Grand Canonical Ensemble

Transcription

1 Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls M<N > and total nrgy M<E>. 3. Lt nr,s numbr of systms that hav, at any tm, th numbr Nr of partcls and th amount Es of nrgy (r, s = 0,,,...). 4. Th nsmbl has th followng constrans:. And Dfn W{nr,s} numbr of dffrnt ways that any st {nr,s} of th numbrs nr,s satsfy th abov condtons. Thn To calculat th most probabl mod of dstrbuton { th functon n rs} as th on that maxmz W{nr,s}, on can dfns *, Ths gvs Nr Es n c * NrEs Consquntly Pr, s, G NrEs. G Dfn th grand canoncal potntal V drvatvs of :,, log G, thn N, E, and th prssur P ar gvn by th

2 N, E, P V In th grand canoncal nsmbl, th ntropy s dfnd by S P log P r, s r, s whch gvs S k E k ln G k N as compard wth th thrmodynamcs: ds kd E kpdv kd N It s asy to prov that: PV kt log G kt whr n grand canoncal nsmbl PV kt log G. Exampl:- Drv th quatons of stat of a monatomc dal gas, usng classcal mchancs and th grand canoncal nsmbl. Soluton Th Hamltonan functon for an N-partcl monatomc gas s

3 . Dnsty Fluctuaton n : Dfn th fugacty (absolut actvty) of th systm as z thn In gnral, N T, whr T s th sothrmal comprssblty of th systm. Thus w s that partcl V VT, dnsty fluctuatons, whch spontanously happn bcaus of ntracton wth a hat-bath, ar ntmatly rlatd to a thrmodynamc proprty of th systm, namly th sothrmal comprssblty. Ths s nglgbl xcpt n a stuaton accompanyng phas transtons. 3

4 . Enrgy Fluctuaton: In th canoncal nsmbl, fluctuatons occur n nrgy bcaus th systm s n qulbrum wth th rsrvor, t has bn provn that log log T ( E) E E E kt C v T k T In grand canoncal nsmbl, wth th sam tchnqu, on can show that (usng th followng) That Wth Usng th xprssons: Thn Last quaton tlls us that th man squar fluctuaton n th nrgy of GCE s qual to th valu n th canoncal plus a contrbuton arsng from th fluctuaton n th numbr of partcls N. Ths also a nglgbl valu xcpt at th phas transton. 4

5 Rlaton btwn canoncal and grand canoncal nsmbls W look at th rlatv partcl numbr fluctuaton n th thrmodynamc lmt, namly whn V, N. N lm lm 0. V N V V What s ths mans? In th thrmodynamc lmt, th fluctuatons ar nglgbl, and th numbr of partcls rmans practcally constant. If th numbr of partcls s almost constant, on can also safly us canoncal nsmbl to dscrb th systm, whr partcl numbr s fxd. So w conclud that th n th thrmodynamc lmt V, canoncal and grand canoncal nsmbls should gv smlar rsults. Summary nsmbl mcrocanoncal canoncal grand canoncal charactrstc systm solatd closd opn Exampl charactrstc systm solatd closd opn ndpndnt varabls dstrbuton functon N, E, V N,, V,, V C W ( N, E, V partton functon ) basc thrmodynamc rlatons E E W S C E C N, N E NE N G k ln W F kt ln C kt ln G (4) Rlaton btwn thr nsmbls In th canoncal nsmbl th most probabl nrgy E* s dntcal to th man valu of all nrgs and corrsponds to th fxd gvn nrgy of th mcro canoncal nsmbl. Th dvatons (fluctuatons) of nrgy from th man valu n th canoncal nsmbl bcom smallr and smallr wth ncrasng partcl numbrs. It mans that at a gvn tmpratur, th systm can assum (up to vry small dvatons) only a crtan nrgy, whch concds wth th total nrgy of mcro canoncal nsmbl. N G E 5