▁of ▁the ▁process . ▁Hence ▁the ▁next ▁time ▁there ▁is ▁a ▁proposal ▁to ▁a ▁state ▁with ▁that ▁particular ▁energy ▁, ▁that ▁proposal ▁is ▁now ▁more ▁likely ▁refused ; ▁in ▁this ▁sense , ▁the ▁algorithm ▁forces ▁the ▁system ▁to ▁visit ▁all ▁of ▁the ▁spectrum ▁equally . ▁The ▁consequence ▁is ▁that ▁the ▁hist ogram ▁ ▁is ▁more ▁and ▁more ▁flat . ▁However , ▁this ▁flat ness ▁depends ▁on ▁how ▁well - appro xim ated ▁the ▁calculated ▁entropy ▁is ▁to ▁the ▁exact ▁entropy , ▁which ▁naturally ▁depends ▁on ▁the ▁value ▁of ▁f . ▁To ▁better ▁and ▁better ▁approximate ▁the ▁exact ▁entropy ▁( and ▁thus ▁hist ogram ' s ▁flat ness ), ▁f ▁is ▁decre ased ▁after ▁M ▁proposal - accept ance ▁steps : ▁ ▁. ▁ ▁It ▁was ▁later ▁shown ▁that ▁updating ▁the ▁f ▁by ▁constantly ▁div iding ▁by ▁two ▁can ▁lead ▁to ▁s atur ation ▁errors . ▁A ▁small ▁modification ▁to ▁the ▁Wang ▁and ▁Land au ▁method ▁to ▁avoid ▁this ▁problem ▁is ▁to ▁use ▁the ▁f ▁factor ▁proportional ▁to ▁, ▁where ▁ ▁is ▁proportional ▁to ▁the ▁number ▁of ▁steps ▁of ▁the ▁simulation . ▁ ▁Test ▁system ▁ ▁We ▁want ▁to ▁obtain ▁the ▁D OS ▁for ▁the ▁harm onic ▁oscill ator ▁potential . ▁ ▁The ▁analyt ical ▁D OS ▁is ▁given ▁by , ▁ ▁by ▁performing ▁the ▁last ▁integral ▁we ▁obtain ▁ ▁In ▁general , ▁the ▁D OS ▁for ▁a ▁mult id im ensional ▁harm onic ▁oscill ator ▁will ▁be ▁given ▁by ▁some ▁power ▁of ▁E , ▁the ▁exponent ▁will ▁be ▁a ▁function ▁of ▁the ▁dimension