Second-Order Strongly Intensive Quantity

• Second-order strongly intensive quantity is defined as a normalized measure combining moments and covariances of two extensive observables to cancel trivial volume fluctuations.
• Its primary representative, Σ, equals unity in independent particle models and deviates under genuine dynamical effects and quantum-statistical influences.
• Applications in collision studies enable robust fluctuation analysis by providing a model-independent diagnostic of intrinsic multi-particle correlations.

A second-order strongly intensive quantity is a combinatoric construct from the moments and covariances of two extensive stochastic observables—such as particle multiplicities or summed particle momenta—that is explicitly independent of both the mean value and fluctuations of the system “volume” (i.e., the unknown number of particle-producing sources). The primary member of this family is the symmetric Σ observable, which systematically cancels all contributions from trivial volume fluctuations, leaving sensitivity exclusively to intrinsic, dynamical correlations of the measured quantities. This property renders Σ—and the related antisymmetric Δ—crucial for robust fluctuation and correlation analyses in high-energy nuclear and hadronic collision studies, where event-by-event system size variability is inevitable and typically unmeasurable.

1. Definition and Mathematical Formalism

Let $A$ and $B$ be two extensive event-level observables (e.g., charged particle multiplicities in two non-overlapping rapidity intervals or total energy and multiplicity). The second-order strongly intensive quantity %%%%2%%%% is defined as

$$\Sigma[A,B] = \frac{\langle B\rangle \omega_A + \langle A\rangle \omega_B - 2\, \mathrm{Cov}(A,B)}{\langle A\rangle + \langle B\rangle}$$

where:

• $\langle X \rangle$ denotes the event ensemble average of $X$,
• $$\omega_X = (\langle X^2 \rangle - \langle X \rangle^2)/\langle X \rangle$$ is the scaled variance of $X$,
• $$\mathrm{Cov}(A,B) = \langle AB \rangle - \langle A \rangle \langle B \rangle$$ is the covariance.

This construction ensures that $\Sigma[A,B]$ depends only on second-order moments and is symmetric under $A \leftrightarrow B$ (Gazdzicki et al., 2013, Kovalenko, 2018, Broniowski et al., 2017).

2. Motivation and Strong Intensivity Criterion

The necessity for strongly intensive quantities arises in event-by-event fluctuation studies, particularly in heavy-ion collisions, where both mean system volume and its event-wise fluctuation are uncontrolled and highly centrality dependent. Traditional fluctuation measures, such as scaled variance alone, inevitably mix physical signal with geometric fluctuations, confounding extraction of dynamical information.

A quantity $I(A,B)$ is strongly intensive if, in any superposition (compound) model—where $A$ and $B$ are sums over a fluctuating number $N$ of independent sources, each contributing random $a_k$ and $b_k$—$I(A,B)$ is independent of both $\langle N \rangle$ and $\mathrm{Var}(N)$, depending only on the single-source (or local) statistics (Gorenstein et al., 2011, Broniowski et al., 2017). For the Σ observable, this criterion is rigorously satisfied in independent-source models, in both statistical and string-theoretic production frameworks.

3. Normalization and Model Baseline

By appropriate normalization, Σ is rendered dimensionless and assumes the value unity in the independent particle model (IPM):

$\Sigma[A,B]_{\mathrm{IPM}} = 1$

This normalization is critical: it sets a transparent, universal reference. Zero event-by-event fluctuations (all events identical) yield Σ = 0. Any deviation from unity directly quantifies genuine dynamical correlations or nontrivial collective phenomena beyond independent emission (Gazdzicki et al., 2013, Gorenstein et al., 2013).

4. Derivation in Superposition and String Models

Consider two measured multiplicities, $N_F$ and $N_B$, e.g. in forward and backward acceptance windows. For superposed independent sources (quark-gluon strings, wounded nucleons, fireballs), one has:

$$N_F = \sum_{i=1}^N \mu_F^{(i)}, \quad N_B = \sum_{i=1}^N \mu_B^{(i)}$$

where $N$ fluctuates from event to event, and $\mu_F$, $\mu_B$ are string-level (or source-level) random variables.

Strongly intensive quantities are constructed so the dependence on $N$ and its variance cancels. Explicitly, in the symmetric case, the observable reduces to

$$\Sigma[N_F, N_B] = [ \langle N_F \rangle \omega_{N_B} + \langle N_B \rangle \omega_{N_F} - 2 \mathrm{Cov}(N_F, N_B) ]/( \langle N_F \rangle + \langle N_B \rangle )$$

which depends only on the cumulative two-particle correlation function of a single source (Andronov et al., 2018, Andronov, 12 Nov 2025). Introducing string fusion or other nontrivial event structure violates the assumptions of independence and can degrade strong intensivity, making Σ weakly dependent on the event class through weighted averages over string types.

5. Physical Interpretation and Applications

Σ quantifies the genuine magnitude of event-by-event correlations, stripped of all contribution from system-size (“volume”) variations. In the context of rapidity-separated multiplicity bins, Σ isolates the true two-particle (long-range) correlation function, and its centrality-independence in models underlines its utility as a probe for dynamical correlation strength, such as arising from string fusion, critical fluctuations, or collectivity (Kovalenko, 2018, Andronov, 12 Nov 2025). In equilibrium quantum systems, statistical effects (Bose/Fermi) manifest as characteristic deviations of Σ from unity, providing means to constrain thermal modeling and distinguish quantum-statistical from dynamical fluctuations (Gorenstein et al., 2013).

A summary of key baselines:

Scenario	Value of Σ	Reference
Independent particle model (IPM)	1	(Gazdzicki et al., 2013, Gorenstein et al., 2013)
No event-by-event fluctuations	0	(Gazdzicki et al., 2013)
Bose/Fermi statistics (massless, $\mu=0$)	see analytic values below	(Gorenstein et al., 2013)

For massless particles at $\mu=0$:

$$\begin{align*} \Sigma[E,N]_{\mathrm{Fermi}} &\approx 0.917, \quad \Sigma[E,N]_{\mathrm{Bose}} \approx 1.499 \ \Sigma[P_T,N]_{\mathrm{Fermi}} &\approx 0.931, \quad \Sigma[P_T,N]_{\mathrm{Bose}} \approx 1.398 \end{align*}$$

6. Behavior Under Model Variations, Extensions, and Limitations

For distinct scenarios (superposed independent sources, fluctuating temperature, mixed emitter types), Σ exhibits distinctive behaviors:

• Independent, identical sources (strings, fireballs): Strictly strongly intensive, depends only on single-source cumulants and correlation structure (Broniowski et al., 2017, Andronov et al., 2018).
• Source-by-source temperature fluctuations: Σ increases relative to unity, quantifying added dynamical fluctuation.
• String fusion/cluster formation: Σ becomes a weighted average over source types, and strict volume independence is lost; Σ then acquires weak dependence on collision geometry or centrality, corresponding to shifts in composition of string types (Andronov et al., 2018).
• Nontrivial rapidity/azimuth separation: Σ varies with acceptance window separation, providing an experimental handle on the range of intrinsic correlations in the underlying particle production process (Andronov, 12 Nov 2025).

7. Comparison With Other Strongly Intensive Families

The second family of strongly intensive measures, denoted $\mathcal{A}_{AB}$, involves only marginal variances (not covariances) and is antisymmetric under $A \leftrightarrow B$. In contrast, Σ is symmetric and probes genuine two-observable correlations (Gorenstein et al., 2011). For applications where correlation quantification is primary (e.g., forward-backward studies, motional variable vs multiplicity), Σ is generally the preferred tool.

• (Gorenstein et al., 2013) "Strongly Intensive Measures for Transverse Momentum and Particle Number Fluctuations"
• (Gazdzicki et al., 2013) "On Normalization of Strongly Intensive Quantities"
• (Kovalenko, 2018) "Strongly intensive fluctuations and correlations in ultrarelativistic nuclear collisions in the model with string fusion"
• (Gorenstein et al., 2013) "Energy and Transverse Momentum Fluctuations in the Equilibrium Quantum Systems"
• (Andronov, 12 Nov 2025) "Strongly intensive quantities for rapidity correlations of multiplicities"
• (Andronov et al., 2018) "Strongly intensive observable between multiplicities in two acceptance windows in a string model"
• (Gorenstein et al., 2011) "Strongly Intensive Quantities"
• (Broniowski et al., 2017) "Statistical moments in superposition models and strongly intensive measures"

This formalism equips experimental and theoretical analyses with model-independent diagnostics for discovering genuinely new dynamical mechanisms in particle production, unaffected by the intractable complications of system size variability.