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Mathematics.

I can generate $N$ random numbers, so that an individual number's distribution (same for all of them) approaches the specified distribution, while enforcing strictly the constraint of the specified sum of these $N$ numbers. This ability is essential to conserve momentum $\sum p_X=0, \sum p_Y=0, \sum p_Z=0,$ and conserve the total multiplicity while simulating individual fireballs, thus reproducing the given multiplicity distribution regardless of the number and size of the individual fireballs. In principle, the constraint of the sum distorts the specified single-number distribution. The degree of the distortion depends on the sum and the parameters of the single-number distribution. The problem of generating a multi-variate distribution is discussed in [83], it is recommended to do a succession of conditional single-variate distributions. The use of the Central Limit Theorem to (approximately) integrate the successive single-variate distributions is as in [84]. Appendix D gives the details. In course of the simulation of an event, we apply the same technique for two problems:

• particle number conservation is enforced when random number of particles is split between clusters. The cluster multiplicity has a Gaussian distribution with RMS much less than the mean. The mean is the ``clustering parameter'' of the model. The number of clusters in a particular event and the random cluster multiplicities are made up based on the ``clustering parameter'' of the run and the total multiplicity of the event.
• momentum conservation for the entire event. The longitudinal expansion prescribes random $p_Z$ momenta of the clusters (see Eq.7.3); $p_X$ and $p_Y$ momenta of each cluster are 0.
• momentum conservation within a fireball. The total momentum is being split between the particles of the fireball.