How to Measure Specific Heat Using Event-by-Event Average $p_T$ Fluctuations.

A simple way to visualize event-by-event average $p_T$ fluctuations is by assuming that each collision has a different temperature parameter (inverse $p_T$ slope) and that the ensemble of events has a temperature distribution about the mean, $\langle T\rangle$, with standard deviation $\sigma_T$ [R.~Korus, {\it et al.} PRC {\bf 64}, 054908\ (2001)]. PHENIX characterizes the non-random fluctuation of $M_{p_T}$, the event-by-event average $p_T$, by $F_{p_T}$, the fractional difference of the standard deviation of the data from that of a random sample obtained with mixed events. This can be related to the temperature fluctuation: \vspace*{-0.13in} \[ F_{p_T}=\sigma^{\rm data}_{M_{p_T}}/\sigma^{\rm random}_{M_{p_T}}-1\simeq(\langle n \rangle -1) \sigma^2_{T}/\langle T\rangle^2 \qquad .\] Combining this with the Gavai, {\it et al.}, [hep-lat/0412036] definition of the specific heat per particle, a simple relationship is obtained: \vspace*{-0.06in} \[ c_v/T^3={{\langle n\rangle}\over {\langle N_{tot}\rangle}} {1\over F_{p_T}} \qquad . \] $F_{p_T}$ is measured with a fraction $\langle n\rangle /\langle N_{tot}\rangle$ of the total particles produced, a purely geometrical factor representing the fractional acceptance, $\sim 1/20$ in PHENIX. The Gavai, {\it et al.} prediction that $c_v/T^3=15$ corresponds to $F_{p_T}\sim 0.33$\%, which may be accessible in PHENIX by measurements of $M_{p_T}$ in the range $0.2\leq p_T\leq 0.6$ GeV/c.