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Generating sequences of random numbers with a fixed sum.

The multifireball event generator used in Chapter 7 is not a true event generator in the sense that it does not propagate the initial state into a final state. Rather, it generates a final state subject to a set of constraints. The problem, expressed in the header of this Appendix, had to be solved in order to

1. impose momentum conservation on the final states created in the multifireball event generator
2. ensure independency of the event multiplicity on the mean fireball size

Formulation of the problem: generate $N$ random numbers $x_i, i=1, ..., N$, subject to the condition

$$\sum_{i=1}^{N}x_i = X,$$ (114)

so that the probability density distribution of an individual $x_i$ approaches given function $f(x)$ ( such that $\int_{-\infty}^{+\infty} f(x) \,dx = 1$ and the variance of $x$ is finite) in the limit of large $N$.

The multivariate probability density distribution can be presented [83] as a product of conditional distributions

$$f(x_1,...,x_N) = f_1(x_1)f_2(x_2\vert x_1) f_3(x_3\vert x_1,x_2)...f_N(x_N\vert x_1,x_2,...,x_{N-1}),$$ (115)

where, e.g., $f_2(x_2\vert x_1)$ denotes distribution for $x_2$ obtained from $f(x_1,...,x_N)$ if one keeps $x_1$ fixed at a certain value. In the following, I will deliberately omit the normalization factor. The extra complexity it brings in is not warranted in this context: you get exactly one random number per subroutine call, therefore normalization factor has no meaning and only the functional shape of the distribution matters. Here ${\mathfrak{D}}[x]$ is variance of $x$. For the $k$-th conditional distribution,

$$f_k (x_k\vert x_1,x_2,...,x_{k-1}) \propto$$

$$\propto f(x_k) \int \delta(x_1+x_2+...+x_{k-1}+x_k+...+x_N-X) \prod_{i=k+1}^{N}f(x_i)\,dx_i \propto$$

$$\propto f(x_k) \exp\left(-\frac{(\sum_1^{k-1}x_i+x_k+X(N-k)/N-X)^2}{(N-k) {\mathfrak{D}}[x]}\right)$$ (116)

The last transition is based on an (approximate and non-rigorous !) use of the Central Limit Theorem (CLT for short). Indeed, having $N$ numbers $x_i$ distributed according to $f(x)$ with variance ${\mathfrak{D}}[x]$ and their sum being $X$, one expects the sum $\sum_{k+1}^{N}x_i$ to be distributed around $X(N-k)/N$, with variance $(N-k){\mathfrak{D}}[x]$. Then, $x_k$ (the only free quantity other than $\sum_{k+1}^{N}x_i$) is distributed around $X-x_k-\sum_{k+1}^{N}x_i$ and the same variance, since the total sum is strictly fixed at $X$. This use of CLT is

• non-rigorous because the CLT is valid for independently sampled random numbers - in contradiction with Eq. D.1.
• approximate because the CLT is a limit of large $N$.

From the practical point of view, these caveats mean that for small $N$, the generated distribution of $x$ may deviate from $f(x)$ somewhat. Nevertheless this technique accomplishes its goal whereas its approximate nature is of no concern for the application in question.