Symmetries and nonequilibrium thermodynamics

Transcription

1 PHYSICAL REVIEW E 95, (207 Symmetres and nonequlbrum thermodynamcs Sergo Bordel * Laboratory o Cell Culture, Insttute o Cardology, Lthuanan Unversty o Health Scences, Sulėlų ave. 7, Kaunas, Lthuana and Department o Chemcal and Bologcal Engneerng, Chalmers Unversty o Technology, Kemvägen 0, SE-4296 Göteborg, Sweden (Receved 6 March 207; revsed manuscrpt receved 27 Aprl 207; publshed 8 June 207 Thermodynamc systems can be dened as composed by many dentcal nteractng subsystems. Here t s shown how the dynamcs o relaxaton toward equlbrum o a thermodynamc system s closely related to the symmetry group o the Hamltonan o the subsystems o whch t s composed. The transtons between states drven by the nteractons between dentcal subsystems correspond to elements o the root system assocated to the symmetry group o ther Hamltonan. Ths mposes constrants on the relaxaton dynamcs o the complete thermodynamc system, whch allow ormulatng ts evoluton toward equlbrum as a system o lnear derental equatons n whch the varables are the thermodynamc orces o the system. The traectory o a thermodynamc system n the space o thermodynamc orces corresponds to the negatve gradent o a potental uncton, whch depends on the symmetry group o the Hamltonan o the ndvdual nteractng subsystems. DOI: 0.03/PhysRevE I. INTRODUCTION Nonequlbrum thermodynamcs s stll open to controversy and derent nterpretatons []. General prncples descrbng the relaxaton o a system toward thermodynamc equlbrum or the steady states o a system that s ept away rom equlbrum due to ts constant exchange o matter or energy wth ts envronment, have been searched wth varable success. Prgogne [2] and Zegler [3] proposed, respectvely, mnmal and maxmal entropy producton rates characterzng steady states, the rst one related to global entropy producton under certan boundary condtons and the second one beng local [4]. Other authors [5] have proposed generalzatons ar away rom the equlbrum to Onsager s relatons between thermodynamc luxes and orces. In prevous wors, we argued that Onsager s recprocty relatons are equvalent to a prncple o steepest entropy ncrease [6,7]. The same conclusons had been prevously reached by Beretta [8 0] by ntroducng a dsspatve term n Schrödnger s equaton. Other authors have also explaned the rreversble relaxaton o thermodynamc systems toward equlbrum by modyng Schrödnger s equaton [ 4]. In act, the apparent contradcton between the second prncple o thermodynamcs and the tme reversblty o Schrödnger s equaton can be explaned based on the theory o quantum measurement (the proecton o a wave uncton on an egenvector o the operator assocated to a measured observable and von Neumann s entropy denton. Indeed, whle von Neumann s entropy does not change over tme when the system evolves ollowng Schrödnger s equaton, t can change the system s proected on one egenvector o some observable wth an assocated operator that does not commute wth the Hamltonan. A recent explanaton ollowng these lnes can be ound n Re. [5]. Ferm s golden rule [6] descrbng transtons between states (derent egenvalues o a Hamltonan as a result o a perturbaton, uses mplctly the theory o quantum measurement to explan the observed transtons. * sergo_bordel@hotmal.com In ths wor, we present the dervaton o a general master equaton descrbng the relaxaton o an solated system toward equlbrum. A thermodynamc system s dened as beng composed o dentcal subsystems that nteract wealy between each other (whch allows usng Ferm s golden rule [6] to descrbe the transtons between states o each subsystem. The only ntal assumptons used n the dervaton are Ferm s golden rule and the denton o the entropy producton rate as the product o thermodynamc luxes and orces [7], whch allows us to dene a nonequlbrum statstcal dstrbuton (Appendx E wthout any o the extra assumptons requently used, such as the maxmzaton o the normaton entropy [7,8]. Thermodynamc luxes are dened as the varaton rates o certan observables, whch are assocated to quantum operators that commute wth the Hamltonan o the ndvdual subsystems. The results n ths paper are very wde but not completely general, because we are treatng the case o solated systems ormed by dentcal subsystems wth the same symmetres. Ths would not cover, or nstance, the approach to equlbrum between sotropc luds and crystallne solds or lqud crystals etc., whch nvolve the nteractons between systems wth derent symmetres. However, the present ormalsm s expected to be generalzed also to these more general stuatons (n partcular, the relaxaton o systems n contact wth an sothermal bath, and to clary the nluence o the derent symmetres on the coecents descrbng the respectve energy exchange among such derent degrees o reedom, n the lght o Appendx A to the paper. However, ths would requre a ull paper and wll be done n the uture. In ths artcle, we show how the symmetres o the Hamltonan o the ndvdual subsystems mpose strngent rules on the allowed transtons between states and ther relatve rates. The states n whch each subsystem can be ound are dened by ts energy level (each energy level corresponds to a representaton o the symmetry group o the Hamltonan and a vector o weghts, whch are egenvalues o a subset o generators o the symmetry group (the Cartan subalgebra. It s proven that the only allowed transtons between states (n a system ormed by ndstngushable subsystems correspond to elements o the root system assocated to the Hamltonan s /207/95(6/06208( Amercan Physcal Socety

2 SERGIO BORDEL PHYSICAL REVIEW E 95, (207 symmetry group. All these propertes allow descrbng the traectory o the system n the space o thermodynamc orces, by a system o lnear derental equatons whose coecents can be obtaned rom the Hamltonan s symmetry group and the nteracton potental between pars o subsystems. II. FORMALISM A. Transtons between states n a system o many dentcal wealy nteractng subsystems We consder a system ormed by a number s o dentcal subsystems (or example, a contaner lled wth s molecules o a certan chemcal compound. To be as general as possble, the subsystems could be large groups o partcles or molecules that orm a Gbbs ensemble. The only necessary crtera or these subsystems s that they are dentcal to each other. Each subsystem s characterzed by a Hamltonan operator h ;as all the subsystems are dentcal, ther derent Hamltonans have the same orm and they only der on the varables that they act on (the poston and momentum coordnates o the components o each subsystem. We wll use the ollowng notaton or the egenvectors o the Hamltonan h :,so that h = ε. ( In Eq. (, ε s a real egenvalue o h. As the Hamltonans o the derent subsystems have the same orm, the values ε are the same or each subsystem and thereore do not depend on. The derent egenvalues ε correspond to the derent energy levels n whch a subsystem can be observed and the set o egenvectors { } orms an orthonormal bass o the Hlbert space H that contans all the possble wave unctons ψ( descrbng the state o the subsystem : ψ( = c. (2 The wave unctons descrbng the whole system are elements o the Hlbert space H S, resultng rom the tensor product o the Hlbert spaces H : s H S = H. (3 = An orthonormal bass o the space H S s ormed by the tensor products o the elements o the bases o H. We wll denote the elements o ths bass as σ = ss. (4 The symbol σ desgnates an s-tuple 2... s o ndexes that correspond to egenvectors o h. Now we can dene N as the number o tmes the ndex s repeated n the s-tuple σ, or n other words, the number o subsystems n the state. Our thermodynamc system has been dened as ormed by dentcal subsystems, whch maes them undamentally ndstngushable rom each other. In ths case, the wave uncton descrbng the whole system has to be symmetrc wth respect to the permutaton o two subsystems. Even the system s composed o ermons, we can dene the subsystems as pars o ermons; thereore, we can assume symmetrc wave unctons wthout loss o generalty. Symmetrc normalzed wave unctons wll tae the ollowng orm: S = N! p(σ, (5 p(σ where p(σ s a permutaton o the ndexes n the s-tuple σ. The Hamltonan o the whole system, ncludng the nteractons between dentcal subsystems, wll tae the orm ˆ H = h +, 2 ˆV = ˆ H 0 + ˆV. (6 For wea nteractons between the subsystems, the transton requency between two states can be descrbed by Ferm s golden rule: W S0 S = 2π h S ˆV S 0 2 δ ( E S E S0. (7 As shown n Appendx A [Eq. (A5], the requency o transtons nvolvng changes n the state o more than two subsystems smultaneously s equal to zero. As the subsystems are undstngushable, there s no derence between a stuaton n whch a subsystem n the state r transts to state r and a subsystem n state l transts to state l or a subsystem n state r transts to state l and a subsystem n state l transts to state r. Thereore, we can ust tal about lr l r transtons, whose requency s equal to ( T rl r l = N 2 Ɣ rl r l N rn l (N r + (N l +, (8 where Ɣ rl r l = Ɣ r l rl = 2π h l ; r ˆV r; l 2 δ(ε r + ε l ε r ε l, (9 r; l = ( r l + l r. 2 From now on the ndexes dentyng the subsystems (, wll not be used. The energy o each state s reerred as ε n Eq. (9. Equaton (8 s derved n Appendx B, and n Appendx D t s also shown that t leads to a Bose-Ensten equlbrum dstrbuton. B. Symmetres and states o a quantum system In ths secton, we remnd some general acts about the descrpton o a quantum system n terms o ts symmetres, whch are textboo materal [9]. So ar, the egenstates o the Hamltonan h have been descrbed usng the ndexes r and l. In act, these egenstates can be descrbed n terms o the symmetry group o the Hamltonan. The symmetry group o h s a Le group whose generators orm a Le algebra and commute wth the Hamltonan h. Among the generators o the symmetry group t s possble to choose a maxmal subset o generators that commute wth each other (the Cartan sub-algebra. The elements o the Cartan subalgebra wll be noted as  : [Â, ] = 0. (

3 SYMMETRIES AND NONEQUILIBRIUM THERMODYNAMICS PHYSICAL REVIEW E 95, (207 The operators n the Cartan algebra have the same egenvectors, as they commute wth each other. The rest o the generators o the Le algebra, whch do not commute wth the elements o the Cartan algebra, can be chosen to be egenvectors o the elements o the Cartan algebra under the operaton o commutaton. They satsy the ollowng commutaton relatons: [Â,Ê α ] = α Ê α. ( The vector α contans the egenvalues o each element o the Cartan algebra and t s reerred to as a root vector. The root vectors are dened by the symmetry o the system. Each energy level o the Hamltonan h s a subspace o the Hlbert space H wth as many dmensons as the degeneracy o the energy level. It s possble to choose an orthonormal bass o ths subspace, n whch each o ts members s an egenvector o the operators n the Cartan subalgebra. The correspondng egenvalues orm a so-called weght vector μ, whch can be used to ndcate each state wthn an energy level. The subspace correspondng to an energy level s a G-module o a representaton o the symmetry group o the Hamltonan. Each energy level corresponds to a group representaton and can be denoted by the vector, whch s equal to the hghest weght vector wthn the energy level. Thereore, each state o the system can be dented by the par o vectors μ and. The operators that orm the Le algebra transorm the states o the system as ollows: Â μ =μ μ, (2 Ê α μ =U α, μ ( μ + α. (3 The operator Ê α s a ladder operator that dsplaces the vector o weghts μ by addng to t the root vector α. Startng rom μ t s possble to operate wth Ê α p tmes untl one o the weghts μ reaches ts maxmal value M (or mnmal value M or the consdered energy level. Once ths lmt has been reached, the operator Ê α wll transorm the nal state n a zero value: Ê α ( μ + p α =0. (4 The coecents U α, μ depend on μ, α, and p n the ollowng way: ( 2 μ α U α, μ = 2 α2 p α α + p +. (5 The maxmal number o tmes n whch t s possble to apply the operator Ê α on the state μ s equal to 2 μ α q = + p. (6 α α We pont out two useul relatons (derved rom the prevous two equatons to be used later: (U α,( μ α 2 (U α, μ 2 = μ α, (7 U α, μ = U α,( μ+ α. (8 As the coecents depend on the nteger p, whch tsel depends on the representaton, strctly speang we should also ntroduce the ndex n the notaton U α, μ. C. Symmetry propertes o two dentcal quantum systems To descrbe the relaxaton o a thermodynamc system, we are nterested n computng the expressons l ; r ˆV r; l that appear n Eq. (9. To acheve ths goal, t s necessary to nd the generators o the symmetry group o a combnaton o two ndstngushable systems. As both systems are undamentally ndstngushable, the symmetry operatons should act dentcally on both; thereore, the symmetry group o the combned system s the same as the symmetry group o a sngle system. The generators o the symmetry group are ust the sums o the generators actng on each ndvdual system: Â = Â + Â2, (9 Ê α = Ê α + Ê2 α. (20 The super ndexes ndcate each o the two subsystems on whch the operators act. The operators transorm the states o the combned system, dened n Eq. (0, n the ollowng way: Â μ ; μ 2 2 =(μ + μ 2 μ ; μ 2 2, (2 Ê α μ ; μ 2 2 =U α, μ ( μ + α ; μ U α, μ2 2 μ ;( μ 2 + α 2. (22 We am to nd a bass o states o the combned system whose components are transormed by the operators n Eqs. (9 and (20 n the same way as n Eqs. (2 and (3. The components o ths new bass, characterzed by the weght vectors μ, are lnear combnatons o all the states o the combned system that satsy the condton μ = μ + μ 2 : μ = μ C(, μ, μ μ ;( μ μ 2. (23 The coecents C(, μ, μ are the so-called Clebsch- Gordan coecents and the summaton runs along all the values o μ or whch there exst a μ 2 that satses μ = μ + μ 2. The vectors μ ;( μ μ 2 orm an orthonormal bass, whch mposes the ollowng condton: ( μ μ 2 ; μ M μ M ;( μ μ 2 = δm δ 2 δ 2 μ μ δ μ μ. (24 From ths property, we obtan a denton o the Clebsch- Gordan coecents: C(, μ, μ = ( μ μ 2 ; μ μ. (25 The vectors μ also orm an orthonormal bass: μ μ =δm δ μ μ. (26 Ths condton, combned wth the denton o the Clebsch- Gordan coecents n Eq. (25, leads to the ollowng

4 SERGIO BORDEL PHYSICAL REVIEW E 95, (207 expresson or the components o the old bass n terms o the new one: μ ;( μ μ 2 = C(, μ, μ μ. (27 From Eqs. (26, (25, and (27 t s possble to obtan the ollowng relatonshp between the Clebsch-Gordan coecents: C(, μ, μ C(, μ, μ = δ μ μ. (28 Operatng on the vector μ wth a ladder operator, and usng Eq. (23 to expand the obtaned vector ( μ + α, we obtan the expresson Ê α μ =U α, μ C(, μ + α, μ μ ;( μ + α μ. μ (29 On the other hand, rst expandng the vector μ and then applyng the operator, Ê α μ = μ C(, μ, μ ( U α, μ ( μ + α ; μ U α, μ2 2 μ ;( μ 2 + α 2. (30 Comparng Eqs. (29 and (30 and usng the orthogonalty condton between the vectors, t s possble to derve the ollowng relaton: U α, μ C(, μ + α, μ = U α, μ2 2 C(, μ, μ + U α,( μ α C(, μ, μ α. (3 D. Symmetry o energy conservng nteractons Ferm s golden rule mposes the condton o energy conservaton or a nonzero transton rate between states. The operator that descrbes the energy nteracton between two subsystems ˆV 2 does not commute wth the Hamltonan h 2 = h + h 2 ; however, t s possble to dene an operator ˆV 2 that satses the ollowng condton: l ; r ˆV 2 r; l = l ; r ˆV 2 r; l δ(ε r l ε rl. (32 Ths new operator commutes wth h 2 and could be expressed as a lnear combnaton o the generators o ts symmetry group: ˆV 2 = κ Â + α κ α Ê α. (33 From the prevous denton and the condton set n Eq. (32, we obtan μ ˆV 2 μ =κ α U α, μ δ δ ( μ+ α μ. (34 Ths means that the only allowed transtons are those satsyng the condton μ = μ + α, or any o the roots o the system. Usng Eqs. (27 and (34, we obtan the ollowng equalty: μ M ;( μ μ 2 ˆV 2 μ ;( μ μ 2 = κ α U α, μ C(, μ + α, μ C(, μ, μ. (35 Equaton (3 can be used to transorm the summaton n Eq. (35 asollows: U α, μ C(, μ + α, μ C(, μ, μ = U α, μ2 2 C(, μ, μ C(, μ, μ + U α,( μ α C(, μ, μ αc(, μ, μ. (36 Usng Eq. (28 we obtan the nal expresson: μ M ;( μ μ 2 ˆV 2 μ ;( μ μ 2 ( = κ α U α, μ2 2 δ μ μ + U α,( μ α δ μ ( μ α. (37 Ths means that only transtons satsyng (or any root o the system μ = μ + α or μ = μ are allowed. The second condton s equvalent to μ 2 = μ 2 + α, as the condton μ = μ + α must also be satsed. Thereore, or nondstngushable subsystems, each elementary transton nvolves the change o a sngle subsystem by one o the roots o the root system that characterzed the symmetry group o ts Hamltonan. The transton s drven by a second subsystem that remans unchanged. To eep the condton o mcroscopc rreversblty, μ M ;( μ μ 2 ˆV 2 μ ;( μ μ 2 = μ ;( μ μ 2 ˆV 2 μ M ;( μ μ 2. (38 The ollowng condton should be satsed: κ α U α, μ = κ α U α,( μ+ α. (39 Tang Eq. (8 nto account, ths s equvalent to κ α = κ α. (40 Ths comes rom the act that the operators Ê α are not Hermtan, but Ê α and Ê α are Hermtan adont and thereore ther sum s Hermtan. So, n order or the operator ˆV 2 to be Hermtan, the operators Ê α and Ê α should appear multpled by the same coecent n Eq. (33. E. Relaxaton dynamcs o a nonequlbrum system Usng the prevously mentoned results, the transton rate rom a state μ to a state ( μ + α can be obtaned rom Eqs. (8 and (37: T μ ( μ+ α = 2 N μ (N ( μ+ α + κ2 α U 2 α, μ. (4 The coecent s the actor that appears n Eq. (8: μ = N μ. (42 The product n Eq. (42 goes through all the states o the system. The coecent accounts or the nteractons wth

5 SYMMETRIES AND NONEQUILIBRIUM THERMODYNAMICS PHYSICAL REVIEW E 95, (207 all the other subsystems and taes the orm = 2π h N μ (N μ +. (43 μ In Appendx E, we show that we dene the thermodynamc luxes o the system as the varaton rates o the observables assocated wth quantum operators that belong to the Cartan subalgebra o the Hamltonan s symmetry group, the ollowng relaton between thermodynamc orces and occupaton numbers can be establshed: ln N μ ( + N ( μ+ α N ( μ+ α ( + N μ = X α. (44 B The notaton n Eq. (44 has been changed rom Appendx E, n order to t our descrpton n terms o weghts and roots. The coecents X are each o the thermodynamc orces o the system. Equaton (44 allows us to obtan the net transton rate rom a state μ to a state ( μ + α ater subtractng the transtons n the opposte sense: T μ ( μ+ α T ( μ+ α μ = 2 ( e X α B N μ (N ( μ+ α + κ2 α U 2 α, μ. (45 Now we are gong to compute the rate o change o the occupaton number N μ : dn μ = ( T( μ α μ T μ ( μ α T μ ( μ+ α + T ( μ+ α μ. (46 The summaton s extended to all the members o the root system (each o them labeled by a derent value o the subndex, excludng the root α α s already ncluded. As t s shown n Appendx E, the occupaton number N μ can be expressed usng the ollowng equaton (usng the current notaton n terms o group representatons, usng the subndex r nstead o μ to smply the expresson: N r = e α+βε B. (47 X μ Assumng the values o α are small, we can mae an approxmaton that wll smply our analyss remarably: N ( μ α (N μ + N μ (N ( μ+ α + N μ (N μ +. (48 Equatons (7, (45, and (48 allow rewrtng Eq. (46na very compact way: dn r = 2 ( e (N r + κ 2 α ( B X α N r μ r α. (49 I we assume that the changes o the occupaton numbers are drven by changes n the thermodynamc orces o the system, we can obtan the prevous dervatve drectly rom Eq. (47: dn r = B N r (N r + dx μ r. (50 Comparng the two prevous equatons, we obtan derental equatons descrbng the relaxaton dynamcs o the system: dx = 2 B ( e B X α κ α 2 α. (5 Equaton (5 s not based n any assumpton about the proxmty o the system to thermodynamc equlbrum and provdes a general soluton to the problem o the relaxaton dynamcs o thermodynamc systems. Lnearzng the term between bracets (whch or many systems s easble even ar rom equlbrum, gven the small values o α, we would obtan the ollowng system o lnear derental equatons: dx = 2 ( α κ 2 α α X. (52 Ths means that the traectory toward equlbrum o the system, n the space o thermodynamc orces, ollows the negatve gradent o the uncton: ( F = α κ 2 α α X X. (53 2 The parameters and are dependent on the thermodynamc orces o the system and reach a mnmum at equlbrum (ther dervatves wth respect to the thermodynamc orces become zero at equlbrum. The thermodynamc luxes (total rates o change o nonconserved observables, as dscussed n Appendx E can be expressed as ollows (or any observable assocated to the operators n the Cartan algebra: J = d μ, (54 n whch μ = μ r N r. (55 r Equaton (54 can be expanded as J = ( μ dx. (56 X We can dene the matrx A as ollows: A = κ 2 α ( α α. (57 Combnng the prevous denton wth Eqs. (52 and (56, the thermodynamc luxes can be rewrtten as J = 2 ( μ A X. (58 X

6 SERGIO BORDEL PHYSICAL REVIEW E 95, (207 Tang the partal dervatve o the lux J wth respect to the orce X l, we obtan ( J = [ ( ] 2 μ A X X l X l X ( 2 μ A l. (59 X I we compute the dervatve n equlbrum, wth all the thermodynamc orces equal to zero, the rst term n Eq. (59 dsappears. On the other hand, dervng Eq. (47, we can obtan the ollowng expresson: ( Nr = N r (N r + μ r. (60 X B Combnng Eqs. (55 and (60, the ollowng equaton s obtaned: ( μ = N r (N r + μ rl μ r. (6 X l B r I the occupaton numbers ollow the equlbrum dstrbuton, they depend only on the energy level (and each energy level corresponds to a representaton o the symmetry group. Usng agan the subndex to label energy levels and usng the subndex r to label derent states wthn the same energy level, we could rewrte Eq. (6 or the partcular case o proxmty to equlbrum: ( μ = NM X l (N + μ rl μ r. (62 B r X= 0 Now we remar that or every weght vector μ r n a representaton, the weght vector μ r also belongs to the same representaton. Ths act maes that the summaton n Eq. (62 becomes zero except or l =. Thereore, Eq. (59 n equlbrum, taes the orm ( ( J = 2 μ A l. (63 X l X= 0 X X= 0 Fnally, consderng that the set o observables {μ } are the egenvalues o the operators ormng the Cartan subalgebra dened n Eq. (0 and that the only propertes requred or these operators are that they commute wth the Hamltonan o the subsystems and between each other, each observable could be multpled by an arbtrary constant wthout loss o generalty. The observables can be chosen so that the ollowng condton s satsed: ( μ =. (64 X X= 0 B Boltzmann s constant has been ntroduced to eep consstent the unts. Wth ths choce o observables, Eq. (63 gves us an expresson or Onsager s coecents, whch are ndeed symmetrc, gven the act that the matrx A s symmetrc: L l = 2 A l. (65 B III. DISCUSSION The strong constrants mposed by the symmetres o the Hamltonan h on the allowed transtons and ther relatve rates allowed us to obtan Eq. (52, whose coecents can be calculated rom the symmetres o the Hamltonan h and the orm o the nteracton potental between subsystems. Equaton (52 ully descrbes the relaxaton toward equlbrum o a thermodynamc system composed by dentcal subsystems (and these subsystems can be chosen to be as complex as we want wthout changng the general conclusons. The thermodynamc luxes (rates o change o macroscopc observables can be computed usng Eq. (55. Equaton (53 shows that there s ndeed a potental whose negatve gradent determnes the traectory o the system and whose mnmal value corresponds to a state o maxmal entropy. However, the partcular shape o ths potental depends on the system beng studed and ts symmetres. Nonequlbrum steady states, resultng rom externally mposed constrants on the thermodynamc orces, could be obtaned by mnmzng the potental F subect to these constrants. Equaton (65 shows a relaton between Onsager s coecents and the structure o the symmetry group o the system (whch determnes the matrx A; ths allows both predctng the coecents ab nto based on the nteracton potental between subsystems or usng expermentally measured coecents (n the proxmty o equlbrum, to obtan matrx A and use t to model the relaxaton o the system also arbtrarly ar rom equlbrum, by usng Eqs. (5 and (52. APPENDIX A: TRANSITIONS BETWEEN STATES FOR DISTINGUISHABLE SUBSYSTEMS In ths secton, we are gong to assume that the subsystems are dstngushable between each other, and thereore the wave uncton σ s derent rom the uncton p(σ, where p(σ s a permutaton o the ndexes n the s-tuple σ. As t wll be shown later, the consderaton o dstngushable subsystems wll lead us to the dervaton o Boltzmann s dstrbuton, whle the assumpton o nondstngushable systems wll lead to the Bose-Ensten dstrbuton. When consderng nonequlbrum thermodynamcs, we wll see that the two approaches lead to derent dynamc propertes o the system. The unctons σ are egenvectors o the operator Hˆ 0 = s = h. In the absence o nteractons between the subsystems ths operator would be the Hamltonan o the whole system, and ths system s ntally observed n the state σ 0, then t wll always be observed n the same state and there would not be entropy change. I there are nteractons between the subsystems, then the Hamltonan o the whole system wll nclude an extra term accountng or these nteractons: ˆ H = h +, 2 ˆV = ˆ H 0 + ˆV. (A The unctons σ are not egenvectors o the complete Hamltonan o the system, and t s thereore possble to observe the system n a state σ 0 and, ater a certan tme t, observe t n a derent state σ wth possbly derent

7 SYMMETRIES AND NONEQUILIBRIUM THERMODYNAMICS PHYSICAL REVIEW E 95, (207 entropy. I we assume the nteractons between subsystems to be wea n comparson to the nteractons wthn each subsystem, we can obtan an expresson or the probablty o ths event n the same way as Ferm s golden rule s derved. Accordng to Ferm s golden rule [,2], the requency o observed transtons per unt o tme between two states s descrbed by the expresson W σ0 σ = 2π h σ ˆV σ 0 2 δ ( E σ E σ0. (A2 The parameters E σ0 and E σ are the correspondng egenvalues o the operator Hˆ 0 and δ s Drac s uncton. We remnd that the unctons σ 0 and σ tae the orm o Eq. (4. We wll use the ollowng notaton: σ 0 = s 0 s, (A3 σ = s s. (A4 The coecents σ ˆV σ 0 can be computed as ollows: σ ˆV σ 0 = 2 ˆV 0 0 u u 0 u u., u, (A5 Note that u 0 u, then u u u u 0 =0. Ths means that three or more subsystems der between the ntal and nal state, all the terms n the summaton are zero and no transtons between both states are observed. In other words, the observed transtons nvolve the change o at most two subsystems, and they are drven by the nteracton between these two subsystems. We can thereore use the ollowng notaton or the transton requences: W σ0 σ = W ( 0, 0 (, = 2π h ˆV 0 2δ ( 0 Eσ E σ0. (A6 The presence o the uncton δ o Drac also mposes the ollowng energy-conservaton condton: or ε 0 + ε 0 = ε ε = ε. + ε, (A7 (A8 Note that as the subsystems have been chosen to be dentcal, the orm o the operators ˆV s the same, and only the varables on whch they act change. So, we can, n general, wrte W ( 0, 0 (, = W (r r,l l. (A9 Ths ust means that the transtons n whch a subsystem I n the state 0 = r passes to a state = r and a second subsystem n the state 0 = l passes to the state = l, occur wth the same requency ndependently o whch are the subsystems I and. Now, we have a system wth N r subsystems n the state r and N l subsystems n the state l. The total requency o (r r,l l transtons wll be T (r r,l l = N r N l W (r r,l l. APPENDIX B: TRANSITIONS BETWEEN STATES FOR NONDISTINGUISHABLE SUBSYSTEMS (A0 In ths case, the wave uncton descrbng the whole system has to be symmetrc wth respect to the permutaton o two subsystems. Even the system s composed o ermons, we can dene the subsystems as pars o ermons; thereore, we can assume symmetrc wave unctons wthout loss o generalty. Symmetrc normalzed wave unctons wll tae the ollowng orm: S = N! p(σ. (B p(σ The transton requency between two states wll be as descrbed beore: In ths case, S ˆV S 0 = W S0 S = 2π h S ˆV S 0 2 δ ( E S E S0. (B2 N N 0 p(σ,p(σ 0 p(σ ˆV p(σ 0. (B3 In the same way as n Appendx A [Eq. (A5], we can see that more than two subsystems change between the ntal and nal states, the transton requency between these states becomes zero. I the change between states nvolves a subsystem n the state r transtng to state r and a subsystem n state l transtng to state l, the summaton n Eq. (B3 taes the orm p(σ ˆV p(σ 0 where p(σ,p(σ 0 = N r N l ( (r r,l l + (r l,l r, (B4 (r r,l l = r l ˆV l r. (B5 The value o ths expresson does not depend on the partcular subsystems I and but only on the states between whch the transtons occur. Equaton (B5 accounts or all the possble combnatons o subsystems n states r and l, respectvely. As the subsystems are now undstngushable, there s no derence between a stuaton n whch a subsystem n the state r transts to state r and a subsystem n state l transts to state l or a subsystem n state r transts to state l and a subsystem n state l transts to state r. That s why both stuatons are grouped n Eq. (B4. In order to smply the expresson, we wll use the ollowng: notaton: (r r,l l + (r l,l r = rl r l. (B6 The parameter rl r l can be redened n a more compact way as ollows: rl r l = l ; r ˆV r; l, (B

8 SERGIO BORDEL PHYSICAL REVIEW E 95, (207 where r; l = 2 ( r l + l r. (B8 To smply Eq. (2, we have to notce the ollowng relatons: N r = N r0 ; N l = N l0 ; N r = N r 0 + ; N l = N l 0 +. For r,l,r,l N = N 0. Ths allows us to rewrte Eq. (B3 n the ollowng way: S ˆV S 0 = N 0 Nr0 N l0 (N r 0 + (N l 0 + rl r l. (B9 Fnally, we can obtan the transton requency between states as W S0 S = 2π h ( N 2 0 N r0n l0(n r 0 + (N l rl r l δ( E S E S0. (B0 In ths case, as the subsystems are ndstngushable, the states are ust dened by the numbers o subsystems n each state; thereore, ths value s dentcal to the requency o lr l r transtons. Thereore, or nondstngushable subsystems, the equvalent o Eq. (A0 sust T rl r l = 2π h ( N 2 0 N r0n l0(n r 0 + (N l rl r l δ( E S E S0. (B From now on we wll drop the subzero ndexes and assume that we are always reerrng to the occupaton numbers o the current state o the system. In a more compact way we can rewrte the prevous equaton usng the ollowng notaton: Ɣ rl r l = Ɣ r l rl = 2π h 2 rl r l δ( E S E S0, (B2 ( T rl r l = N 2 Ɣ rl r rn l (N r +(N l +. (B3 APPENDIX C: EQUILIBRIUM CONDITION FOR DISTINGUISHABLE SUBSYSTEMS AND BOLTZMANN S DISTRIBUTION Now we wll analyze n whch condtons the requency o each transton (r r,l l s equal to the requency o ts opposte transton (r r,l l, whch mples that the numbers o subsystems n any state do not change (n average over tme. The equlbrum condton can be expressed as ollows: T (r r,l l = T (r r,l l. From Eq. (A6, we see that by denton, W (r r,l l = W (r r,l l. (C (C2 Equaton (C2 s ust an expresson or the property o mcroscopc reversblty. The equlbrum condton then becomes ust equvalent to N r N l = N r N l, (C3 or N r N r = N l. (C4 N l By tang logarthms, we can ust rewrte the equlbrum condton as ollows: where ln N r = ln N l, ln N r = ln N r ln N r. (C5 (C6 I all the possble transtons between states satsy Eq. (C5, the system s n equlbrum. As t s stated n Eq. (A7, all the observable transtons between states satsy also the energy conservaton condton: ε r = ε l. (C7 Equatons (C5 and (C7 mply that, n equlbrum, the derence between the energy levels o two states s proportonal to the derence between the logarthms o the numbers o subsystems n these states. Ths mples that there s a lnear relaton between the logarthm o the number o subsystems n a partcular state and the energy o ths state: or ln N r = α βε r r, N r = exp(αexp( βε r. (C8 (C9 The values o the parameters α and β are xed by specyng the total number o subsystems and the total energy o the system (the sum o the observed egenvalues o the Hamltonans o each subsystem. Ths s dentcal to Boltzmann s dstrbuton: exp (α = s Z β = B T. (C0 The varable s s the total number o subsystems and Z the partton uncton o the system. As t s well nown, Boltzmann s equaton results rom maxmzng the entropy o a system by eepng the total energy constraned; thereore, we have ust shown that the state o equlbrum (the numbers o subsystems n any state do not change over tme s equvalent to a state o maxmal entropy. APPENDIX D: EQUILIBRIUM CONDITION FOR NONDISTINGUISHABLE SUBSYSTEMS AND BOSE-EINSTEIN DISTRIBUTION In the same way as we dd or the case o dstngushable subsystems, we set as equlbrum condton the equalty between the requences o transtons lr l r and l r lr. It s very mportant to remar that ths second requency s not the requency o transtons bac rom S to S 0 (whch would be the same as the prevous one but the requency o transtons rom S 0 to a state derent rom S 0 and S n whch two subsystems n the states r and l have been substtuted by two subsystems n states r and l. Equaton (B3 reveals that the equlbrum condton mples that or all the allowed transtons (satsyng the energy conservaton

9 SYMMETRIES AND NONEQUILIBRIUM THERMODYNAMICS PHYSICAL REVIEW E 95, (207 condton the ollowng relaton has to be satsed or every energy conservng transton: N r N l (N r + (N l + = N r N l (N r + (N l +. (D Usng the same notaton as n Eq. (C6, we can rewrte the prevous expresson as N r N l ln = ln N r + N l +. Ths leads us to the ollowng relaton: or alternatvely, ln N r N r + = α βε r r, (D2 (D3 N r =. (D4 e α+βε r Ths corresponds to Bose-Ensten s dstrbuton and the value o α s adusted so that the total number o subsystems s equal to the sum o the numbers o subsystems n each state. As all the degenerate states o a certan energy level have the same occupaton numbers, the total number o subsystems n a certan energy level s ust g ε N ε = e α+βε. (D5 The parameter g ε s the number o states n the correspondng energy level. Ths dstrbuton, as t s well nown, corresponds to the maxmzaton at constant energy and partcle number (n our case number o subsystems o the entropy o a system o nondstngushable partcles (subsystems, whch s dened as S = B ln ε (g ε + N ε! (g ε!n ε!. (D6 APPENDIX E: STATISTICS OF NONEQUILIBRIUM SYSTEMS The content o ths Appendx has been largely presented n Re. [6]. We start by denng thermodynamc luxes as tme dervatves o certan nonconserved observables: dφ = J. (E In ths artcle, we are gong to restrct ourselves to observables assocated to quantum operators ˆ that commute wth the Hamltonan Hˆ 0 : [ Hˆ 0, ˆ ] = 0. (E2 I ths was not the case, the tme evoluton o the observable would be due also n part drven by Hˆ 0 and not only by the nteractons between subsystems and the transtons between states that we descrbe n Eqs. (A0 and (B3. Ths also means that each state correspondng to an egenvector o Hˆ 0 has a well-dened value o the observables φ and any transton between egenvectors s assocated to a well-dened ncrement o the observables φ. We are gong to proceed n the same way as prevously, by descrbng the cases o dstngushable and nondstngushable subsystems.. Dstngushable subsystems A transton (r r,l l s assocated to the ollowng entropy change (drectly derved rom Boltzmann s denton o entropy: as N r N l S (r r,l l = B ln (N r + (N l +. (E3 For large values o the occupaton numbers, we can smply S (r r,l l B ln N rn l. (E4 N r N l The entropy ncrease rate can be obtaned as a uncton o all the transton rates: ds = B T (r r,l l ln N rn l. (E5 N r,r,l,l r N l On the other hand, usng a descrpton n terms o thermodynamc luxes and orces, the entropy producton rate wll be ds = X J. (E6 I we rewrte the thermodynamc luxes as unctons o the transtons between states, we obtan J = T (r r,l l φ (r r,l l. (E7 r,r,l,l Combnng the two prevous equatons we obtan ds = X T (r r,l l φ (r r,l l r,r,l,l = X φ (r r,l l. (E8 r,r,l,l T (r r,l l Comparng Eqs. (E5 and (E8, we can ner that n a thermodynamc system not n equlbrum and characterzed by a set o thermodynamc orces, the ollowng equalty s satsed: ln N rn l = X φ (r r N r N l,l l. (E9 B For the prevous equaton to be satsed n all the possble transtons, the occupaton numbers o the derent states should ollow the ollowng dstrbuton: N r = e α βε r + X φ B r. (E0 2. Nondstngushable subsystems In ths case, the entropy results rom the number o ways n whch a certan number o nondstngushable subsystems can be arranged wthn derent boxes contanng sets o g states. In Eq. (D6, each o these boxes corresponds to an energy level, that s why the ndex ε has been used. Here we dene

10 SERGIO BORDEL PHYSICAL REVIEW E 95, (207 the boxes as sets o states wth the same energy and the same values o the other observables φ used to descrbe the system. For ths reason, we use the more general ndex λ: S = B ln λ (g λ + N λ! (g λ!n λ!. (E Now let s consder the transton rl r l n whch the states r,r,l, and l belong to the boxes ρ,ρ,λ, and λ, respectvely. Usng Eq. (E, we can compute the entropy change assocated to ths transton: S rl r l = B ln N ρ (g ρ + N ρ N λ (g λ + N λ (N ρ + (g ρ + N ρ (N λ +(g λ + N λ. For large occupaton numbers, we can smply as S rl r l B ln N ρ(g ρ + N ρ N λ (g λ + N λ N ρ (g ρ + N ρ N λ (g λ + N λ. (E2 (E3 I all the states wthn each box have the same occupaton numbers, we can wrte N λ = g λ N l. (E4 Dong ths substtuton or each o the states nvolved n the transton, we obtan S rl r l B ln N r( + N r N l ( + N l N r ( + N r N l ( + N l. (E5 Now we ust need to proceed n an dentcal way as we dd or the case o dstngushable subsystems to obtan the dentty: ln N r( + N r N l ( + N l N r ( + N r N l ( + N l = X φ (rl r l. (E6 B As n the case o dstngushable subsystems, or the prevous equaton to be satsed n all the possble transtons, the dstrbuton o the occupaton numbers o each state should ollow the equaton N r = e α+βε r B X φ r. (E7 [] G. Lebon, D. Jou and J. Casas-Vázquez. Understandng Nonequlbrum Thermodynamcs (Sprnger, Berln, [2] I. Prgogne, Scence 20, 777 (978. [3] H. Zegler, Ing. Arch. 30, 40 (96. [4] G. P. Beretta, Phys. Rev. E 90, 0423 (204. [5] J. Hurley and C. Garrod, Phys. Rev. Lett. 48, 575 (982. [6] S. Bordel, Physca A 389, 4564 (200. [7] S. Bordel, J. Stat. Mech. (20 P0503. [8] G. P. Beretta, Found. Phys. 7, 365 (987. [9] G. P. Beretta, Rep. Math. Phys. 64, 39 (2009. [0] G. P. Beretta, Entropy 0, 60 (2008. [] M. D. Kostn, J. Chem. Phys. 57, 3589 (972. [2] I. Balyn-Brula and J. Mycels, Ann. Phys. 00, 62 (976. [3] H. D. Doebner and G. A. Goldn, Phys. Rev. A 54, 3764 (996. [4] J. J. Hallwell and J. Perez-Mercader, and W. G. Zure, Physcal Orgns o Tme Asymmetry (Cambrdge Unversty Press, Cambrdge, 996. [5] L. Maccone, Phys. Rev. Lett. 03, (2009. [6] E. Ferm, Nuclear Physcs (Unversty o Chcago Press, Chcago, 950. [7] D. N. Zubarev and V. P. Kalashnov, Physca 46, 550 (970. [8] R. C. Dewar, J. Phys. A: Math. Gen. 36, 63 (2003. [9] H. F. Jones, Groups, Representatons, and Physcs (Taylor & Francs, London,