▁breaking ▁and ▁phase ▁co ex istence . ▁ ▁Statist ical ▁physics ▁The ▁set ▁of ▁Gib bs ▁measures ▁on ▁a ▁system ▁is ▁always ▁convex , ▁so ▁there ▁is ▁either ▁a ▁unique ▁Gib bs ▁measure ▁( in ▁which ▁case ▁the ▁system ▁is ▁said ▁to ▁be ▁" erg od ic "), ▁or ▁there ▁are ▁infinitely ▁many ▁( and ▁the ▁system ▁is ▁called ▁" non erg od ic "). ▁In ▁the ▁non erg od ic ▁case , ▁the ▁Gib bs ▁measures ▁can ▁be ▁expressed ▁as ▁the ▁set ▁of ▁convex ▁combinations ▁of ▁a ▁much ▁smaller ▁number ▁of ▁special ▁Gib bs ▁measures ▁known ▁as ▁" p ure ▁states " ▁( not ▁to ▁be ▁confused ▁with ▁the ▁related ▁but ▁distinct ▁notion ▁of ▁pure ▁states ▁in ▁quantum ▁mechan ics ). ▁In ▁physical ▁applications , ▁the ▁Hamiltonian ▁( the ▁energy ▁function ) ▁usually ▁has ▁some ▁sense ▁of ▁local ity , ▁and ▁the ▁pure ▁states ▁have ▁the ▁cluster ▁decomposition ▁property ▁that ▁" far - separ ated ▁sub system s " ▁are ▁independent . ▁In ▁practice , ▁phys ically ▁real istic ▁systems ▁are ▁found ▁in ▁one ▁of ▁these ▁pure ▁states . ▁ ▁If ▁the ▁Hamiltonian ▁poss esses ▁a ▁symmetry , ▁then ▁a ▁unique ▁( i . e . ▁er god ic ) ▁Gib bs ▁measure ▁will ▁necessarily ▁be ▁invariant ▁under ▁the ▁symmetry . ▁But ▁in ▁the ▁case ▁of ▁multiple ▁( i . e . ▁non erg od ic ) ▁Gib bs ▁measures , ▁the ▁pure ▁states ▁are ▁typically ▁not ▁invariant ▁under ▁the ▁Hamiltonian ' s ▁symmetry . ▁For ▁example , ▁in ▁the ▁infinite ▁fer