Energy, entropy and work function in a molecule with degeneracy

Transcription

1 Avill oli t WS 97 (08) EISS 39-9 SHOR COMMICAIO Ergy, tropy d work futio i molul with dgry Mul Mlvr Dprtmt of si Sis, Mritim ivrsity of th Cri, Cti l Mr, ul E-mil ddrss: mmf.um@gmil.om ASRAC I this ppr, w hv ddud som thrmodymi proprtis for molulr systm with dgry sd i th postults of th sttistil mhis. W oti lytil xprssios for itrl rgy, ht pity C v, tropy S d work futio o Hlmholt fr rgy A. It is prditd i this rsrh th thrmodymi hvior of, C v, S d A i th limit of high d low tmprtur. ywords: Sttistil mhis, dgry, itrl rgy, ht pity, tropy, work futio, limit of high d low tmprtur. IRODCIO I th sttistil d mirosopi dsriptio of y systm, th prtitio futio plys dtrmit rol d is dfid s th totl sum of stts of th systm []: ( Rivd 9 Mrh 08; Aptd April 08; Dt of Pulitio 9 April 08 )

2 World Sitifi ws 97 (08) g( E ) E () whr: lls th totl rgy E, is th oltm s ostt d g(e ) it is th umr of dgrt stts with th sm rgy E. h us of th qutio () [,] llows to oti sttistil d mirosopi dsriptio of th rgy d th thrml pity. h opts of th sttistil mhis must osidrd if w wt to do mirosopi dsriptio of physil systm. Rtly Mäklä [3,4] ostrutd mirosopi modl of Strthd Horio of Shwrshild d Rissr-ordström lk hols d otid lytil xprssio for th prtitio futio from th poit of viw of osrvr o its strthd horio. Mlvr [5-7] studid th hvior of th thrml pity C for Shwrshild d Rissr-ordström lk hols wh >> C d << C whr C is th hrtristi tmprtur d foud tht th vlu for C if >> C is th sm tht would otid i idl ditomi gs if r osidrd th rottiol d trsltiol dgrs of frdom, rsptivly. iggiu [8] prst sttistil lysis i grvitos d drivd qutios for th prtitio futio d th m rgy. Mlvr [9] otid lytil xprssio for th thrml pity for grvitos d w studid th, hvior of C i th limit of high d low tmprtur (Artg, [0], Arms []). h im of this ppr is studis th thrmodymi hvior of molulr systm with dgry usd th mthods of th sttistil mhis. his ppr is outlid i th followig mr: th stio II w prst th qutios for sttistil mhis i th limit of high tmprturs. I stio III w prst lysis of th flututios of th rgy with xtrmly high tmprturs. I stio I it is show th thrmodymi hvior of, C, S d A for low tmprturs. Filly i stio, w olud.. SAISICAL MECHAICS I HE LIMI OF HIGH EMPERARE Aordig Rif [] d sh [], th oil prtitio futio for idtil prtils with itrl dgrs of frdom writt s () whr: is th prtitio futio for y o of ssmly of idtil uit. W hv dfid s g ) ( (3) h trms rprst th rgis ssoitd with vry possil qutum stt of th sigl uit d th dgrt rgy stts. I this ppr, w hv tk th form of th rgy trm for molul s, whr d α r positiv rl ostts d positiv itgr. Evry rgti lvl hv dgrt rgy stts whr, β d γ r positiv ostts. h prtitio futio (3) is giv y for -5-

3 World Sitifi ws 97 (08) (4) For moluls with xtrmly high tmprturs th summtio i (4) rpld y th orrspodig itgrl d 0 0 (5) Sustitutig th hg of vril y / d y i th itgrl (5) d rplig / w hv dy y y 0 (6) With th dfiit itgrl (6) th followig rsult is otid (7) whr: is th gmm futio ssoitd with th prtitio futio. h oil prtitio futio writt s (8) Dfiig th sttistil itrl rgy with th followig xprssio []

4 World Sitifi ws 97 (08) (9) h thrml rgy of gs tht otis of ths moluls is (0) For th ht pity to ostt volum C () I sttistil mhis, th work futio o Hlmholt fr rgy A is dfid s A () Sustitutig q. (8) i () w hv for th work futio A (3) h rltio tw A d th thrmodymi tropy S is S A (4) With (3) d th q. (0) i q. (4) w oti S (5)

5 World Sitifi ws 97 (08) I this ppr, w hv lultd th vlus of, C, A d S with α = d th prtiulr ss β =, β =, γ = 0. For th s α=, β=, γ=0, usig (6), (7) d q. (8) w oti (6) Of th q. (0) w hv for th itrl rgy (7) With (), (3) d (5) th lytil xprssios for C, A d S, writt s A C (8) (9) S (0) For th s α =, β =, γ = 0, w hv for th prtitio futio 3 4 () d for, C, A d S w foud 3 () 3 C (3) 3 A (4) -54-

6 World Sitifi ws 97 (08) S (5) 3. FLCAIOS OF EERGY WHE I grmt with (), th disprsio of th rgy writt i th form [] : E E (6) wht is quivt to d th rltiv rgy flututio is giv y E C (7) C (8) sustitutig (0) d () i q. (6) w oti th flututio of rgy writt s (9) 0 wh (30) h vlu of th flututio dpds o th prmtrs α, β, γ ssoitd with th futio gmm. 4. EHAIOR OF, C, S AD A I HE LIMI OF LOW EMPERARE For low tmprturs th prtitio futio summtio (4) writt s -55-

7 World Sitifi ws 97 (08) (3) whr:, With th q. (9) w oti for th itrl rgy (3) From (), (4), () d (), th xprssios for C, A d S r giv y for C (33) A (34) S (35)

8 World Sitifi ws 97 (08) COCLSIOS I this ppr, w hv ddud xprssios for,, C, S d A i molulr systm with dgry i trms of th gmm futio wh d hv studid th thrmodymi hvior for 0. For xtrmly high tmprturs th vlus of th gmm futio d th flututios of rgy will dpd o th prmtrs α, β, γ ssoitd with th rgy trm d with th dgry ftor. h sttistil mhis rih th ourss of thrmodymis, whih otriuts to ttr omprssio of th thrml phom. h thrmodymi qutios ddud from th postults of th sttistil mhis r trtl mthmtilly d offr wid xptio of my physil systms of itrst. Rfrs [] Rif. F. (975). Sttistil Physis, rkly physis ours-volum 5, MGrw-Hill ook Compy; IS [] sh, L (970). Elmts of Clssil d Sttistil hrmodymis, Addiso Wsly Pulishig Compy, Rdyg, Msshustts. [3] Mäklä, J. (0). Prtitio Futio of th Shwrshild lk Hol. Etropy, 3, , DOI: / [4] Mäklä, J. (04). Prtitio Futio of th Rissr-ordström lk Hol. It. J. Mod. Phys. D, 3, 45000, DOI: 0.4/S [5] Mlvr, M. (0). w of Eisti s thrml pity for Shwrshild s lk hol? IJRRAS,, [6] Mlvr, M.(03) Comprtiv Alysis of hrml Cpity i Shwrshild d Rissr- ordström lk hols, World Applid Progrmmig, 3, [7] Mlvr, M. (03). lk Hols, Wormhols d Drk Ergy Strs i Grl Rltivity. Lmrt Admi Pulishig, rli; IS [8] iggiu, S. (07). Sttistil mhis of grvitos i ox d th lk hol tropy, Physi A, 473, 4-4. [9] Mlvr, M. (07). A lysis of th thrml pity i Grvitos, World Sitifi ws 7, [0] Dil Artg, Rud Prti, d Eri rdgur. Propgtio i thrml grvito kgroud. Phys. Rv. D 70, 04409, 004. [] Jy Arms, rols Hrmrk, ils A. Ors, Mrt Orslli, Adrs igd Pdrs. hrml Git Grvitos. Jourl of High Ergy Physis ovmr 0, 0: 3-57-