# Physics:PP/9/Basic Descriptive Statistics

## Basic Descriptive Statistics

## Multiplicity distributions

## Moments and cumulants

## Bunching parameters

Similar to cumulants, Poisson statistics in particle collisions can be probed by calculating the so-called bunching parameters (BP) ^{[1]} ^{[2]}. They are defined as:

[math]\displaystyle{ \eta_{m} = \frac{m}{m-1} \frac{P_m P_{m-2}} {P^2_{m-1}}, \qquad m\gt 1 }[/math]

where [math]\displaystyle{ P_m }[/math] is the probability of observing *m* particles. For a Poisson distribution, [math]\displaystyle{ \eta_m = 1 }[/math], so that mth order BP measures deviations of the observed multiplicity distribution from the Poisson distribution.
If BPs are larger than 1, the corresponding multiplicity distribution
is broader than the Poisson distribution. On the other hand, if [math]\displaystyle{ \eta_m\lt 1 }[/math],
the corresponding multiplicity distribution is narrower than the Poisson distribution.

For counting experiments, one advantage of the BPs is that they prob a multiplicity distribution locally, i.e. a BP of the order *m* is defined
by events with the multiplicities *m-2*, *m-1* and *m*.
As the result of this feature, BPs are not
affected by statistical fluctuations from large-multiplicity ``tails* that*
normally contribute to the moments or cumulants.
Experimental definitions of the BPs can be found in ^{[1]}, ^{[2]}

## Long-range correlations

## Short-range correlations

## Intermittency

**Intermittency** in particle physics ^{[3]} is the intermittent pattern of fluctuations in high energy which can be observed by calculating the scaled factorial moments. The term is analogies to Intermittency, which is a manifestation of scale invariance and randomness in physical systems.

Multiplicity distributions and correlations between final-state particles are an important testing ground for perturbative QCD and for phenomenological models
describing the hadronic final state ^{[4]}
Short-range correlations in particle multiplicities for a given observable [math]\displaystyle{ \Omega }[/math] can studied in terms of the normalized factorial moments:

[math]\displaystyle{ F_q(\Delta\Omega)=\langle n(n-1)\ldots (n-q+1)\rangle / \langle n \rangle^q, \qquad q=2,3 .. }[/math]

where [math]\displaystyle{ n }[/math] is the number of particles inside a phase-space region
of size [math]\displaystyle{ \Delta \Omega }[/math] and
angled brackets [math]\displaystyle{ \langle \ldots\rangle }[/math] denote the average over all events. The factorial
moments, along with cumulants ^{[5]} and bunching parameters ^{[6]} ^{[7]}
are convenient tools to characterize the multiplicity distributions when [math]\displaystyle{ \Delta\Omega }[/math] becomes small.
For uncorrelated particle production within [math]\displaystyle{ \Delta\Omega }[/math],
Poisson statistics holds and [math]\displaystyle{ F_q=1 }[/math] for all [math]\displaystyle{ q }[/math]. Correlations
between particles lead to broadening of the multiplicity distribution
or dynamical fluctuations. In this case the factorial moments increase
with decreasing [math]\displaystyle{ \Delta \Omega }[/math]. If this rise follows a power law,
this effect is frequently called intermittency.

The intermittency in this case means that the normalized factorial moments, normalized comulants or bunching parameters exhibit the power law behavior vs. the resolution :

[math]\displaystyle{ F_q(\Delta\Omega) \simeq (1/\Delta\Omega)^f_{q} }[/math]

where the strength of intermittency is characterized by the intermittency exponents [math]\displaystyle{ f_q }[/math], which are directly related to the anomalous fractal dimensions [math]\displaystyle{ d_q }[/math]

[math]\displaystyle{ d_q = f_q / (q-1) }[/math]

The theoretical understanding of intermittency in high energy experiments is missing as yet because we are unable to relate this phenomenon to the fundamental properties of the theory of strong interactions. However this because can be reproduced by various self-similar random cascading models and Monte Carlo simulations of particle collisions that include short-range correlations in the parton cascade, fragmentation and Bose-Einstein correlations between final-state particles.

## References

- ↑
^{1.0}^{1.1}Chekanov, S. V.Kuvshinov, V. I. "Bunching parameter and intermittency in high-energy collisions" Acta Phys. Polon. (1994) B25, p.1189 arXiv: https://arxiv.org/abs/hep-ph/9605379 - ↑
^{2.0}^{2.1}Chekanov, S. V.Kittel, W.Kuvshinov, V. I. "Generalized bunching parameters and multiplicity fluctuations in restricted phase-space bins" Z. Phys. (1997) C74, p.517 arXiv: https://arxiv.org/abs/hep-ph/9606335 - ↑ A. Bialas, R. B. Peschanski, Nucl. Phys.B 308 (1988) 857-867, DOI: 10.1016/0550-3213(88)90131-9
- ↑ E.A. De Wolf, I.M. Dremin and W. Kittel, Physics Review 270 (1996) 1.
- ↑ A.H. Mueller, Physics Review D 4 (1971) 150.
- ↑ S.V.~Chekanov and V.I.~Kuvshinov, Acta Phys. Pol. B 25 (1994) 1189
- ↑ S.V. Chekanov, W. Kittel and V.I. Kuvshinov, Z.f.Physics C 73 (1997) 517.