Energy and Transverse Momentum Fluctuations in the Equilibrium Quantum Systems

The fluctuations in the ideal quantum gases are studied using the strongly intensive measures $\Delta[A,B]$ and $\Sigma[A,B]$ defined in terms of two extensive quantities $A$ and $B$. In the present paper, these extensive quantities are taken as the motional variable, $A=X$, the system energy $E$ or transverse momentum $P_T$, and number of particles, $B=N$. This choice is most often considered in studying the event-by-event fluctuations and correlations in high energy nucleus-nucleus collisions. The recently proposed special normalization ensures that $\Delta$ and $\Sigma$ are dimensionless and equal to unity for fluctuations given by the independent particle model. In statistical mechanics, the grand canonical ensemble formulation within the Boltzmann approximation gives an example of independent particle model. Our results demonstrate the effects due to the Bose and Fermi statistics. Estimates of the effects of quantum statistics in the hadron gas at temperatures and chemical potentials typical for thermal models of hadron production in high energy collisions are presented. In the case of massless particles and zero chemical potential the $\Delta$ and $\Sigma$ measures are calculated analytically.


I. INTRODUCTION
Experimental and theoretical studies of the event-by-event (e-by-e) fluctuations in nucleusnucleus (A+A) collisions give new information about properties of the strongly interacting matter and its phases. A possibility to observe signatures of the QCD matter critical point inspired the energy and system size scan program of the NA61/SHINE collaboration at the SPS CERN [1] and the low energy scan program of the STAR and PHENIX collaborations at the RHIC BNL [2]. In these studies one measures and then compares the e-by-e fluctuations in collisions of different nuclei at different collision energies. The average sizes of the created physical systems and their e-by-e fluctuations are expected to be rather different [3]. This strongly affect the observed hadron fluctuations, i.e. the measured quantities would not describe the local physical properties of the system but rather reflect the system size fluctuations. For instance, A+A collisions with different centralities may produce a system with approximately the same local properties (e.g., the same temperature and baryonic chemical potential) but with the volume changing significantly from interaction to interaction. Note that in high energy collisions the average volume of created matter and its variations from collision to collision are usually out of experimental control (i.e. these volume variations are difficult or even impossible to measure).
In the statistical mechanics the extensive quantity A is proportional to the system volume V , whereas intensive quantity has a fixed finite value in the thermodynamical limit V → ∞. The intensive quantities are used to describe the local properties of a physical system. In particular, an equation of state of the matter is usually formulated in terms of the intensive physical quantities, e.g., the pressure is considered as a function of temperature and chemical potentials.
In the statistical systems outside of the phase transition regions, a mean value of fluctuating extensive quantity, A , and its variance, Var(A) = A 2 − A 2 , are both proportional to the volume V in the limit of large volumes. The scaled variance, is therefore an intensive quantity. However, the scaled variance being an intensive quantity depends on the system size fluctuations.
Strongly intensive quantities introduced in Ref. [4] are independent of the average volume and of volume fluctuations. These quantities were suggested for and are used in studies of e-by-e fluctuations of hadron production in A+A collisions. Strongly intensive measures of fluctuations are defined in terms of two arbitrary extensive quantities A and B. In the present study we consider a pair of extensive variables -the motional extensive variable X = x 1 +. . . x N as a sum of single particle variables x j , with j = 1, . . . , N, and number of particles N. These measures were recently studied within the UrQMD simulations in Ref. [5]. The case of two hadron multiplicities A and B in A+A collisions has been considered within the HSD transport model in Ref. [6]. At the beginning we identify a single particle variable x with the particle energy ǫ and then consider the particle transverse momentum p T .
The strongly intensive measure ∆[X, N] and Σ[X, N] are defined as [4]: where C ∆ and C Σ are the normalization factors, and the scaled variances ω[X] and ω[N] are given by Eq. (1).
In Ref. [7] a special normalization for the strongly intensive measures ∆ and Σ has been proposed. Namely, the properly normalized strongly intensive quantities assume the value one for fluctuations given by the independent particle model (IPM). For the X and N extensive quantities the proposed normalization reads [7]: Note that the overline denotes averaging over a single particle inclusive distribution, whereas . . . represents averaging over multiparticle states of the system.
The first strongly intensive measure for fluctuations, the so-called Φ measure, was introduced a long time ago in Ref. [8]. The Φ quantity for the ideal quantum gases was considered in Ref. [9].
There were numerous attempts to use the Φ measure describing fluctuations in experimental data [10] and models [11]. In general, however, Φ is a dimensional quantity and it does not have a characteristic scale for a quantitative analysis of e-by-e fluctuations for different observables.
Note that the latter properties were clearly disturbing. The Φ measure can be expressed in terms of Σ [4]. A presence of additional fluctuation measure ∆ and utilization of special normalization conditions for both ∆ and Σ give essential advantages in application to the data analysis in A+A collisions.
In the present paper we study the strongly intensive measures (2) and (3) with normalization factors (4) for the relativistic ideal quantum gases in the grand canonical ensemble. The paper is organized as follows. In Section II we calculate the ∆[X, N] and Σ[X, N] quantities for the ideal quantum gases in the grand canonical ensemble. Analytical and numerical results suitable for the hadron gas created in A+A collisions are presented in Section III. A summary in Section IV closes the article. The calculation details are given in the Appendix.

II. IDEAL QUANTUM GAS
The grand canonical ensemble (GCE) partition function reads: where V is the system volume, β ≡ T −1 is the inverse system temperature, λ ≡ exp(βµ) denotes the fugacity and µ the chemical potential. The index α numerates the system quantum states, and N is the number of particles. The ensemble average values of the k th moments (k = 1, 2, . . . ) of any state quantity A are calculated as: The GCE partition function (5) can be presented in the form where d is the number of particle internal degrees of freedom and ǫ ≡ m 2 + p 2 is the particle energy with m being the particle mass and p its momentum. The values η = −1 and η = 1 correspond to the Bose and Fermi statistics, respectively, whereas η = 0 to the Boltzmann approximation. Using the presentation (7) one can calculate the averages (6) for the 1 st and 2 nd moments of the energy E and number of particles N: where ρ ≡ N /V and ε ≡ E /V denote the particle number density and the energy density, respectively.
From Eqs. (8)(9)(10)(11)(12) one finds for the scaled variances: They describe the fluctuations of the number of particles and the system energy at fixed volume V . The scaled variances in Eq. (13) are intensive quantities, they depend only on T and µ.
The quantities (13) are independent of the particle degeneracy factor d. Note that there is a (positive) correlation between the energy E and particle number N: The moments of single particle energy ǫ (k = 1, 2) are and the scaled variance ω[ǫ] is Note that our choice of the normalization (4) makes ∆[E, N] and Σ[E, N] dimensionless. These quantities are also independent of the degeneracy factor d.
The GCE within Boltzmann approximation satisfies the assumptions of IPM. Thus, one expects for the Boltzmann gas This can be easily proven, as for the η = 0 one finds from Eq. (8)(9)(10)(11)(12): and Eqs. (17) and (18) i.e. Bose statistics makes ∆[E, N] to be smaller and Σ[E, N] larger than unity, whereas Fermi statistics works in exactly opposite way.
The strongly intensive measures ∆ and Σ are independent of the volume and of its fluctuations. This is valid within the GCE, when the temperature and chemical potentials are volume independent. In a presence of volume fluctuations, the second moments E 2 , N 2 , and EN will include the terms proportional to V 2 which describe the contributions of the volume fluctuations to the fluctuations of E and N. The full averaging will then include both the GCE averaging (8)(9)(10)(11)(12) at fixed volume V and an additional averaging over the volume fluctuations.

III. HADRON GAS
In this Section we consider the measures ∆ and Σ for the hadron gas with thermodynamical parameters typical for the thermal models of A+A collisions.

A. Massless Particles
We start from the system with m = µ = 0. In the case, the calculations of quantities entering Eqs. (8)(9)(10)(11)(12) can be performed analytically. Using Eqs. (A1,A2) one finds: where ζ(s) is the Riemann zeta function: ζ(2) = π 2 /6 ∼ = 1.645, ζ ( Finally, one obtains The strongly intensive measures ∆[E, N] and Σ[E, N] (30) possess the values which are independent of T . This is evident as the temperature is the only dimensional variable for the system with m = µ = 0, and ∆ and Σ measures are dimensionless quantities due to our normalization.

B. Pion Gas
The pion gas corresponds to the Bose statistics (η = −1) and m π ∼ = 140 MeV. We consider the chemical equilibrium pion gas (µ π = 0) and one example of chemical non-equilibrium (µ π = 100 MeV). Calculating the quantities ρ, ǫ, ǫ 2 , I E , I N , I EN according to Eqs. (A4-A9) and  Fig. 1. The two lines are presented: the solid line for µ π = 0 and the dotted line for µ π = 100 MeV. The horizontal line in Fig. 1 (a) and (b) corresponds to the Boltzmann approximation (19). This approximation appears to be always valid at T ≪ m π . In the ultra-relativistic limit T ≫ m π the results from Eq. from the IPM results (19) are quite significant, about 25% and 10%, respectively. These deviations are strongly enlarged for the chemical non-equilibrium pion gas with µ π > 0. The Bose statistics lead to the singular behavior of fluctuations at µ π → m π , which corresponds to the Bose-Einstein condensation of pions. We do not touch this problem in the present study. For the fluctuations of pion multiplicity this was considered in Ref. [12].
In applications to A+A collisions one should however take into account that a substantial fraction of the final state pions come from the resonance decays. These pions do not 'feel' [9] the Bose statistics, and thus the Bose statistics contribution to ∆ (c) we introduce the ratio where A Bose and A Boltz are calculated for Bose statistics and within the Boltzmann approximation, respectively. The Bose and Boltzmann A-values will be calculated at the same T and µ π values. In Fig. 2

C. Transverse Momentum Fluctuations
Using the above equations one can easily calculate the measures ∆[P T , N] and Σ[P T , N] for the transverse momentum fluctuations. Since p T = p · sin(θ) with p = |p| and θ being the angle between the 'beam' z-axis, one gets for an arbitrary f (p) function: First, one needs to calculate the integrals (A4-A9) for P and N quantities, i.e. with p and p 2 instead of p 2 + m 2 and p 2 + m 2 , respectively. The integrals (A4) for ρ and (A7) for I N remain unchanged. There are two new integrals: instead of ǫ and I EN , respectively. To calculate p 2 and I P the following relations can be used: Using Eq. (35) one then finds: The These values can be compared with the corresponding results for E and N, Eq. (30).

D. Connection to the Φ Measure
The well-known fluctuation measure Φ was introduced in Ref. [8]. In a general case, when x i represents any motional extensive quantity as a sum of single particle quantities, one gets [4]: where Σ[X, N] is given by Eq. (3) and C Σ by Eq. (4). Therefore, the Φ quantity can be expressed via measure Σ. At m = µ = 0 it then follows: These results are in agreement with those obtained in Ref. [9].

IV. SUMMARY
The strongly intensive fluctuation measures ∆ and Σ have been studied for the ideal Bose and Fermi gases within the grand canonical ensemble. In the present paper, the ∆ and Σ quantities are considered for two specific extensive quantities -motional variable X (either the system energy E or transverse momentum P T ) and number of particles N. We have used the normalization of the strongly intensive measures which makes them dimensionless and equal to unity for fluctuations given by the independent particle model. The grand canonical ensemble within the Boltzmann approximation satisfies the conditions of independent particle model.
Our results demonstrate deviations from the independent particle model due to the Bose and Fermi statistics. We present estimates of these quantum statistics effects for the hadron gas with thermodynamical parameters typical for the thermal models of A+A collisions. In the case of massless particles and zero chemical potential the ∆ and Σ measures are calculated analytically. Numerical estimates for the Bose effects in the pion gas at the temperatures from 0 to 200 MeV are presented. For the Fermi gas of protons the quantum effects appear to be quite small.
The measures ∆ and Σ are used to study the event-by-event fluctuations and correlations in high energy nucleus-nucleus and proton-proton collisions. From our results it follows that the Bose effects in the pion gas can be an important source of the transverse momentum fluctuations, especially in chemically non-equilibrium case with µ π > 0. However, other sources of dynamical fluctuations and correlations (e.g., exact conservation laws within micro-canonical ensemble, resonance decays, transverse collective flow, fluctuations of temperature, correlations between temperature and particle multiplicity, etc.) should be considered to make a realistic comparison of theoretical models with the data. (−η) n−1 n K 4 (n y) − K 0 (n y) exp n µ T , (−η) n−1 n K 5 (n y) + K 3 (n y) − 2K 1 (n y) exp n µ T , where K l (z) are the modified Bessel functions.