▁Jos iah ▁Will ard ▁Gib bs , ▁is ▁a ▁probability ▁measure ▁frequently ▁seen ▁in ▁many ▁problems ▁of ▁probability ▁theory ▁and ▁statistical ▁mechan ics . ▁It ▁is ▁a ▁general ization ▁of ▁the ▁canonical ▁ensemble ▁to ▁infinite ▁systems . ▁ ▁The ▁canonical ▁ensemble ▁gives ▁the ▁probability ▁of ▁the ▁system ▁X ▁being ▁in ▁state ▁x ▁( equ ival ently , ▁of ▁the ▁random ▁variable ▁X ▁having ▁value ▁x ) ▁as ▁ ▁Here , ▁ ▁is ▁a ▁function ▁from ▁the ▁space ▁of ▁states ▁to ▁the ▁real ▁numbers ; ▁in ▁physics ▁applications , ▁ ▁is ▁interpreted ▁as ▁the ▁energy ▁of ▁the ▁configuration ▁x . ▁The ▁parameter ▁ ▁is ▁a ▁free ▁parameter ; ▁in ▁physics , ▁it ▁is ▁the ▁inverse ▁temperature . ▁The ▁normal izing ▁constant ▁ ▁is ▁the ▁partition ▁function . ▁However , ▁in ▁infinite ▁systems , ▁the ▁total ▁energy ▁is ▁no ▁longer ▁a ▁finite ▁number ▁and ▁cannot ▁be ▁used ▁in ▁the ▁traditional ▁construction ▁of ▁the ▁probability ▁distribution ▁of ▁a ▁canonical ▁ensemble . ▁Trad itional ▁approaches ▁in ▁statistical ▁physics ▁studied ▁the ▁limit ▁of ▁int ensive ▁properties ▁as ▁the ▁size ▁of ▁a ▁finite ▁system ▁approaches ▁infinity ▁( the ▁therm od ynamic ▁limit ). ▁When ▁the ▁energy ▁function ▁can ▁be ▁written ▁as ▁a ▁sum ▁of ▁terms ▁that ▁each ▁involve ▁only ▁variables ▁from ▁a ▁finite ▁sub system , ▁the ▁notion ▁of ▁a ▁Gib bs ▁measure ▁provides ▁an ▁alternative ▁approach . ▁Gib bs ▁measures ▁were ▁proposed ▁by ▁probability ▁the or ists ▁such ▁as ▁Dob r ush in , ▁Lan ford , ▁and ▁Ru elle ▁and ▁provided ▁a ▁framework ▁to ▁directly ▁study ▁infinite