Thermoequilibrium statistics for a finite system with energy nonextensivity

Transcription

1 Artcle Statstcal Physcs and Mathematcs for Complex Systems December 0 Vol.56 No.34: do: 0.007/s SPCIAL TOPICS: Thermoequlbrum statstcs for a fnte system wth energy nonextensvty ZHNG Lang & LI We,* Insttute of Partcle Physcs and Complexty Scence Center, Central Chna Normal Unversty, Wuhan , Chna; Max-Planc-Insttute for Mathematcs n the Scences, Lepzg 0403, Germany Receved Aprl 9, 0; accepted May, 0 A frst-prncples dervaton s presented of canoncal dstrbutons for a fnte thermostat tang nto account nonextensve energy. Parameterzng ths energy by, we derve an explct form for the dstrbuton functons by regulatng, and then explore the nontrval relatonshp between these functons and energy nonextensvty, as well other system parameters such as system sze. A varatonal entropy functon s also derved from these dstrbuton functons.. fnte system, energy nonextensvty, frst-prncples method, canoncal dstrbuton functon Ctaton: Zheng L, L W. Thermoequlbrum statstcs for a fnte system wth energy nonextensvty. Chnese Sc Bull, 0, 56: , do: 0.007/ s *Correspondng author (emal: lwe@ms.mpg.de) Studes nto the famly of small or fnte-sze systems reveal that these systems show complex statstcal characterstcs very dfferent from large systems, and expose falngs n our understandng of thermostatstcs for fnte systems and ther nonequlbrum dynamcs [ 4]. One of these characterstcs s nonextensvty (refer to [] and references theren for examples), whch means that macroscopc quanttes of such systems may not be proportonal to system sze. A thermodynamc quantty of a nonextensve system s called nonaddtve f ths quantty s not the sum of that of ts subsystems. Nonextensvty mght arse from surface effects or nteractons between subsystems. Thus, extensve energy and entropy may become napproprate n treatng fnte systems. There has already been much dscusson on the propertes of fnte systems that has rased many questons and controverses [5 7]. To descrbe the behavors of these systems, one pont worth notng s that the Boltzmann-Gbbs statstcal mechancs, whch s based on the concept of extensve entropy, s not applcable [,8,9]. Consequently, t s questonable to assume an exponental dstrbuton for fnte systems. Therefore, one of the frst thngs to wor out for a statstcal descrpton of fnte systems s to fnd the approprate probablty dstrbuton. Some publshed results obtaned by frst prncples (see [8] for example) have shown that a small system, n equlbrum wth a fnte reservor, may follow a q-exponental dstrbuton of nonextensve statstcal mechancs (NSM), as proposed by Tsalls [9]. Ths NSM has been used wdely n many areas, and consdered effectve n solvng many physcal problems [0 3]. Results from mathematcal proofs are able to demonstrate a connecton between the system s fnteness and nonextensvty of the theory. However, most proofs have reled on the addtvty of energy. Clearly, for large systems, ths assumpton s acceptable and provdes a helpful approxmaton n obtanng the statstcs n the thermodynamc lmt. The assumpton s, however, questonable when establshng statstcs for fnte systems. Related problems arsng from ths addtve energy assumpton can be found n [4 7]. Wor n [8] represents a frst attempt to buld statstcs for fnte systems based on energy nonaddtvty. A result proved n ths wor was that addtve energy s unnecessary n the development of a NSM for small systems by mean feld theory. In our present wor, we follow on from [8], and provde results that have greater generalty. We have ntroduced the hgh-temperature approxmaton for the sae of smplcty. In the followng sectons, the possble canon- The Author(s) 0. Ths artcle s publshed wth open access at Sprngerln.com csb.scchna.com

2 Zheng L, et al. Chnese Sc Bull December (0) Vol.56 No cal dstrbuton and thermodynamc entropy are derved. Based on the results from these calculatons, a summary and some nterestng nferences are gven. Probablty dstrbuton for the model We present here ntroductory materal and bref outlnes of proofs requred n later sectons. As n [9], let us begn wth an adabatcally and mechancally solated system, wth fnte N partcles and energy. The probablty dstrbuton functon can be gven by px ( ) [ HX ( )]. ( ) () Next, dvde ths system nto two nteractng subsystems and wth Hamltonans H (X ) and H (X ). Assumng the nonaddtve rule of energy suggested n [0,], we can wrte the energy of the system as H( X) H( X, X ) () H ( X ) H ( X ) H ( X ) H ( X ), where λ s a couplng constant, whch denotes the nonextensvty of the system. Here s the generalzed form of the addtve energy; H( X) H( X) s the nteracton energy between and ts thermostat system. Obvously, we stll have the assumpton H( X), even f the thermodynamc lmt N s nvald. If we substtute eq. () nto eq. (), the probablty dstrbuton of s gven by p( X ) [ H ( X ) H ( X ) ( X ) ( ) H ( X ) H ( X )]dx ( ) ( X ) ( R ) { H ( X ) [ H ( X )][ K( P ) V( R )]}dx ( ) { H ( X ) [ H ( X )] V( R )}d R, where K(P ) s the netc energy and V(R ) s the potental energy of the partcles n the thermostat system, wth the set of momenta and coordnates denoted by P and R, respectvely. For smplcty, we let H, H and H represent H (X ), H (X ), and H(X) n the followng wor. Hence {} can be gven n the form { y } [ yu (, H, P )]d P, (4) ( P ) n whch we have set y H( H) V( R) and u(, H, P ) [ H ( X )] K( P ). Here {} y can be regarded (3) as the hyper-surface correspondng to u=y. Naturally, t s equal to the dervaton of the hyper-volume of the momentum space assocated wth P by the quantty u(, H, P),.e. {} y y wth ( y) d P. (5) u y After ntroducng new varables, H D ( P ) n, (6) mn wth =3(n)+α and the mass of n-th partcle m n, the equaton u=y can be wrtten as D y. The ntegraton (4) fnally gves m { y} b y. (7) H Here b depends only on N. Thus, we can get the followng dstrbuton: b H p( X ) ( ) { H [ H ] V( R )} dr b ( H ) ( H ) H V R R (8) ( ) d, H where b s the normalzaton constant f the mass of the partcle s fxed. Ths equaton s exact, because we have not made any approxmaton so far. For an adabatcally system, t s easy to see the followng relatonshp: H V( R ) H V V. (9) H H H H If the hgh-temperature approxmaton s ntroduced, we mmedately get H >>V(R ), whch ndcates that the netc energy s much greater than the potental energy. Next, the ntegral n eq. (8) can be expanded as follows: 3 N H V R R H ( ) d 3 H N ( ) d V R R H N 3 H V N V( R)d R, H (0) where V represents the volume of the system. Consderng the result above, the energy dstrbuton functon for fnte

3 3668 Zheng L, et al. Chnese Sc Bull December (0) Vol.56 No.34 systems can be wrtten as 3 N p() ( ) ( ). () N Here, we have replaced H by ε and V( R)d R / V whch nclude potental energy contrbutons. Moreover, we wll fnd that obtaned n the hgh- temperature approxmaton. By employng the same trc as eq. (0), eq. () can be replaced by p() ( ) ( ). Thus, the fnal dstrbuton can be gven as () p() ( ). (3) In the above dervaton, we have taen 3 N. From eq. (3), we conclude that the dstrbuton of a fnte system wth nonaddtve energy s dependent on the sze of the system and the nonextensve parameter. Havng as- sumed that, we can expand to the frst order, as long as the factor λ s not much larger than, mplyng that the nonextensve energy λh H s relatvely small compared wth H. Thus, we obtan p(). (4) In seeng to explctly reveal the nfluence of nonextensvty, we neglect the potental energy to reduce the complexty of the equaton;.e. we are consderng the dstrbuton for an deal non-nteractng classcal system. Hence the fnal dstrbuton eq. (3) reduces to p() C, (5) where C s a normalzaton constant. Moreover, because nteractons are gnored, t s reasonable to assume that, where θ/ represents the mean netc energy per degree of freedom n the reservor. By employng the normalzaton condton p ()d, (6) 0 one can easly get the expresson for C. Fgure shows curves of the probablty dstrbuton n eq. (5). These curves suggest that probabltes assocated wth for hgh-energy states ncreases as nonextensve energy declnes. If λ=0, curves reduce to exponental decays as N. However, f λ 0, no trend to a Boltzmann dstrbuton wll occur because of nonextensve energy Fgure Probablty dstrbuton versus energy for fxed partcle number N =00 and dfferent values of. consderatons. Notably, for wea potental energy, we can derve ths energy dstrbuton n an alternatve way. Adoptng the concluson of [6], we get H( X) px ( ) C. (7) As s dscussed above, both H and λh H are much smaller than H, and hence we obtan the concluson that H. Imposng ths constrant, we can rewrte the Hamltonan n eq. () as H( X) H( X, X ) H( X ) H ( X ), (8) where H ( X) ( ) H( X). If we substtute H ( X ) for the H (X ) n eq. (7), the orgnal dstrbuton transforms exactly nto the form of eq. (5), whch justfes eq. (5) from another standpont. The above calculatons suggest that NSM dstrbutons for small systems based on nonaddtve energy exst. Addtonally, as we have not taen account of specfc forms of nteractons, ths dstrbuton mght be applcable for varous specfc forms of the potental energy. ntropy functon n deal case Wth entropy as defned n [] as a measure of dsorder or randomness n a thermodynamc system, we can gve the thermodynamc entropy by a varatonal relaton: d S (d d ), (9) where β s the nverse temperature. Ths expresson for the varaton n entropy can be vewed as a generc entropy defnton of an uncertanty measure, and the specfc form can be derved from ths entropy expresson f the probablty dstrbuton s gven. Let {p } be the set of probabltes correspondng to the spectrum {ε }. In the deal case, we can wrte the probablty dstrbuton for a fnte system wth nonaddtve energy as

4 Zheng L, et al. Chnese Sc Bull December (0) Vol.56 No p p( ) Z, (0) where Z s gven n the form Z. () From the defnton eq. (9) d S (d d ) d p, () where (p ) can be solved from the energy probablty dstrbuton eq. (0) ( p) ( pz), (3) we have a very specfc form for the entropy, namely ds d p ( pz) d p. (4) Consderng the varatonal condton dp 0, obtaned from p, one can get ds Z p d p. (5) After ntegraton, the expresson for entropy s S Z p const, (6) n whch the const ensures that S vanshes n the absence of uncertanty. The dependence of S for a two-state system on a snglestate probablty for dfferent values of s gven n Fgure, from whch S s seen to be maxmal for states of equal probablty. Moreover, the value of entropy decreases as ncreases; the explanaton s that an ncreasng nteracton energy reduces the resdual uncertanty for the system of nterest. If =0, we fnd that entropy reduces to the same form as the Tsalls entropy q q S Z p, q (7) where q. Ths nonextensve ndex concdes wth that of [3], the ndex deduced on the bass of addtve energy. 3 Dscusson and concluson In summary, under the assumpton of nonaddtve energy, we have derved the energy probablty dstrbuton for fnte sze systems from frst-prncples. The Boltzmann dstrbuton can be obtaned by tang λ=0 and the thermodynamc lmt, N. From eq. (3), the nonextensve energy precludes the dstrbuton from reducng to Boltzmann form even f N. These calculatons have been made wthout consderng explct expressons for the potental energy. Thus, the valdty of the conclusons mght be general and ndependent of dfferences n the potental energy. The result s vald to descrbe the general thermostatstcs of fnte systems. The Boltzmann dstrbuton appears as a specal case wthn the framewor. However, some questons wth the procedure stll reman. For example, the basc assumpton that the energy nonextensvty of fnte systems can be parameterzed by a fxed constant, unrelated to the propertes of the system, s not a sound hypothess. Ths nonextensve parameter depends on the characterstcs, such as sze and nteractons of the system. Some adjustment of ths model can be brought by ntroducng a related to sze, but calls for further research. These nvestgatons can be appled to the analyss of varous systems [4 7]. By deepenng our understandng of dstrbuton functons and ther statstcs, we would have a very powerful tool for the study of complex systems. Ths wor was supported by the Natonal Natural Scence Foundaton of Chna (06475, , , ) and the Program of Introducng Talents of Dscplne to Unverstes (B08033). Fgure ntropy S for two-state system versus probablty for fxed partcle number N =00 and dfferent values of. Combes F, Robert R, Pfennger D, et al. Statstcal mechancs of non-extensve systems. C R Phys, 006, 7: Wang R, Nganso D S, Kaabouch A, et al. Investgaton of an energy nonaddtvty for nonextensve system. Chnese Sc Bull, 0, 56, do: 0.007/s Wang L N, Mn J C. Thermodynamc analyss of adsorpton process at a non-equlbrum steady state. Chnese Sc Bull, 00, 55: Shao Y Z, Zhong W R, He Z H. Nonequlbrum dynamc transton n a netc Isng model drven by both determnstc modulaton and correlated stochastc noses. Chnese Sc Bull, 005, 50: Almeda M P. Thermodynamcal entropy (and ts addtvty) wthn

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