Correspondence of multiplicity and energy distributions

Abstract

The evaluation of the number of ways we can distribute energy among a collection of particles in a system is important in many branches of modern science. In particular, in multiparticle production processes the measurements of particle yields and kinematic distributions are essential for characterizing their global properties and to develop an understanding of the mechanism for particle production. We demonstrate that energy distributions are connected with multiplicity distributions by their generating functions.

Appendix A: Boltzmann-Gibbs energy distribution and Poissonian multiplicity distribution

Suppose that one has N independently produced particles with energies $\{E_{1,\ldots ,N}\}$, distributed according to Boltzmann distribution,

$$\begin{aligned} F\left( E\right) =\frac{1}{T}\exp \left( -\frac{E}{T}\right) \end{aligned}$$

(A.1)

with “temperature” parameter $T=\langle E\rangle$. The sum of energies, $U=\sum _{i=1}^{N}E_{i}$ is then distributed according to gamma distribution

$$\begin{aligned} F_{N}\left( U\right)&=\frac{1}{T\left( N-1\right) !}\left( \frac{U}{T}\right) ^{N-1} \exp \left( -\frac{U}{T}\right) \nonumber \\&=F_{N-1}\left( U\right) \frac{U}{N-1} \end{aligned}$$

(A.2)

with cumulative distribution equal to:

$$\begin{aligned} F_{N}\left( >U\right) =1-\sum _{i=1}^{N-1}\frac{1}{\left( i-1\right) !} \left( \frac{U}{T}\right) ^{i-1}\exp \left( -\frac{U}{T}\right) .\nonumber \\ \end{aligned}$$

(A.3)

Looking for such N that $\sum _{i=0}^{N}E_{i}\le U\le \sum _{i=0}^{N+1}E_{i}$ we find its distribution. which has known Poissonian form

$$\begin{aligned} P\left( N\right)&=F_{N+1}\left(>U\right) -F_{N}\left( >U\right) \nonumber \\&=\frac{\left( U/T\right) ^{N}}{N!}\exp \left( -\frac{U}{T}\right) \nonumber \\&=\frac{\langle N\rangle ^{N}}{N!}\exp \left( -\langle N\rangle \right) \end{aligned}$$

(A.4)

with $\langle N\rangle =U/T$.

For the constrained systems (if the available energy is limited, $U=\mathrm{const}$), whenever we have independent variables $\{E_{1,\ldots ,N}\}$ taken from the exponential distribution (A.1), the corresponding multiplicity N has Poissonian distribution (A.4)⁴. However, if the multiplicity is limited, $N=\mathrm{const}$, the resulting conditional probability becomes:

$$\begin{aligned} F\left( E|N\right)&=\frac{F_{1}\left( E\right) F_{N-1}\left( U-E\right) }{F_{N} \left( U\right) }\nonumber \\&=\frac{N-1}{U}\left( 1-\frac{E}{U}\right) ^{N-2} \end{aligned}$$

(A.5)

the same as given by Eq. (5), and only in the limit $N\rightarrow \infty$ the energy distribution goes to the Boltzmann distribution (A.1). For fluctuating multiplicity according to Poisson distribution, the energy distribution is given by (A.1).

In the same way, as demonstrated in Ref. [17], Tsallis energy distribution is connected with the NBD of multiplicity.