Nonextensibility of energy in Tsallis statistics and the zeroth law of

Transcription

1 Nonextensbty of energy n Tsas statstcs and the zeroth a of thermodynamcs onge Ou and Jncan hen* T Word Laboratory, P. O. 870, eng 00080, Peoe s Reubc of hna and Deartment of Physcs, Xamen nversty, Xamen 6005, Peoe s Reubc of hna To mortant robems exstng n Tsas statstcs are nvestgated, here one s hether energy s extensve or not, and the other s hether t s necessary to ntroduce the so-caed generazed zeroth a of thermodynamcs or not. The resuts obtaned sho ceary that ke entroy, energy s aso nonextensve n Tsas statstcs, and that the zeroth a of thermodynamcs has been mcty used n Tsas statstcs snce 988. Moreover, t s exounded that the standard energy addtvty rue adoted by a great number of researchers s not sutabe n Tsas statstcs, because t not ony voates the a of energy conservaton but aso ts coroary s n contradcton th the zeroth a of thermodynamcs. P numbers: d; 05.0.J; y; a *uthor to hom a corresondence shoud be addressed. Mang address: Deartment of Physcs, Xamen nversty, Xamen 6005, Peoe s Reubc of hna Ema: cchen@xmu.edu.cn

2 Temerature, nterna energy and entroy are three of the most mortant arameters n thermodynamcs. The concets of temerature and nterna energy become nontrva hen entroy aears to be nonextensve. nce the generazed statstca entroy as roosed by Tsas n 988, the non-extensbty of entroy n some comex systems th ong-range nteractons and/or ong-duraton memory has been dey recognzed -8. Hoever, there are st to robems of hysca mortance n Tsas statstcs, here one s hether the nterna energes of these comex systems are extensve or not, and the other s hether t s necessary to ntroduce the generazed zeroth a of thermodynamcs or not. though the to robems have been dscussed for many years 9-5, they have not been soved u to no, and conseuenty, have affected the deveoment and mrovement of Tsas statstcs. Thus, t s very mortant and urgent to sove the to robems and reach some usefu concusons. In nonextensve statstca mechancs deveoed from Tsas entroy, the frst choce s very tte used n the terature snce t coud not sove the reevant mathematca dffcutes athough there are three dfferent choces for the nterna energy constrant 6. For the second choce -, 8,, the dstrbuton functon 7 [ ] / can be derved from the generazed statstca entroy here k /,, R / ] [, k s a ostve constant, s the tota number of mcroscoc ossbtes of the system, s the energy of the system at the state. mary, for the thrd choce 4,6,8, /, one can obtan the dstrbuton functon as 6

3 / ] / [ th / ] / [. It can be roved -4,6,8,6-8 that,, / / T k, here T s the absoute temerature. For the sake of convenence, s reaced by beo. For an soated system comosed of to subsystems and of hch the dstrbutons satsfy,6-8, 4 usng the reatons and and Es. -4, e can obtan the seudo-addtvty entroy rue, 5, 8, 6 k 5 and the seudo-addtvty energy rues, 6 ] ][ [ ] [ ] [ ] [. 7 It s orthhe ontng out that Es. 6 and 7 are to mortant resuts that have never aeared n Tsas statstcs and one of the mortant bases for dscussng and sovng to robems mentoned n ths aer. When one mortant condton 8 s adoted, Es. 6 and 7 may be, resectvey, smfed as 9,8, 9

4 4 ] ][ [ ] [ ] [ ] [. 0 It s mortant to note the fact that E. 8 has been mcty used n nonextensve statstca mechancs 8-8 snce the generazed statstca entroy as advanced n 988, athough t has never been obvousy gven n terature of nonextensve statstca mechancs. One fnd from the foong anayss that E. 8 s essentay the mathematca exresson of the zero a of thermodynamcs. sng the as of entroy and energy conservaton and the above euatons, e can strcty rove, 0, ] [ ] [ k k δ δ and, 0, δ δ. From Es., and, one obtans,,, or. It s ust the zeroth a of thermodynamcs. Obvousy, the hysca essence of E. s cometey dentca th that of E. 8. It mes the fact that startng from E. 8, one get E. hch s the same resut as E. 8. It s thus cear that the dervatve rocess of E. s ony of a sef-consstent cacuaton, but s not a roof for the zeroth a of thermodynamcs n nonextensve statstca mechancs. Ths shos ceary that the zeroth a of thermodynamcs st hods n nonextensve statstca mechancs, but t can not be roved from theory. onseuenty, the concet of temerature s aso sutabe n nonextensve statstca mechancs.

5 It s aso mortant to note the other fact that n nonextensve statstca mechancs, f E. 8, has not been mcty used, one can not get the standard energy addtvty rue, 4 even though the thrd term on the rght hand sde of Es. 6 and 7 s not consdered. It s thus obvous that E. 8 s a necessary condton for the vadty of E. 4. Hoever, many researchers have not exounded the ueston and drecty used Es. -5 and 4 to nvestgate some mortant robems. For exame, they have been used to derve the so caed generazed zeroth a of thermodynamcs n nonextensve statstca mechancs. y comarng the exresson of the so caed generazed zeroth a of thermodynamcs 8,-6,8 or 5 k k [ / k] [ / k] 6 th E. 8, t can be seen thout dffcuty that E. 5 or 6 s obvousy n contradcton th E. 8, because and are not, n genera, eua to and, resectvey. Ths shos ceary that the standard energy addtvty rue 4 hch has been dey used by a ot of researchers may not be sutabe n nonextensve statstca mechancs because ts coroary voates the zeroth a of thermodynamcs 9. Therefore, t s unnecessary to ntroduce the so caed generazed zeroth a of thermodynamcs n nonextensve statstca mechancs and the ne concet of the hysca temerature 8,-6,0. The above resuts sho ceary that, Es. 9 and 0 can be derved n nonextensve statstca mechancs,based on Es. -4 and the zeroth a of thermodynamcs. They are the concrete mathematca exressons of the energy conservaton n nonextensve 5

6 thermodynamcs. Just as onted out n Ref. [8], f the correaton nonextensve terms of hatever observabe or nteractons can be negected, hat s the orgn of the nonextensvty of entroy? It s e-knon that entroy shoud be a contnuous functon of the observabes. For exame, for a sme nonextensve system th, T, f the nterna energy s extensve, s entroy ossby nonextensve? In addton, f E. 4 s true, one ose Es. 0[or 9] and 4, hch are cruca for the nonextensve theory. If E. 4 fas, e cannot, n fact, fnd even the entroy correaton gven by E. 5. It s thus cear that ke entroy, energy s aso nonextensve n Tsas statstcs, he the standard energy addtvty rue 4 s not sutabe n Tsas statstcs because t drecty voates the a of energy conservaton. ummng u, e have soved to ong-standng robems n nonextensve statstca mechancs. The zeroth a of thermodynamcs cannot be roved from theory, but t has been mcty used n nonextensve statstca mechancs, he the so-caed generazed zeroth a of thermodynamcs derved by severa authors may not be correct because t s derved on the basc of E. 8 and ts coroary s obvousy n contradcton th E. 8. The zeroth a of thermodynamcs s a base of nonextensve statstca mechancs, he t s unnecessary to ntroduce the so-caed generazed zeroth a of thermodynamcs and the ne concet of the hysca temerature. The standard energy addtvty rue 4 used dey by many researchers may not be sutabe n nonextensve statstca mechancs because t voates the a of energy conservaton and ts coroary s obvousy n contradcton th ts remse. In nonextensve statstca mechancs, one has to use the seudo-addtvty energy rue hch s consstent th the zeroth a of thermodynamcs and satsfes the a of energy conservaton. Fnay, t s onted out that the concusons obtaned here conform to be s standont 6, 6

7 statstca mechancs may be modfed but thermodynamcs shoud reman unchanged. cknoedgements Ths ork as suorted by the Natona Natura cence Foundaton No.07505, Peoe s Reubc of hna. 7

8 References. Tsas, J. tat. Phys. 5, V. H. Hamty and D. E. arraco, Phys. Rev. Lett. 76, E. K. Lenz, L.. Maacarne and R.. Mendes, Phys. Rev. Lett. 80, R. aazar and R. Tora, Phys. Rev. Lett. 8, E. Vves and. Panes, Phys. Rev. Lett. 88, be and. K. Raagoa, Phys. Rev. Lett. 9, Taruya and M. akagam, Phys. Rev. Lett. 90, be and Y. Okamoto, Nonextensve statstca mechanchs and ts acatons,rnger, Hedeberg, Q.. Wang and. L. Méhauté, haos, otons and Fractas, 5, R. D. sto,. Martíneza, F. Pennn,. Pastno and H. Vucetchb, Phys. Lett. 0, be,. Martínez, F. Pennn and. Pastno, Phys. Lett. 8, Martínez, F. Pennn and. Pastno, Physca 95, be, Physca. 00, Martínez, F. Pennn, and. Pastno, Physca 95, be, Physca. 69, Tsas, R.. Mendes and. R. Pastno, Physca 6, E. M. F. urado and. Tsas, J. Phys. 4, L Q.. Wang, Eur. Phys. J. 6, M. Nauenberg, Phys. Rev. E, 67, Martínez and. Pastno, Physca 45,