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From Harmonic Oscillators to Multiplicity Distributions in Particle Production Processes

This paper proposes an information-theoretic oscillator framework to derive negative binomial and generalized negative binomial distributions for modeling multi-particle production, demonstrating that the latter effectively describes high-multiplicity data and introducing a novel Lerch-type generalization of the Barnes zeta function for non-degenerate systems.

Original authors: Trambak Bhattacharyya, Maciej Rybczyński, Zbigniew Włodarczyk

Published 2026-10-21

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CC BY 4.0

Original authors: Trambak Bhattacharyya, Maciej Rybczyński, Zbigniew Włodarczyk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the chaotic heart of a particle accelerator, where beams of protons smash together at speeds nearing that of light, a fundamental question arises: how many new particles are born from the collision? When these high-energy events occur, they do not produce a fixed number of particles every time. Instead, the total count fluctuates wildly from one crash to the next. Physicists have long sought to describe this randomness with mathematical patterns. For decades, the standard tool for this job has been a specific statistical curve known as the negative binomial distribution. This curve works well for describing the average behavior of particle production, capturing the fact that some collisions yield a few particles while others yield many. However, as scientists have looked closer at the most extreme events—those that produce an unusually high number of particles—the standard curve begins to fail, drifting away from what the detectors actually see.

To solve this puzzle, a team of researchers at the Jan Kochanowski University in Poland has proposed a new way of thinking about these collisions. Instead of treating the particles as independent entities, they modeled the entire production process as a system of vibrating strings, or harmonic oscillators. Imagine a collection of tiny, invisible springs, each capable of holding a certain amount of energy. In this framework, every time a particle is created, it is equivalent to adding a single unit of vibration, or a "quantum," to one of these springs. The total number of particles in an event is simply the sum of all the vibrations across the entire system. By applying the rules of information theory to this vibrating system, the researchers could derive the probability of finding a specific number of particles.

When the researchers applied the standard rules of physics to this oscillator model, they successfully reproduced the familiar negative binomial distribution. This confirmed that their vibrating spring picture was a valid way to understand the basic mechanics of particle production. However, the true breakthrough came when they introduced a more complex set of rules to account for the fact that these oscillators might not be entirely independent. In the real world of high-energy collisions, particles often influence one another over long distances, and the system retains a kind of memory of past interactions. By using a generalized statistical approach that allows for these connections, the team derived a new, more flexible version of the distribution. They call this the generalized negative binomial distribution.

When they tested this new formula against real data from the Large Hadron Collider, the results were striking. While the standard model worked perfectly for events with a low or moderate number of particles, it began to lose accuracy as the particle count grew very high. The new generalized formula, however, continued to track the experimental data with remarkable precision, even in the most extreme, high-multiplicity collisions. This suggests that the standard assumption of independence between particles is insufficient for describing the most energetic events. The new model implies that the particles are more deeply interconnected than previously thought, behaving as a collective system rather than a crowd of strangers.

The researchers also discovered that their mathematical framework led to the appearance of a very rare and complex mathematical function, one that had not been previously studied in the context of particle physics. This function, a specialized type of zeta function, emerged naturally when they tried to calculate the probabilities for systems with many different types of oscillators. While the full mathematical structure of this function remains to be explored, its appearance suggests a deep and previously uncharted link between the statistical behavior of vibrating systems and the fundamental laws governing particle creation.

The study also revealed something unexpected about the nature of these collisions. The data fit the new model best when a specific parameter in the equation was less than one. In the language of statistical physics, this value usually indicates that the system is interacting with a finite "reservoir" of energy, rather than an infinite one. This could mean that the observed particles are only a small part of a larger, unobserved system, and the limitations of this finite environment are what cause the unusual patterns seen in the data. The researchers suggest that as the range of particles measured in an experiment gets wider, these finite effects become more prominent, explaining why the new model is so effective at describing the most complex collisions.

Ultimately, this work offers a fresh perspective on how matter is created in the universe's most violent events. By viewing particle production through the lens of vibrating oscillators and generalized statistics, the team has provided a tool that not only matches existing data better but also opens the door to understanding the hidden correlations between particles. The findings suggest that the chaotic spray of particles emerging from a collision is not merely random noise, but the result of a highly structured, interconnected process that can be described by a single, elegant mathematical framework. This approach may soon help physicists decode other mysterious quantities in particle physics, such as how energy is distributed or how particles correlate across vast distances, turning the chaotic aftermath of a collision into a clear story of physical law.

Technical Summary: From Harmonic Oscillators to Multiplicity Distributions in Particle Production Processes

Problem Statement
In high-energy collision events, the total number of produced particles ( $N$ ) fluctuates from event to event, forming a multiplicity distribution $P(N)$ . While the Poisson distribution serves as a baseline for independent sources, experimental data from $pp$, $pA$ , and $AA$ collisions consistently exhibit fluctuations larger than Poissonian, typically described by the Negative Binomial Distribution (NBD). However, existing phenomenological studies often resort to multi-component distributions to describe deviations observed in high-multiplicity regions. This paper addresses the need for a unified theoretical framework that can explain both the standard NBD behavior and the deviations at high multiplicities using information theory and statistical mechanics, without necessarily invoking complex multi-component models.

Methodology
The authors propose a framework based on harmonic oscillators, where the energy levels $\omega_i$ correspond to the central values of momentum bins, and the occupation numbers $n_i$ represent the number of particles in those bins. The total multiplicity is defined as $N = \sum n_i$ .

The study proceeds in two main statistical regimes:

Boltzmann-Gibbs (BG) Statistics: The authors maximize the Shannon entropy subject to probability normalization and fixed mean energy constraints. They analyze both single-mode and multi-mode oscillators.
Nonadditive (Tsallis) Statistics: To account for long-range correlations and memory effects prevalent in high-energy collisions, the framework is generalized using Tsallis entropy. This introduces a non-additive parameter $q$ and relaxes the assumption of independent micro-states.

The derivation involves calculating partition functions and probability distributions for systems of degenerate (identical frequency) and non-degenerate oscillators. The authors also investigate the generating functions $G(z)$ for these distributions, exploring their mathematical structures, including connections to special functions like the Barnes zeta function and the Lerch transcendent.

Key Contributions and Results

Derivation of the NBD from BG Statistics:
Starting from Shannon entropy, the authors demonstrate that a system of $k$ degenerate harmonic oscillators in BG statistics yields the Negative Binomial Distribution (Eq. 8).
$P_D(N) = \frac{(N+k-1)!}{N!(k-1)!} p^N (1-p)^k$
where $p = e^{-\beta\omega}$ . They further show that as the number of oscillators $k \to \infty$ with a fixed mean multiplicity, this distribution converges to the Poisson distribution.
Generalized NBD (gNBD) from Nonadditive Statistics:
By maximizing Tsallis entropy, the authors derive a generalized Negative Binomial Distribution (gNBD) for degenerate oscillators (Eq. 21):
$P_D(N) = \frac{1}{Z_q} \frac{(N+k-1)!}{N!(k-1)!} [1 + (q-1)\beta\omega N]^{-\frac{1}{q-1}}$
This distribution incorporates the entropic parameter $q$ . The authors observe that for specific datasets, the gNBD provides a superior description of particle multiplicity in the high-multiplicity region compared to the standard NBD, offering a potential alternative to the multi-component distributions currently used in phenomenology.
Mathematical Generalization (Lerch-type Barnes Zeta Function):
In the analysis of non-degenerate multi-mode oscillators under Tsallis statistics, the authors encounter a generating function involving a summation they identify as a "Lerch-type generalization" of the Barnes zeta function (Eq. 24).
$G(z) = \frac{\Phi_B(z, 1/(q-1), \lambda|\{a_i\})}{\zeta_B(1/(q-1), \lambda|\{a_i\})}$
The paper notes that while special cases of these functions exist in literature, this specific generalized form has not previously been studied in the context of nonadditive statistics or multiplicity generation.
Phenomenological Application:
The framework is tested against CMS experimental data for $pp$ collisions at $\sqrt{s} = 900$ GeV and $7$ TeV across different rapidity windows ( $|\eta| < 0.5$ and $|\eta| < 2$ ).
- Low Multiplicity: Both the standard NBD and gNBD describe the data satisfactorily.
- High Multiplicity: The standard NBD deviates from experimental observations at higher $N$ values. The gNBD, however, successfully describes these high-multiplicity data points.
- Parameter $q$ : The fits yield values of $q < 1$ . The authors note this is a rare instance in high-energy physics where the entropic parameter is sub-unity. They suggest this may be explained by a "finite reservoir" picture, where the observed system interacts with an unobserved part acting as a reservoir with a finite number of degrees of freedom.

Significance and Claims
The paper claims to provide a unified information-theoretic origin for multiplicity distributions, linking the phenomenological NBD directly to a system of degenerate harmonic oscillators. Its primary significance lies in demonstrating that the gNBD, derived from nonadditive statistics, can describe high-multiplicity data that typically requires multi-component fits, thereby simplifying the phenomenological description.

The authors modestly state that while the gNBD offers a promising alternative, "more comprehensive analysis is required to ascertain this conclusion." They also highlight the mathematical novelty of the derived Lerch-type generalization of the Barnes zeta function, suggesting it warrants further analytical study. Finally, they propose that the framework could be extended to other observables such as inelasticity and forward-backward correlations, though these are reserved for future work.

Technical Summary: From Harmonic Oscillators to Multiplicity Distributions in Particle Production Processes

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