▁set ▁A . ▁ ▁Then ▁ ▁as ▁the ▁contribution ▁to ▁the ▁total ▁energy ▁of ▁all ▁the ▁finite ▁sets ▁A ▁that ▁meet ▁. ▁Note ▁that ▁the ▁total ▁energy ▁is ▁typically ▁infinite , ▁but ▁when ▁we ▁" local ize " ▁to ▁each ▁ ▁it ▁may ▁be ▁finite , ▁we ▁hope . ▁▁ ▁The ▁Hamiltonian ▁in ▁ ▁with ▁boundary ▁conditions ▁, ▁for ▁the ▁potential ▁, ▁is ▁defined ▁by ▁ ▁where ▁. ▁▁ ▁The ▁partition ▁function ▁in ▁ ▁with ▁boundary ▁conditions ▁ ▁and ▁inverse ▁temperature ▁ ▁( for ▁the ▁potential ▁ ▁and ▁) ▁is ▁defined ▁by ▁ ▁where ▁▁▁ ▁is ▁the ▁product ▁measure ▁A ▁potential ▁ ▁is ▁- ad miss ible ▁if ▁ ▁is ▁finite ▁for ▁all ▁ ▁and ▁. ▁A ▁probability ▁measure ▁ ▁on ▁ ▁is ▁a ▁Gib bs ▁measure ▁for ▁a ▁- ad miss ible ▁potential ▁ ▁if ▁it ▁satisfies ▁the ▁Dob r ush in – L an ford – R uelle ▁( DL R ) ▁equation ▁ ▁for ▁all ▁ ▁and ▁. ▁ ▁An ▁example ▁To ▁help ▁understand ▁the ▁above ▁definitions , ▁here ▁are ▁the ▁corresponding ▁quantities ▁in ▁the ▁important ▁example ▁of ▁the ▁Is ing ▁model ▁with ▁nearest - ne igh bor ▁interactions ▁( cou pling ▁constant ▁) ▁and ▁a ▁magnetic ▁field ▁( ), ▁on ▁: ▁▁ ▁The ▁lattice ▁is ▁simply ▁. ▁ ▁The ▁single - sp in ▁space ▁is ▁▁ ▁The ▁potential ▁is ▁given ▁by ▁ ▁See ▁also ▁ ▁Bol tz mann ▁distribution ▁ ▁Ex pon ential ▁family ▁ ▁Gib bs ▁algorithm ▁ ▁Gib bs ▁sampling ▁ ▁Inter act ing ▁particle ▁system ▁