The ROC model. Heinz Müller

Transcription

1 The ROC model Heinz Müller nd June 004

2 Chapter 1 Introduction The Rossendorf collision (ROC) model [1,, 3, 4, 5, 6, 7, 8] is implemented as a Monte- Carlo generator which samples complete events for hadronic as well as nuclear interactions. It can be considered as the further development of first attempts to understand multiple hadron production in hadronic interactions which were based on statistical considerations and the observation of excited intermediate subsystems called fireballs (F) (see the reviews [9, 10, 11]). A modern version of such a thermodynamical approach can be found, e.g., in ref. [1]. Contrary to the older statistical approaches the ROC model is, however, founded on the quark picture of hadrons. In this sense the ROC model seems to be a basically statistical approach in deriving the relative contributions of the various final channels by calculating their statistical weights from the phase-space factors. Decisive, however, is the dynamical input in form of empirical transition matrix elements which strongly modify the the population of the final states. The ROC approach does not need any parameterisation of elementary cross sections. Its applicability is not restricted at higher energies by the growing number of unknown elementary cross sections. All dynamic information is gathered in a few parameters which are either constant or change smoothly with energy and/or target mass. In the realm of hadronic interactions the ROC model is able to describe soft hadron production in the energy region between particle production thresholds and ISR energies. For this purpose the basic ingredients of hadron production models are reformulated in such a way that they are applicable at low energies as well. The ROC model is built as a minimal approach in the sense that the number of parameters is restricted to the minimum necessary to well reproduce the main features of the available data. Special emphasis is put on the consideration of short-range correlations, because the F concept yields a natural explanation of the observed phenomena. In the case of nuclear reactions the participant-spectator picture is applied. Observables are calculated as an incoherent sum over contributions from a varying number of nucleons participating in the interaction. The interaction of the participants is treated in complete analogy to hadronic reactions.

3 Chapter The model The basic idea of a statistical approach consists in the assumption that the probabilities of formation of the various final states are proportional to their statistical weights. This idea was implemented by Fermi[13] fifty years ago, but his model turned out to be applicable only at relatively low energies. At higher energies it is no longer a good approximation to assume that the whole initial energy is randomly distributed among the final particles. Particles with high transverse momentum, e.g., are produced with extremely low probability indicating that the final states are dynamically linked with the initial state. In contrast to this early attempt the ROC model is based on the following modified statistical approach. Instead of calculating the statistical weights from the whole phasespace the dynamics of the interaction is implemented in form of empirical functions which either suppress certain regions of the phase-space or impose additional non-statistical weights. We define a channel α by the number n, masses m i and quantum numbers of the final particles. The relative probability of populating a channel α is calculated as the product of the Lorentz-invariant phase-space factor d L n (s; α) with the square of an empirical matrix element T, which describes the dynamics of the interaction process. Here, s = M = P denotes the square of the total energy with P being the total fourmomentum..1 Hadron-hadron interactions.1.1 Decomposition of phase-space The phase-space of n particles dl n is given by dl n (s) = = n n d 4 p i δ(p i m i ) δ 4 (P p i ) n d 3 p i n δ 4 (P p i ) (.1) e i with s = M = P. In (.1) P = (E, P ) is the total four-momentum and p i = (e i, p i ) the four-momentum of the i-th particle with rest mass m i = p i. For numerical calculations the δ-function in eq. (.1) is removed by introducing a new set of n 4 variables to replace the 3n three-momentum components. It is reasonable to choose a set of variables, which reflects the underlying physical picture of the interaction process In fig..1 the reaction a + b F F N h h n (.) between hadrons a and b resulting in the production of n particles is schematically depicted. First, N n intermediate particle groups called Fs are produced, which decay into so-called primary particles. The primary particles define the channels for which the weights (.4) are calculated. Among them are resonances, which decay subsequently into stable 3

4 Figure.1: Phase-space decomposition of a two-step process. N Fs with masses M I (I = 1... N) are produced in the interaction of two hadrons a and b with four-momenta p a and p b. The Fs decay in the second step, where the I-th F disintegrates into n I primary particles with n I=1 n I = n. Resonances among the primary particles decay afterwards into stable hadrons. A possible decay chain is shown. hadrons. The phase-space factor corresponding to the diagram of fig..1 can be calculated according to ref. [14] dl n (s; α) = [ N ] dmi dl ni (M I ; α I ) dl N (s; M 1... M N ) (.3) I=1 with the invariant masses of the Fs equal to M I = PI = ( n I p i) and the final channel defined as the vector α = ( α 1,..., α N ) of the decay channels of the individual Fs. The probability of populating the channel α is given by dw (s; α) dl n (s; α) T (.4) Here, the square of the matrix element T contains the dynamical input and is split into factors T = Ti Tqs Tex Tt Tl Tst (.5) which describe the interaction Ti resulting in the production of N Fs, the production of hadrons Tqs via the creation of quark-anti-quark (q q) pairs, the invariant-mass distribution of the Fs Tex, the transverse Tt and longitudinal Tl momentum distribution of the Fs, and, finally, some additional factors Tst necessary for the calculation of the statistical weights. In the following subsections these factors will be discussed in more detail..1. The interaction T i In models like DPM [15] or VENUS [16] the colliding hadrons are considered as extended and composite objects consisting of an indefinite number of partons, the interaction of which is assumed to proceed via color exchange. If, e.g., the color exchange between the valence quarks of two baryons takes place, then two strings are produced. Each of them consists of the remaining diquark and the valence quark removed from the other baryon. Since more complicated exchanges are possible and the collision may proceed at different impact parameters a varying number of strings is produced. The corresponding probabilities are derived from Gribov-Regge theory [17] in the limit of high energies in 4

5 combination with the use of profile functions for integrating over the impact parameter. PYTHIA [18] describes low-p t events on the basis of the multiple interaction model of ref. [19], which extends a high-p t picture down into the low-p t region by regularizing the p t scale. The number of (independent) parton-parton collisions in one event depends on the impact parameter and on the assumed matter distribution inside the interacting hadrons. In the ROC model, non-statistical weights Ti (N) for producing a definite number N of Fs are introduced. This is a phenomenological parameterization of the contributions from the different color exchange diagrams and of the integral over the impact parameter, which we apply also at low energies. Such a dynamical input is necessary, because the phase-space factor alone tends to overestimate the number of Fs simply due to the fact that the number of states increases the more Fs are produced. A thermodynamical approach using a chemical potential as the only parameter to regulate the number of Fs turned out to be not flexible enough. Therefore, the negative binomial distribution is applied where we have two free parameters v and q T i (N; q, v) = ( v N ) ( q) N (1 q) v. (.6) In the calculations we use the mean N and the ratio of the variance to the mean D = σ / N as parameters from which q = (D 1)/D and v = N/(D 1) follow. y this means, the whole complicated interaction scenario is described by altogether two parameters in conjunction with the corresponding phase-space factors..1.3 Quark statistics T qs The factor Tqs stands symbolically for the algorithm applied to sample the possible final states. At first the valence quarks of the interacting hadrons are redistributed among the Fs and then the final hadrons are produced via the creation of new q q pairs. All internal quantum numbers are conserved automatically by this procedure. The complicated details of color exchange diagrams and string drawings are replaced by statistical considerations. For describing the interaction process the notion of quark removal is borrowed from ref. [16]. It is assumed that the multiple interaction of the partonic constituents of the incoming hadrons leads to the creation of N Fs. Color exchange between the constituents results in the removal of the involved quarks and gluons from the incoming hadrons. Removed quarks are found in any of the other Fs with equal probability. The remaining partons of the interacting hadrons form the two leading Fs, the scattered partons the N central Fs. In a next step q q pairs are produced and randomly (the q s and q s independently) distributed between the Fs such that each F becomes color neutral and contains the minimal number of q s and q s necessary for building at least one hadron (meson or baryon). This procedure is the equivalent for the sum over the possible color exchange diagrams with the restriction that the removal of two or three valence quarks or of a single sea quark is neglected. Only the removal of one valence quark with probability W v is considered. The remaining probability 1 W v is understood as gluon or quark-pair removal. As equivalent to string fragmentation each F is then filled separately with an arbitrary number n qi of additional q q pairs where n qi 0. Up, down, strange and charm quarks are produced in the ratios u : d : s : c = 1 : 1 : λ s : λ c (.7) with λ s and λ c being suppression factors due to the heavier masses of the strange and charm quarks. The creation of top quarks can be neglected in the considered energy range. The final hadrons are built up in each F independently according to the rules of quark statistics [0]by randomly selecting sequences of q s and q s. A q q gives a meson, while baryons or antibaryons are formed from qqq or q q q. From a given sequence of quarks the different hadrons are formed according to the tables of the particle data group [1]. 5

6 All baryons marked in the tables with three or four stars, the meson nonets built from u, d and s quarks with angular momenta zero and one ( 1 S 0, 3 S 1, 1 P 1, 3 P 0, 3 P 1, 3 P ) as well as all charmed hadrons are taken into account. An empirical probability distribution The described algorithm together with the parameters λ s, λ c and Θ h is the equivalent of the usually much larger number of parameters describing the fragmentation of strings. It should be stressed that the ROC model has no parameter fixing the probability of diffractive processes. A diffractive process is usually assumed to proceed via the exchange of a Pomeron, a fictitious particle which does not affect the quantum numbers of the involved particles. In the string models [15, 16] diffractive scattering is treated as a special process whose probability is determined by a free parameter adapted to data. In the ROC model diffractive scattering is one of the possible final channels, because there is a certain probability that one or even both leading Fs are identical with the initial protons. This happens if the valence quark content of the considered F remains unchanged, if no additional q q pairs are produced, and if a proton, and not a resonance, is built from the available quarks uud in the recombination phase..1.4 Mass distribution of Fs T ex Until now we have explained how the hadrons forming the final state are sampled. In a next step the integral over the invariant masses of the Fs M I [see eq. (.3)] is performed. Again phase-space alone produces too large F masses because of their corresponding large numbers of states. To restrict the invariant masses the Fs are assumed to be characterized by a temperature Θ. As matrix element squared the function with the asymptotic behavior T ex = (M I /Θ) K 1 (M I /Θ) (M I /Θ) K 1 (M I /Θ) N (M I /Θ) K 1 (M I /Θ) (.8) I=1 M I /Θ M I /Θ exp( M I /Θ) M I /Θ 0 1 (.9) is used. The expression (M I /Θ) K 1 (M I /Θ) is the kernel of the so-called K-transformation (see [14]) used to transform a micro-canonical phase-space distribution depending on the total energy M I of the I-th F into a canonical one, which is characterized by a temperature Θ. In eq. (.8) K 1 stands for the modified essel function. For increasing invariant masses M I the function (M I /Θ) K 1 (M I /Θ) strongly decreases, while the phase-space factor L ni (M I ) of the I-th F becomes larger. Their product has a maximum at a value of M I determined by the parameter Θ, which thus fixes the average internal excitation energy of the I-th F. Since Fs consist of only a few particles, we are far from the thermodynamical limit. Therefore, the momenta of the hadrons are calculated from the decay of the Fs according to phase-space and not from macro-canonical distributions.. Hadron-nucleus interactions..1 The partial cross sections The interaction of an incoming hadron a with a target nucleus (, Z ) consisting of Z neutrons and Z protons is considered in the participant-spectator picture. Observables are calculated as an incoherent sum over contributions from a varying number b of nucleons (thereof z b protons) participating in the interaction d σ(s) = b=1 min(b,z ) z b =max(0,b N ) σ b,zb W b,zb (s; α) Wb,zb (s; α). (.10) Here, s = P denotes the square of the centre-of-mass energy of the projectile-target system with P = (E, P ) being the total four-momentum. The summation extends over 6

7 all possible charge states ranging from clusters consisting completely of neutrons to those composed totally of protons. This range is restricted in cases where the number of available protons Z or neutrons N = Z is smaller than the number of cluster nucleons. The partial cross sections σ b,zb are calculated using a modified version of a Monte-Carlo code [] which is based on a probabilistic interpretation of the Glauber theory [3]. They account for sequential collisions between the projectile and b target nucleons capable of sharing their energy in close analogy with the virtual clusters of the cooperative model [4, 5, 6, 7, 8, 9, 30, 31] (see [3] for a discussion of the novel features of the ROC model compared to the cooperative model). We use the profile function [ ] Γ (d) = 1 (1 p i ) ρ ( r i ) d 3 r i of the considered nucleus, which depends on the nucleon density ρ ( r i ) and the probability p i = exp( d i π/σ hn ) for an interaction of the projectile and the ith target nucleon with d i being the distance between the interacting particles. The nucleon density [33] ρ ( r) (1 + η [1.5 (b a )/b + a r /b 4 ]) exp( r /b ) (.11) of light nuclei < 0 can be derived from a standard shell model wave function with η = ( 4)/6, b = a (1 1/) and b = 1.55 fm. Then the hadron-nucleon cross section σ hn is adapted such that the integral of the profile function over the impact parameter d reproduces the total inelastic h cross section σh in = d d Γ (d) The same calculation yields also the partial cross sections σ b,zb further details see []). we are interested in (for.. Phase-space decomposition For a hadron-nucleus (a) reaction the total four-momentum is given by P = p a +P with p a and P being the four-momenta of the interacting hadron and nucleus, respectively. We differentiate between target nucleons participating in the interaction and spectators. In a first step the spectators (or, what is the same, the decay products of the residual nucleus) are separated by introducing the invariant mass M and the four-momentum P of the group of n particles arising from the decay of the residue of the target nucleus via the following identities: 1 = dm δ(p M) (.1) with the definition 1 = d 4 P δ 4 (P P = Inserting (.1)... (.14) into (.1) the phase-space factor becomes n n dl n (s) = dm d 4 P δ(p M ) n n C p i ) (.13) p i. (.14) d 4 p i δ(p i m i ) δ 4 (P d 4 p i δ(p i m i ) δ 4 (P C 7 n n C p i ) p i ). (.15)

8 Taking into account that d d 4 p i δ(p i m 3 p i i ) = (.16) e i and the definition (.1) of the n-particle phase-space Eq. (.15) can be written as dl n (s) = dm d 3 P dl n (M E ) dl nc (PC) (.17) with n C = n n being the number of particles emerging from the participant interaction. The invariant mass of this participant group is given by M C = PC with the fourmomentum P C = P P C. Eq. (.17) makes sense, because the distribution of the momentum P can be calculated from the internal momentum distribution ρ( P ) of the nucleus...3 Direct sampling of internal motion In the ROC model a final channel is selected by creating new particles in the participant system via the production of an arbitrary number of q q pairs and by sampling the masses of the fragments the spectator decays into. Eq. (.17) is the usual way to carry out the integral over the internal momentum distribution if there is sufficient kinetic energy available in the whole system (if the kinetic energy carried away by the residual nucleus is small compared to the available energy). In fact, the momentum distribution of the excited residual nucleus of invariant mass M is sampled. This has the advantage that we always deal with particles on the mass shell. In case of small excess energies the sampling of the momentum P of the target residue may result in an invariant mass M C µ C with µ C being the sum of the rest masses µ C = m i of all particles emerging from the participant interaction. Such nonphysical events must be discarded or it is necessary to fix the limits of the integral over the internal motion on the basis of the given masses M (or µ ) and µ C...4 Participant mass M C as variable Another way consists in replacing the integral d 3 P = dp P d cos Θ dϕ by dp dϕ d M C in the following way. According to [14, p. 198 eq. (4.7)] the particles of the final state are separated into two groups, the spectator particles and the participants C dl n (s) = dm dm C dl (P ; M, M C) dl n (M ) dl nc (M C). (.18) The invariant masses M and M C ly within the triangle defined by M µ M C µ C M + M C s (.19) and can be sampled accordingly. For given masses it remains to consider the two-body phase-space dl (P ; M, M C ) in an appropriate reference system. The simplest possibility, the center-of-mass system, might be reasonable in case of very low excess energy. If, however, the strong damping of the spectator momentum due to the nuclear wave function still plays a role, then the rest system of the target nucleus is surely more appropriate, although the mathematics is slightly more complicated dl (E, P ) = d3 P E d 3 P C E C δ(e E E C )δ 3 ( P P P C ). (.0) The integral over the second δ-function yields P C = P P and eq. (.0) becomes dl (E, P dp P ) = d cos Θ dϕ 4E (E E ) δ(e E [( P P ) + MC] 1/ ). (.1) 8

9 The remaining δ -function is used for carrying out one of the integrals by considering the argument of the δ -function either as a function of P P or of cos Θ. f(cos Θ ) = E E [( P P ) + M C] 1/ = E E [ P + P P P cos Θ + M ]1/ C (.) The condition f(cos Θ ) = 0 yields then cos Θ as function of P. From (.) it follows (E E ) = P + P P P cos Θ + M C cos Θ = E EE + E P P M C P P = M + M M C EE P P (.3) In order to integrate over cos Θ in eq. (.1) the following property of the δ -function is used 1 δ(f(x)) = f (x 0 ) δ(x x 0), f(x 0 ) = 0. (.4) The derivative of (.) is inserted in (.1) resulting in f (cos Θ) = P P E C dl (E, P ) = dϕ d P P 4E (E E ) E C P P = dϕ d P P. (.5) 4 E P In (.5) the limits of the integral over d ϕ are given by 0 ϕ π. The limits of d P follow from momentum and energy of the residue E = M + M M C, P M = E M in the center-of -mass system by Lorentz-transformation into the target rest system P min P max = ( E P + P E )/M = (E P + P E )/M. In case of v = P /E < v = P /E the minimal residue momentum P min becomes negative. That means that all values of Θ are possible and the lower limit of the integral is given by P min. For v v there exists a maximal value for Θ and it holds P min 0. Inserting (.5) into (.18) yields finally dl n (s) = dm dm C dϕ d P P dl n (M) 4 E P dl nc (MC). (.6)..5 Spectator-participant interaction So far only the kinematics was considered where the two discussed ways, see (.17) and (.6), were selected with regard to the application of the internal momentum distribution ρ ( P F ) of the nucleons within the nucleus. In the versions of ROC the momentum P of the nuclear residue was assumed to be equal to the internal momentum P F sampled 9

10 according to ρ ( P F ) (the momentum of the participants is then P F ). Together with the sampled mass M of the spectator system the four-momentum P = (E, P F ) = ( M + P F, P F ) (.7) is thus fixed. The four-momentum of the participants follows from P b = P P, and the four-momentum of the participants from P C = p a + P b resulting in P = p a + P = p a + P b + P = P C + P. (.8) As consequence of this approach the momentum distribution of the target residue is isotropic in the target rest frame. If further an isotropic decay of the residue without any interaction with the participant system is assumed, then the fragment spectra become isotropic, too. There are, however, experimental results [34] which clearly contradict such an assumption. In [34] the energy spectra of fragments arising from the reaction p(.1 GeV) + 1 C were measured at angles between 0 and 160. For the fragment 7 e the differential cross section is at 0 nearly a factor of four larger than at 160. The origin of this obvious discrepancy between experimental facts and the result of model calculations might be the treatment of the internal motion. The separation of the target nucleus in participants and spectators is done so far in a rather asymmetric way. Since the mass of the spectator system is immediately increased to its final value, the mass squared of the participants Pb = (m E ) P F with E from (.7) becomes more and more off-shell with increasing internal momentum P F. However, one can argue that the excitation of the target residue has its origin in the passage of participants and newly produced particles through the target residue during the interaction. Therefore, it seems to be more appropriate to consider the interaction as a two-step process. First, the nucleus is virtualy separated into two nucleon groups, participants and spectators, which have a relative momentum. The energy of the mother nucleus is distributed between these two groups as symmetrically as possible. Then, the final four-momenta of participants and spectators are assumed to be the result of secondary interactions during the separation of these two particle groups. In this picture the target nucleus consists of participants and spectators P = P b + P, where the prime indicates that the particle systems are considered as intermediate ones. In the rest system of the target nucleus the internal momentum is sampled according to ρ ( P F ) and the energy is distributed in dependence on the number of nucleons in the two nucleon groups. So, the intermediate four-momenta of target participants and spectators are given by P b = ( b M, P F ) and P = ( M, P F ) with = + b, respectively, while the intermediate four-momentum of the whole participant system follows from P C = p a +P b. The following spectator-participant interaction (SPI) is assumed to be a soft process with only minimal changes of the kinematics of the interacting particle groups. This can be deduced from the fact that already the simple treatment of the internal motion without any secondary interactions yields a satisfactory description of a large amount of experimental findings. The minimal demand we have to fulfill consists in increasing the invariant mass of the spectator system from M = ( M ) P F to its final value M. Further, it seems to be reasonable to assume that any changes of the momentum directions of the interacting systems remain negligibly small. For the calculations it is desirable to implement the interaction in such a way that a smooth transition between the two extreme pictures becomes possible. The above discussed SPI is equivalent to introducing additional integrals over the intermediate particles and C into (.17) according to d 4 P d 4 P C δ 4 (P P P C) ρ( P F ) T spi, (.9) where the δ -function ensures energy-momentum conservation, the internal momentum distribution ρ( P F ) applies here to the intermediate particle with P F = P and the empirical matrix element Tspi is responsible for the details of the interaction to be discussed in the following. We use the δ -function in (.9) to carry out the integral over d 4 P C yielding P C = P P. It remains the integral d 4 P ρ( P F ) T spi = de d 3 P ρ( P F ) T spi (.30) 10

11 with independent integrations over energy and momentum of the virtual particle. Inserting (.30) into (.17) yields dl n (s) = de d 3 P ρ( P F ) Tspi dm d 3 P dl n dl nc (.31) E containing four additional integrals compared to the original expression (.17) For the matrix element squared we assume Tspi = W (E ) δ( ˆP ˆP ) W (M C ) (.3) and use either W (E ) = δ(e M ) (.33) or a function W (E ) exp(η E / E ) (.34) with one parameter η and restrict the limits of the probability distribution W (E ) to M E E = P F + M. The expression E = P F + M M in (.34) diminishes the dependence of the parameter η from the actual energy values. Expression (.33) yields the one extreme picture, virtual separation with subsequent secondary interaction, while distribution (.34) yields a smooth transition between the this pictures, (η < 0 and large) and the other extreme, immediate creation of the final spectator system(η > 0 and large). In the rest system of the nucleus the relevant four-momenta are then given by P = (E, PF ) P b = (M E, P F ) P C = (e a + E b, p a + P b) (e a + M E, p a + P F ) (.35) The interaction of participants and spectators is considered in their cms, and the fourmomenta P and P C in this system are denoted by P and P C. In (.17) we have to solve the integral over the final spectator momentum P under consideration of the remaining terms from the matrix element (.3). The invariant expression d 3 P /E is considered in the cms and we get d 3 P E δ( ˆP ˆP ) W (M C ) = d P E P W (M C ) = de P W (M C ). (.36) The function δ( ˆP ˆP ) fixes the direction of of the residue and reduces thus the threedimensional integral d 3 P to a one-dimensional one over the magnitude P of the threemomentum of. From it follows de P E = M + M M C. (.37) M W (M C ) = dm C P 4M W (M C). (.38) For the probability distribution of the final participant mass M C we may take as the simplest assumption W (M C ) = δ(m C M C ) or again an exponential for integrating M C in the region M C < M C < M M. Inserting (.30), (.3) and (.38) into (.17) yields finally dl n (s) = 1 4M dm dmc d 4 (P ) ρ( P )W (E ) W (M C ) P dl n (M) dl nc (PC). (.39) 11

12 .3 Nucleus-nucleus interactions In the case of a nucleus-nucleus reaction A + the identities 1 = dmaδ(p A MA) (.40) 1 = d 4 P A δ 4 (P A with the definition P A = n A n A p i ) (.41) p i (.4) can be used in complete analogy to (.1)... (.14) and the phase-space factor becomes dl n (s) = dma d 4 P A δ(pa MA) dm d 4 P δ(p M) n A n n C what can be finally written as d 4 p i δ(p i m i ) δ 4 (P A d 4 p i δ(p i m i ) δ 4 (P d 4 p i δ(p i m i ) δ 4 (P C n A n n C p i ) p i ) p i ), (.43) dl n (s) = dm A d 3 P A E A dl na (MA) dm d 3 P dl n (M E ) dl nc (PC). (.44) Here, the number of particles arising from the collision of the participants and their total four-momentum n C = n n A n (.45) P C = P P A P (.46) are used. Again P A and P are related to the internal momentum distributions of the colliding nuclei. 1

13 Appendix A Mathematical details A.1 The integral over the invariant mass of the Fs In calculating the phase-space integral we have to integrate over the invariant masses of the Fs produced during the interaction. From (.3) together with the corresponding matrix element (.8) the considered partial integral becomes di = dm dl n (M ) (M/Θ) K 1 (M/Θ). (A.1) According to [14, page 300] a F can be considered (in the thermodynamical limit) as a microcanonical ensemble. In the non-relativistic limit the kinetic energy per degree of freedom is fixed by the temperature and is equal to 1 Θ. Also for small numbers of particles in the ensemble we are dealing with the maximum is expected at Ekin max = 3 (n 1) Θ with a fluctuation of σ = 3 (n 1) Θ. For convenience we use for the MC-evaluation the distribution W MC E kin exp( (E kin Ekin max) ). (A.) σ The normalization of (A.) follows from C E up kin E kin =0 W MC (E kin ) de kin = 13

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