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.