▁of ▁the ▁gra ins ▁size ▁distribution . ▁In sp ired ▁by ▁the ▁work ▁of ▁L if sh itz ▁and ▁S ly oz ov ▁on ▁Ost wald ▁rip ening , ▁Hill ert ▁has ▁suggested ▁that ▁in ▁a ▁normal ▁gra in ▁growth ▁process ▁the ▁size ▁distribution ▁function ▁must ▁conver ge ▁to ▁a ▁self - similar ▁solution , ▁i . e . ▁it ▁becomes ▁invariant ▁when ▁the ▁gra in ▁size ▁is ▁scale d ▁with ▁a ▁characteristic ▁length ▁of ▁the ▁system ▁that ▁is ▁proportional ▁to ▁the ▁average ▁gra in ▁size ▁. ▁ ▁Several ▁simulation ▁studies , ▁however , ▁have ▁shown ▁that ▁the ▁size ▁distribution ▁devi ates ▁from ▁the ▁Hill ert ' s ▁self - similar ▁solution . ▁Hence ▁a ▁search ▁for ▁a ▁new ▁possible ▁self - similar ▁solution ▁was ▁initi ated ▁that ▁indeed ▁led ▁to ▁a ▁new ▁class ▁of ▁self - similar ▁distribution ▁functions . ▁Lar ge - scale ▁phase ▁field ▁simulations ▁have ▁shown ▁that ▁there ▁is ▁indeed ▁a ▁self - similar ▁behavior ▁possible ▁within ▁the ▁new ▁distribution ▁functions . ▁It ▁was ▁shown ▁that ▁the ▁origin ▁of ▁the ▁deviation ▁from ▁Hill ert ' s ▁distribution ▁is ▁indeed ▁the ▁geometry ▁of ▁gra ins ▁spe cially ▁when ▁they ▁are ▁shr inking . ▁ ▁Normal ▁vs ▁ab normal ▁▁ ▁In ▁common ▁with ▁recovery ▁and ▁rec ry st all isation , ▁growth ▁phen omena ▁can ▁be ▁separated ▁into ▁continuous ▁and ▁dis continu ous ▁mechan isms . ▁In ▁the ▁former ▁the ▁micro structure ▁evol ves ▁from ▁state ▁A ▁to ▁B ▁( in ▁this ▁case ▁the ▁gra ins ▁get ▁larger ) ▁in ▁a