▁energy ▁particle ▁production , ▁but ▁also ▁worked ▁out ▁a ▁formula ▁for ▁the ▁had ron ic ▁mass ▁spectrum ▁and ▁predicted ▁the ▁limit ing ▁temperature ▁for ▁hot ▁had ron ic ▁systems . ▁ ▁After ▁some ▁time ▁this ▁limit ing ▁temperature ▁was ▁shown ▁by ▁N . ▁Cab ib bo ▁and ▁G . ▁Paris i ▁to ▁be ▁related ▁to ▁a ▁phase ▁transition , ▁which ▁character izes ▁by ▁the ▁de con fin ement ▁of ▁qu arks ▁at ▁high ▁energ ies . ▁The ▁mass ▁spectrum ▁was ▁further ▁analyz ed ▁by ▁Steven ▁Fra utsch i . ▁ ▁Q - ex pon ential ▁function ▁The ▁H aged orn ▁theory ▁was ▁able ▁to ▁describe ▁correctly ▁the ▁experimental ▁data ▁from ▁collision ▁with ▁center - of - mass ▁energ ies ▁up ▁to ▁approximately ▁ 1 0 ▁Ge V , ▁but ▁above ▁this ▁region ▁it ▁failed . ▁In ▁ 2 0 0 0 ▁I . ▁Bed i aga , ▁E . ▁M . ▁F . ▁Cur ado ▁and ▁J . ▁M . ▁de ▁Mir anda ▁proposed ▁a ▁phenomen ological ▁general ization ▁of ▁H aged orn ' s ▁theory ▁by ▁replacing ▁the ▁exponential ▁function ▁that ▁appears ▁in ▁the ▁partition ▁function ▁by ▁the ▁q - ex pon ential ▁function ▁from ▁the ▁Ts all is ▁non - ext ensive ▁statistics . ▁With ▁this ▁modification ▁the ▁generalized ▁theory ▁was ▁able ▁again ▁to ▁describe ▁the ▁extended ▁experimental ▁data . ▁ ▁In ▁ 2 0 1 2 ▁A . ▁De pp man ▁proposed ▁a ▁non - ext ensive ▁self - cons istent ▁therm od ynam ical ▁theory ▁that ▁includes ▁the ▁self