We formulate the kinetic master equation describing the production of charged
particles which are created or destroyed only in pairs due to the conservation
of their Abelian charge.Our equation applies to arbitrary particle
multiplicities and reproduces the equilibrium results for both canonical (rare
particles) and grand canonical (abundant particles) systems. For canonical
systems, the equilibrium multiplicity is much lower and the relaxation time is
much shorter than the naive extrapolation from the grand canonical ensemble
results. Implications for particle chemical equilibration in heavy-ion
collisions are discussed.Comment: 4 Pages in RevTe

B. Müller

C. M. Ko

D. Yen

E. V. Shuryak

F. Becattini

H.-Th. Elze

J. Cleymans

J. E. L. Bratkovskaya

J. Letessier

J. Rafelski

J. S. Hamieh

K. Redlich

L. Turko

L. V. Bravina

M. Stephanov

P. Braun-Munzinger

P. Koch

R. Hagedorn

R. Stock

T. Matsui

T. S. Biro

U. Heinz

V. Koch

W. S. Chung

X. S. Fang

Xin-Nian Wang

Zi-wei Lin

English

arXiv

Ko, C. M.

Koch, V.

Lin, Zi-wei

Redlich, K.

Stephanov, M.

Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

Kinetic Equation with Exact Charge Conservation

Crossref

Physical Review Letters

Ko, CM

Koch, V

Redlich, K

Stephanov, M

Wang, Xin-Nian

Texas A&amp;M Repository

arXiv:nucl-th/0010004v1  1 Oct 2000LBL-46896Kinetic equation with exact charge conservationC.M. Koa, V. Kochb, Zi-wei Lina, K. Redlichb,c, M. Stephanovd,e, and Xin-Nian WangbaCyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843-3366bNuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720cInstitute of Theoretical Physics, University of Wroc law, PL-50204 Wroc law, PolanddDepartment of Physics, University of Illinois, Chicago, IL 60607-7059eRIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973(August 7, 2018)We formulate the kinetic master equation describing the production of charged particles which arecreated or destroyed only in pairs due to the conservation of their Abelian charge. Our equationapplies to arbitrary particle multiplicities and reproduces the equilibrium results for both canoni-cal (rare particles) and grand canonical (abundant particles) systems. For canonical systems, theequilibrium multiplicity is much lower and the relaxation time is much shorter than the naive extrap-olation from the grand canonical ensemble results. Implications for particle chemical equilibrationin heavy-ion collisions are discussed.I. INTRODUCTIONRelativistic statistical thermodynamics has long beenused as a tool to describe particle production in heavy-ion and in high-energy particle collisions [1–3]. Recentanalyses have shown that the statistical models can in-deed give a satisfactory description of the multiplicitiesof most hadrons measured in A-A collisions at AGS andSPS energies [4,5]. However, the dynamics of particleequilibration, and in particular chemical equilibration, isstill not well understood [6–8]. In this Letter, we shalladdress the problem of chemical equilibration within thestatistical kinetic approach.Within this approach, particle production is commonlydescribed using the grand canonical ensemble, whereevent-averaged multiplicities are controlled by chemicalpotentials. In this description the net value of a givenU(1) charge (e.g., electric, baryon, strangeness, charm,etc.) fluctuates. These fluctuations can be neglectedonly if the particles carrying the charge in question areabundant. In this case the charge will be conserved onthe average. In the opposite limit of rare particle pro-duction, conservation laws must be implemented locallyon an event-by-event basis [1,2,9,10]; i.e.,a canonical en-semble must be used.The local conservation of quantum numbers in thecanonical approach severely reduces the phase spaceavailable for particle production [1,2,5,9–12]. Recently,it has been shown that the canonical statistical modelprovides a good description of particle yields measuredin low-energy heavy-ion [13] and high-energy hadron-nucleus, hadron-hadron and e+e− reactions [5,14,15].In view of these results it is of particular importanceto formulate a kinetic theory for the time evolution ofparticle production in order to investigate the approachto the canonical chemical equilibrium. Obviously, if par-ticles are abundantly produced, the equilibrium result forthe particle multiplicity should coincide with the grandcanonical one. For rare processes, however, the parti-cle production is strongly correlated and the canonicalequilibrium result is expected. The kinetic formulationfor the production of strongly correlated particles wasstudied in the literature [16,17]. However, no completesolution has been obtained.In this letter we consider the time evolution of the mul-tiplicity of particles that carry the charges correspondingto an U(1) internal symmetry. We formulate a kineticmaster equation valid for arbitrary particle multiplicity.It reproduces the canonical equilibrium solution for rareparticle production, and reduces to the standard grandcanonical rate equation for abundant particle production.II. RATE EQUATIONIn the standard formulation [16,18–20], the rate equa-tion for a binary process a1a2 → b1b2 with a 6= b isdescribed by the following population equation:d〈Nb1〉dτ=GV〈Na1〉〈Na2〉 −LV〈Nb1〉〈Nb2〉, (1)where G ≡ 〈σGv〉 and L ≡ 〈σLv〉 give the momentum-averaged cross sections for the gain process a1a2 → b1b2and the loss process b1b2 → a1a2, respectively. Nk repre-sents the total number of particles k, and V is the propervolume. Among such processes, a typical example is thekaon production/annihilation via π+π− ↔ K+K−. Theabove rate equation, however, cannot be applied to the1situation where particle production is rare and is stronglycorrelated by exact charge conservation.To account for the correlation between the produc-tion/annihilation of particles b1 and b2, let us define Pi,jas the probability to find a number i of particle b1 anda number j of particle b2 in an event. We also denoteas Pi the probability to find a number i of particle b inan event. The average number of particles b per event isthen defined as:〈Nb〉 =∞∑i=0iPi. (2)We can now write the following general rate equationfor the average particle multiplicity:d〈Nb1〉dτ=GV〈Na1〉〈Na2〉 −LV∑i,jijPi,j . (3)Further, let particles b1 and b2 carry opposite units ofa charge, corresponding to an U(1) internal symmetry(strangeness in the case of kaons). Then the U(1) chargeneutrality of the system gives N ≡ Nb1 = Nb2 . We havethen,Pi,j = Pi δij ,∑i,jijPi,j =∑ii2Pi ≡ 〈N2〉 = 〈N〉2 + 〈δN2〉, (4)where 〈δN2〉 represents the event-by-event fluctuation ofthe number of b1b2 pairs. Note that we always consider a1and a2 particles abundant (such as, e.g., pions) so thatwe can neglect the event-by-event fluctuations of theirmultiplicity and the change of their number due to theconsidered processes.Following Eqs.(3) and (4), the general rate equationfor the average number of b1b2 pairs can be written asd〈N〉dτ=GV〈Na1〉〈Na2〉 −LV〈N2〉. (5)For abundant production of b1b2 pairs, where 〈N〉 ≫ 1,〈N2〉 ≈ 〈N〉2, (6)and Eq.(5) obviously reduces to the standard form, i.e.,d〈N〉dτ≈ GV〈Na1〉〈Na2〉 −LV〈N〉2. (7)However, for rare production of b1b2 pairs, where 〈N〉 ≪1, the rate equations (1) and (7) are no longer valid. Wehave instead〈N2〉 ≈ 〈N〉, (8)which reduces Eq.(5) to the following form:d〈N〉dτ≈ GV〈Na1〉〈Na2〉 −LV〈N〉. (9)Thus, in the limit 〈N〉 ≪ 1, the absorption term dependson the pair number only linearly, instead of quadraticallyfor the limit 〈N〉 ≫ 1.III. EQUILIBRIUM MULTIPLICITY ANDRELAXATION TIMETo illustrate the differences in the time evolution ofparticle abundance, we consider the two limiting cases,〈N〉 ≫ 1 and 〈N〉 ≪ 1, and present their equilibriumvalues and relaxation times for the production of b1b2pairs. As an example, we consider a system at fixedtemperature and volume and with no initial b1b2 pairs,i.e., 〈N〉(τ = 0) = 0.In the limit when 〈N〉 ≫ 1, the standard Eq.(7) is validand has the following well-known solution:〈N〉GC(τ) = NGCeq tanh(τ/τGC0), (10)where the equilibrium value for the number of b1b2 pairsNGCeq and the relaxation time constant τGC0 are given byNGCeq =√ǫ, τGC0 =VL√ǫ, (11)respectively, with ǫ ≡ G〈Na1〉〈Na2〉/L.In the particular case where particle momentum dis-tributions are thermal, the gain (G) and loss (L) termsjust represent the thermal averages of the production andabsorption cross sections withGL=db1α2b1K2(αb1)db2α2b2K2(αb2)da1α2a1K2(αa1)da2α2a2K2(αa2), (12)where dk’s denote the degeneracy factors, and αk ≡mk/T . The equilibrium value for the number of b1b2pairs in Eq.(11) now reads asNGCeq =db12π2V T 3α2b1K2(αb1). (13)Thus it is described by the Grand Canonical (GC) resultwith vanishing chemical potential due to our requirementof the (average) U(1) charge neutrality of the system.In the opposite limit where 〈N〉 ≪ 1, the time evo-lution is described by Eq.(9), which has the followingsolution:〈N〉C(τ) = NCeq(1− e−τ/τC0), (14)with the equilibrium value and relaxation time given byNCeq = ǫ, τC0 =VL. (15)With a thermal momentum distribution the equilib-rium value of b1b2 pair multiplicity has the followingform:NCeq =[db12π2V T 3α2b1K2(αb1)]·[db22π2V T 3α2b2K2(αb2)]. (16)This equation demonstrates the locality of the U(1)charge conservation. With each particle b1, a particle b22with the opposite charge is produced in the same eventin order to conserve charge locally. This is the resultexpected from the Canonical (C) formulation of conser-vation laws [10,11].We note that Eq.(16) is just the leading term in theexpansion of the canonical result for the multiplicity ofparticles carrying the U(1) charges. The general expres-sion is known to have the following form [10,11]:NCeq = NGCeqI1(2NGCeq )I0(2NGCeq ), (17)where NGCeq is given by Eq.(13) and Ii’s are the modifiedBessel functions.Comparing Eq.(11) and Eq.(15), we first find that, for〈N〉 ≪ 1, the equilibrium multiplicity in the canonicalformulation is much lower than what is expected fromthe grand canonical result,NCeq = (NGCeq )2 ≪ NGCeq . (18)This shows the importance of the canonical descriptionof charge conservation when the multiplicity of chargedparticles is small. We also note that the volume depen-dence in the two cases is different. The particle densityin the GC (abundant) limit is independent of V , whereasin the opposite canonical (rare) limit the density scaleslinearly with V .Secondly, the relaxation time for a canonical system isfar shorter than what is expected from the grand canon-ical result,τC0 = τGC0 NGCeq ≪ τGC0 , (19)due to small number of particles (NGCeq ≪ 1). For ex-ample, the total number of produced kaons in Au + Aucollisions at 1 GeV/A is of the order of 0.02. Thus thecanonical relaxation time is a factor of 50 shorter thanwhat is expected from the grand canonical formulation.We note from Eq.(5) that these two limits are essen-tially determined by the size of 〈δN2〉, the event-by-event fluctuation of the number of b1b2 pairs. The grandcanonical results correspond to small fluctuations, i.e.,〈δN2〉/〈N〉2 << 1, while the canonical description is nec-essary in the opposite limit.IV. MASTER EQUATIONIn this section, we formulate the general evolutionequation which is valid for an arbitrary value of 〈N〉.It is a master equation for Pn(τ), the probability of find-ing n pairs of b1b2 at time τ . This probability increaseswith time due to transitions from n− 1 and n+ 1 statesto the n state, while it also decreases due to transitionsfrom the n state to n−1 and n+1 states. The transitionprobability n→ n+1 per unit time due to pair creation isG〈Na1〉〈Na2〉/V and the transition probability n→ n−1due to pair annihilation is n2L/V . Therefore, the masterequation set has the form:dPndτ=GV〈Na1〉〈Na2〉 [Pn−1 − Pn]− LV[n2Pn − (n+ 1)2Pn+1], (20)where n = 0, 1, 2, 3, · · ·. Multiplying the above equationby n and summing over n, one recovers Eq.(5), the gen-eral rate equation for the time evolution of the averagenumber of b1b2 pairs. However, the master equation (20)contains much more information than the rate equation(5). It contains enough information to solve for the evo-lution of 〈N〉 (and all moments of N) for arbitrary 〈N〉.For example, one can also obtain the time evolution ofparticle fluctuations, 〈δN2〉, which are of physical impor-tance for rarely produced particles.We can convert the iterative equations (20) for Pn’sinto a partial differential equation for the generatingfunctiong(x, τ) =∞∑n=0xnPn(τ). (21)Multiplying Eq.(20) by xn and summing over n, we find∂g(x, τ)∂τ=LV(1− x) (xg′′ + g′ − ǫg) , (22)where g′ ≡ ∂g/∂x. Note that the g(1, τ) does not changewith time, which is equivalent to the conservation of totalprobability evident in Eq.(20).The equilibrium solution, geq(x), thus, obeys the fol-lowing equation:xg′′eq + g′eq − ǫgeq = 0. (23)By variable substitution (x = y2/(4ǫ)) this equation canbe reduced to the Bessel equation. The solution that isregular at x = 0 (since g(0) = P0 ≤ 1) is given bygeq(x) =1I0(2√ǫ)I0(2√ǫx), (24)where the normalization is fixed by g(1) =∑Pn = 1.The equilibrium probability distribution Pn can nowbe found from Eqs.(21,24) asPn,eq =ǫnI0(2√ǫ) (n!)2, (25)and the value of the average number of b1b2 pairs perevent at equilibrium is given by〈N〉eq = g′(1) =√ǫI1(2√ǫ)I0(2√ǫ), (26)which obviously coincides with the expected result forparticle multiplicity in the canonical ensemble given byEq.(17) and also reduces to the results for the two limit-ing cases in Sec.III.3V. CONCLUSIONSWe have formulated the kinetic master equation forstrongly correlated production of particles, where the cor-relation is due to the local charge conservation requiredby an U(1) internal symmetry. Our general rate equationis valid for arbitrary value of 〈N〉, thus it reduces to thegrand canonical results for large 〈N〉 and to the canonicalresults for small 〈N〉. Therefore, our equation providesa generalization of the standard rate equation beyondthe grand canonical limit. We have shown that for rareparticle production the equilibrium multiplicity is muchsmaller and the relaxation time is much shorter than ex-pected from the standard rate equation. For abundantparticle production, where the standard rate equation ap-plies to first order, one can use the general rate equationsto study finite-size corrections to the grand canonical re-sults.Our results could be of importance in the descriptionand understanding of equilibration phenomena and equi-librium properties of partonic or hadronic matter cre-ated in heavy-ion collisions. For example, it could pro-vide insights into the equilibration time of strange parti-cles produced at SIS, or open charm and charmoniumproductions at SPS and higher energies. It may alsobe meaningful for transport model studies of rare par-ticle production, especially the perturbative procedurefor rare processes [21], where the local charge conserva-tion requires that the production probability should beassigned to the pair instead to each particle separately.ACKNOWLEDGMENTSWe acknowledge stimulating discussions with M. Gyu-lassy, S. Jeon and L.D. McLerran. K.R. also acknowl-edges comments from C. Greiner, J. Knoll, and H. Feld-meier. M.S. and Z.L. are grateful to the nuclear the-ory group at LBNL for supporting the summer programwhere this work was started. This work was supportedby the Director, Office of Energy Research, Office of HighEnergy and Nuclear Physics, Division of Nuclear Physics,the Office of Basic Energy Science, Division of NuclearScience, of the U.S. Department of Energy under Con-tract No. DE-AC03-76SF00098, the Committee for Sci-entific Research (KBN-2P03B 03018), the National Sci-ence Foundation under Grant No. PHY-9870038, theWelch Foundation under Grant No. A-1358, and theTexas Advanced Research Program under Grant No.FY99-010366-0081.[1] R. Hagedorn, CERN yellow report 71-12, 101 (1971).[2] E.V. Shuryak, Phys. Lett. B42, 357 (1972); Sov. J. Nucl.Phys. 20, 295 (1975).[3] U. Heinz, Nucl. Phys. A661, 349 (1999); R. Stock, Phys.Lett. B456, 277 (1999).[4] P. Braun-Munzinger, I. Heppe and J. Stachel, Phys. Lett.B465, 15 (1999); P. Braun-Munzinger, J. Stachel, J. P.Wessels and N. Xu, Phys. Lett. B344, 43 (1995); Phys.Lett. B365, 1 (1996); J. Cleymans and K. Redlich, Phys.Rev. Lett. 81, 5284 (1998); D. Yen and M.I. 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Kinetic equation with exact charge conservation

Kinetic equation with exact charge conservation

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Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

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Texas A&M Repository

Kinetic equation with exact charge conservation

Abstract

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Full text

Available Versions

Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

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Texas A&amp;M Repository

Texas A&M Repository