slope = 1˜ q ˜ T + (cid:18) − q (cid:19) E. (122)We observe that the qualitative behavior is the same, butthe interpretation of the fit parameter is different in thesedifferent approaches. The correspondence between the en-ergy spectrum fit parameters is given as T = ˜ T / ˜ q,q = 2 − / ˜ q. (123)In a sense q and ˜ q are double-duals of each other, bothusing the 1 /q - and the 2 − q -duality. Also the estimatedtemperature parameter differ. Typical values from rela-tivistic heavy ion experiments are q ≈ . q ≈ . T = 1 . T . In conclusion we reviewed basic concepts of non-extensivethermodynamics which may be relevant in understandingparticular features of hadronic spectra stemming from rel-ativistic heavy ion collisions. The overall presence of rel-ativistic speeds of particles in the physical system underinvestigation on the other hand offers a unique possibility to study and - whenever necessary - to generalize familiarthermodynamics.We presented some general arguments for a possibleneed to face with total energy and entropy not beingproportional to the particle number even in the large N limit. These arguments are based on the long range na-ture of pair interactions. This phenomenon, called non-extensivity, was then related to the generalization of com-position rules of the familiar thermodynamical extensives,like energy and entropy. We have mathematically provedthat abstract composition rules become symmetric andassociative in the large N limit, provided that the compo-sition function, h ( x, y ), is at least right-sided differentiableat y = 0 + . This means that associative composition rulesconstitute attractors among all rules when approachingthe thermodynamical limit. As a consequence the associa-tivity of the composition rule is a thermodynamical re-quirement.The key quantity in this proof, the formal logarithm,relates the abstract composition rule to the addition ofthe system size indicator, to the particle number N . Wegave the formula how to construct it. Based on the formallogarithm the widely used deformed exponential and log-arithm functions can easily be derived. While the formerdescribes the energy distribution in canonical equilibrium,the latter defines a generalized formula for the entropy. Anadditive entropy can be always gained from this expressionby taking its formal logarithm.We presented some often used composition rules to-gether with the corresponding equilibrium energy distri-butions and entropy formulas including the traditionalBoltzmann-Gibbs formula derived from the simple addi-tion (extensivity), the Tsallis rule, the Kaniadakis ruleand - for the sake of demonstration - the Einstein rulefor composing relativistic velocities. Our general methodin this case leads to the rapidity as the additive formallogarithm. Among non-associative composition rules theclass of h ( x, y ) = x + y + G ( xy ) is found to be particularlyinteresting in high energy physics, since it asymptoticallyapproaches the Tsallis rule leading to power-law tailedenergy distributions in canonical equilibrium. We havedemonstrated that such a composition rule may emergein the extreme relativistic kinematics limit from an en-ergy correction to a pair of particles in a medium whichis a function of the Lorentz invariant relative momentumsquared variable Q . In fact this is frequently the casewhen following several interactions among partons accord-ing to the formulas derived from (or at least motivated by)QCD. Finally it is interesting to note that the elementaryproperty, h ( x,
0) = x is related to σ (1) = 0 property of theentropy density function if the composition rule h ( x, y )is assumed for composite states with factorizing proba-bilities. In this case the general result σ ( p ) = ln a (1 /p )emerges in the thermodynamical limit, with ln a = L − ◦ lnbeing the corresponding deformed logarithm function.The - in some sense opposite - requirement, i.e. aimingat an additive entropy formula while the probabilities donot factorize, but their logarithms follow a general com-position rule h ( x, y ) instead of the usual addition, leads to a more complex relation between the formal logarithmof the rule, L , and the entropy density σ .In the second part of this review we compiled the mostimportant numerical results on parton cascade simulationsof non-extensive systems. Following the presentation ofa class of generalized Boltzmann equations, and provingthat the second law of thermodynamics can only be ful-filled if the derivative of the entropy density is a linearexpression of the deformed logarithm of the one-particlephase space density (cf. eq.(80)), we presented some de-tails of the kinematical description of relativistic particlesin such simulations. We payed special attention to the ran-dom choice of particle momenta after an, in energy non-additive, pair collision (which can have a physical reasonin the influence of third or further particles, or fields in adense medium).Results on the phase space evolution under non-exten-sive energy composition rules were presented demonstrat-ing the ability of such a computer simulation to generatepower-law tailed energy spectra in the detailed balancestate of the non-extensive Boltzmann equation. The im-portant question of equilibration between two large sub-systems, related to the zeroth theorem of thermodynam-ics, was also investigated by us numerically in this frame-work. We found that non-extensive systems with the power-law tailed Tsallis-Pareto energy distributions do behaveas they should, just the thermodynamic temperature, T is related to the microcanonical equation of state, S ( E ),by receiving corrections due to the formal logarithms ofthe entropy and energy composition rules (cf. eq.(109)).Finally our studies on the hadronization of quark mat-ter in relativistic heavy ion collisions revealed that if thequark coalescence is a dominant mechanism, then the non-extensivity parameter, q − aT also must show thequark number scaling. This assumption can be and shouldbe tested on experimental data and should be related toother information on quark coalescence, e.g. to those ob-tained from studies of the elliptic flow.Certainly there remain open questions for further re-search. Among them the study of the quark matter equa-tion of state with elementary field theory means, as lat-tice QCD, in a non-extensive canonical state is still a hardchallenge. Also the determination of the pair correlationfunction, g ( r ), from first principles in microscopical cal-culations should help to identify those physical situationswhere the concepts and formalism of non-extensive ther-modynamics have to be used. Meanwhile the physical rea-son for a non-exponential energy distribution can be nu-merous. The quark gluon plasma is a wonderful candidatefor finding non-extensive behavior, since long range effectsare there at any finite temperature. Acknowledgments
This work has been supported by the Hungarian Na-tional Science Fund, OTKA (K49466, K68108). Discus-sions with C. Tsallis, G. Wilk, T. Kodama, G. Kaniadakis,P. V´an and A. L´aszl´o are gratefully acknowledged.
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