ON THE DYNAMICS AND THERMODYNAMICS OF SMALL MARKOW-TYPE MATERIAL SYSTEMS

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1 ON THE DYNAMICS AND THERMODYNAMICS OF SMALL MARKOW-TYPE MATERIAL SYSTEMS Andrzej Trzęsows Deprmen of Theory of Connuous Med, Insue of Fundmenl Technologcl Reserch, Polsh Acdemy of Scences, Pwńsego 5B, Wrsw, Polnd E-ml: Absrc. The collecve properes of smll merl sysems consdered s semdynmcl sysems revelng he Mrov-ype rreversble evoluon, re nvesged. I s shown h hese merl sysems dm her remen s hermodynmc sysems n dherml nd soherml condons. A nec equon descrbng sscl regulres of he Mrov-ype merl sysems nd consrned by he compbly condon wh he frs nd second lws of hermodynmcs nd wh he relxon posule, s proposed. The nfluence of exernl prmeers on he Gbbs dsrbuon of smll merl sysems s dscussed. PACS clssfcon: y, Ln, G 1. Inroducon The problem of comprehensve descrpon of collecve properes of merl sysems, observed no on he mcroscopc scle bu on dfferen mesoscles, ppers n he heory of nnosrucured merls [1, 2, 3]. Prculrly, s ndspensble o clrfy he pplcbly of he conceps of hermodynmcs o merl sysems on he smll lengh scles [2]. For exmple, f we re delng wh he mcroscopc observon level scle, hen he sndrd procedure o show he exsence of hermodynmc lm, nd herefore emperure, s bsed on he de h, s he spl exenson ncreses, he surfce of regon n spce grows slower hn s volume [2]. However, s no he cse of smll merl sysems [1, 3]. The m of hs pper s o show h he mehod of descrpon of collecve properes of merl sysems, whch hs been formuled n [4] for sochsc sysems wh he counble spce of ses, cn be helpful, fer s modfcon nd generlzon, for beer undersndng of he dynmcs nd nonequlbrum hermodynmcs of smll merl sysems wh he se spce beng counble or of he crdnly of he connuum. The sysems cn be deermnsc (wh he dynmcs descrbed by semdynmcl sysems) or sochsc bu revel he Mrov-ype rreversble evoluon (Secon 2) nd dm her remen s hermodynmc sysems n dherml nd soherml condons (Secons 3-5). Noe lso h he proposed mehod of he descrpon of collecve properes of hese sysems s conssen wh he so-clled Prgogne s selecon rule for rreversble dynmcl processes (Secon 6). 2. Probblsc represenons of dynmcl sysems Le us consder merl sysem h dynmcl behvor s defned by opologcl spce (counble or of he crdnly of he connuum) of ll s dmssble ses nd by deermnsc opologcl dynmcl sysem [5] defned s connuous Abeln semgroup or S S :, T T = R =, + or group = { } of connuous rnsformons, where 1

2 [ ) T = R+ = 0, + (wh he nernl operon 1 ± 2 T for every 1 2 T ) for he group or semgroup, respecvely, nd S S S, S d,,. (2.1) s = + s 0 = s T We cn lso consder semgroup S wh T R (, 0] If x0 = = s he semgroup of prmeers. s dsngushed se of he merl sysem, clled s nl se, hen x = x x ; S x (2.2) 0 0 denoes he nsnneous se of hs sysem he nsn T. For exmple, f s df- x x ; : T s he generl soluon of he dfferenl equon ferenl mnfold nd 0 ( x) x0 x ɺ = v, x 0 =, (2.3) where xɺ = dx/d nd v s vecor feld on ngen o, hen Eq.(2.2) cn be consdered s defnon of (deermnsc) smooh dynmcl sysem genered by he dfferenl equon of Eq.(2.3) [5, 6]. We ssume h he spce s ddonlly endowed wh dsngushed σ-fne, nonnegve mesure : Γ R+ where Γ denoes σ-lgebr of subses of [6]. In prculr, f s counble se ( crd ℵ 0 ) nd s Γ s en he se of ll subses of, hen s he so-clled counng mesure on defned for A Γ by he rule: ( A) crd A for crd A<ℵ0, = for crd A =ℵ 0. Furher on mesure spce (,, ) spce s lso clled phse spce of he sysem. By L (2.4) = Γ [6] s clled he se spce of sysem. The we wll denoe he lner Bnch spce of -mesurble funcons f : R such h [6]: Prculrly, f s he counng mesure, hen Noe h L d f = f x x <. (2.5) x f = f x <. (2.6) s he so-clled Bnch lgebr wh respec o he nernl ponwse mulplcon of funcons belongng o hs spce [6]. Now, he semgroup (or group) S of rnsformons should conss of mesurble rnsformons [5], h s he so-clled condon of double mesuremen of hese rnsformons should be fulflled [7]: ( 1 ) A Γ, T, S A,S A Γ. (2.7) In he lerure deermnsc (nd opologcl or smooh) dynmcl sysem S= S :, T fulfllng he condon of double mesuremen s frequenly { } clled semdynmcl sysem f s semgroup, h s he condon (2.1) wh T = R+ (or T = R ) s fulflled [7]. ` 2

3 The mos promnen exmple of dynmcl sysems genered by dfferenl equons nd fulfllng he bove condon of double mesuremen s he Hmlonn dynmcl sysem descrbng he dynmcs of merl sysem conssng of N dencl prcles. In hs 2 R n cse =, n= 3N, s he Lebesgue mesure, nd Eq.(2.3) wh v ( x) = J H( x) x 0 I J=, I= dg(1, 1,...,1) GL ( n,r ), I 0 where GL ( n,r) s he group of nonsngulr n n, (2.8) rel mrces, H : R s he Hmlonn of he sysem nd x denoes he grden operor wh respec o vrbles x, s consdered [8]. Le S denoe he Hmlonn dynmcl sysem defned by Eqs. (2.1) - (2.3) nd (2.8). I cn be shown h S s he one-prmeer group of rnsformons of he spce 2 R n =, clled cnoncl rnsformons, h preserve he Lebesgue mesure (h s preserve he volume n ) [9, 10]. If he nl se x 0 of hs merl sysem s nown up o he probbly of s loclzon n subse A Γ : ( ) = P x0 A P A p x d x, p L, p 0, P = 1, A (2.9) hen he probbly h n nsnneous se of he sysem s loclzed he nsn T n he se A cn be defned s: ( 0 ) -1 ( ) = = P x A P A p x d x P x S A. (2.10) I resuls from he defnon of cnoncl rnsformons, h A -1 p = p S D, (2.11) where ws denoed: D = p L : p 0, p( x) d ( x) = 1. (2.12) I follows h he consdered Hmlonn dynmcl sysem generes semgroup U = U : D D, T of rnsformons cng ccordng o he rule: { } 1 U p= p S, p= p0, U0 = d D, exensble o he lner mppng U : (2.13) L L nd such h he probblsc denses p, T, of Eq.(2.11) fulfll he so-clled Louvlle equon: p( x, ) = Lp( x, ), = = p x, p x, p x, 0 p x, where L s he Louvlle operor cng ccordng o he rule (2.14) 3

4 { } { } Lf = f, H = H, f, f, H C, 1, (2.15) nd {, } denoes he Posson brces [9]. Noe h snce n order o solve he Louvlle equon we ough o nown, n generl, soluon of he Hmlonn equons, s he problem unrelzble for mcroscopc sysems. Consequenly, n he sscl physcs re consdered pproxme soluons of he Louvlle equon h descrbe sscl regulres of he Hmlonn sysem [9]. If we re delng wh he orened n me evoluon of sochsc merl sysem wh he se spce, hen he rndomness of nsnneous ses of he sysem cn be descrbed by fmly x = { x : T} of mppngs T ( Ω ) [ ] P x :, P, T, P : Ω 0, 1, Ω = 1, (2.16) where ( Ω, P) denoes probblsc spce of elemenry evens nd T ( = R + or R ) s n one-prmeer, ddve nd Abeln semgroup. If ( ω ( ω ) ) P x A P Ω : x A = p x d x, (2.17) A hen we cn ssume he exsence of n Abeln semgroup U { U, T} 0 = of mppngs U : D D, U = d, (2.18) L such h f x 0 s rndom vrble wh probbly densy funcon p of Eq.(2.9), hen T, U p= p, (2.19) nd h he mppngs re exensble o he lner operors U : L L. I ough o be sressed h he wy n whch he probblsc represenon of dynmcl sysem s nroduced depends on he nd of rndomness ssoced wh he dynmcs. For exmple, n he cse of deermnsc dynmcl sysem of Eq.(2.3), he only wy o nroduce s probblsc represenon s he rndomness of s nl condons. Anoher suon es plce e.g. n qunum sysems, n whch he rndomness cn be reled wh he Hesenberg uncerny relon. More generlly, we cn consder semgroup U of lner operors n L fulfllng he condon (2.18) bu no necessrly genered by he fmly x T of rndom vrbles. If hese lner operors re he so-clled Mrov operors, h s: ( ( )) f L, f 0 T, U f 0 nd U f = f, (2.20) hen U s clled sochsc semgroup [7]. I follows from he condon (2.20) h should be [7]: f L, T, U f f. (2.21) Thus, he Mrov operors re conrcons. Le D densy, defned by [7]: π be he so-clled sonry T, U π = π. (2.22) ` 4

5 A sochsc semgroup s clled sympoclly sble f here exss excly one sonry densy π such h [7] p D, lm U p = 0. (2.23) π For exmple, le us consder fmly K { K : R, T} = of he so-clled sochsc ernels ssfyng, for every s, T nd lmos everywhere on, he so-clled Chpmn-Kolmogorov equon [7, 11]: K x, y K x, z K z, y d z, + s = s K x, y 0, K x, y d y = 1. Gven K we cn defne sochsc semgroup by seng for ny f L Noe h for every 0, T, > 0, we hve [7]: : (2.24) U f x = K x, y f y d y. (2.25) 0 0 = p D, U p x h x, h x nf K x, z, x. z I cn be shown [7] h f K s fmly of sochsc ernels such h 0 (2.26) h0 x d x > 0 (2.27) for some 0 T, hen he semgroup defned by Eq.(2.25) s sympoclly sble. The probblsc nerpreon of he bove sochsc semgroup s defned by Eqs.(2.16)-(2.19) nd (2.25). For exmple, s he cse of he so-clled Mrov chns (.e. Mrov processes wh he counble se spces [11]) wh he connuous me whch hs been consdered n [4]. Frher on sochsc semgroup genered by fmly of sochsc ernels s clled he Chpmn-Kolmogorov semgroup. Sochsc semgroups correspondng o he rndomness of nsnneous ses of merl sysems (Eqs.(2.16)-(2.20)) pper mnly n pure probblsc problems such s rndom wls, sochsc dfferenl equons nd mny ohers (e.g. n he problem of Mrovn descrpon of collecve properes of sysems wh he counble se spce [4]). However, hey cn ll be genered lso by deermnsc semdynmcl sysems [7]. A semdynmcl sysem s clled nonsngulr f n ddon ( ) 1 ( ) A Γ, A = 0 T, S A = S A = 0. (2.28) { } For ny nonsngulr semdynmcl sysem we cn unvoclly defne he sochsc sem- U = U : L L, T ssumng h for every mesurble se A, group S we hve (cf. Eqs.(2.9) nd (2.10)):,, U d d. (2.29) f L T f x x = f x x A 1 S ( A) 5

6 The such defned sochsc semgroup fulflls ddonlly he followng condon: where supp f { x : f ( x) 0} A= supp f, hen U f ( x ) = 0 for x ( A) ( f) S ( f) supp U supp, (2.30) = s he suppor of f [7]. I follows from Eq.(2.30) h f [7]. A nonsngulr sem-dynmcl sysem S s S clled ssclly sble f he correspondng sochsc semgroup s sympoclly sble. The behvor of U llows o deermne mny properes of he semdynmcl sysem S. S For exmple, le us consder he problem of he exsence of mesure 0 nvrn under S, h s such h [7] ( S 1 ( A) ) ( A) = (2.31) 0 0 for every mesurble se A nd T (s e.g. n he cse of Hmlonn dynmcl sysems). Assume now h mesure 0 s normlzed ( 0 = 1) nd nvrn under S. The pr ( S ), s clled exc f for every mesurble se A he followng condon 0 s fulflled: A > 0 lm S A = 1. (2.32) ( ) 0 0 Le S be nonsngulr semdynmcl sysem S nd le ssoced wh s. If f L s nvrn under S f nd only f, hen he mesure U S denoes he sochsc semgroup f A = f x d x, A Γ, (2.33) A T, U f = f, (2.34) where Eq.(2.29) ws en no ccoun [7]. Moreover, f he semgroup U S s sympoclly sble, f D s s unque sonry densy nd f s he mesure gven by Eq.(2.33), hen he pr (, S f ) s exc nd f s he unque bsoluely connuous normlzed (nonnegve) mesure nvrn under S [7]. The probblsc nerpreon of nonsngulr semdynmcl sysem S cn be formuled f he correspondng sochsc semgroup U s conssen wh sochsc process defned by Eqs. (2.16)-(2.19). I cn be e.g. he cse of Mrov processes [11]. We wll ssume ddonlly, generlzng he cse of Mrov chns [4, 11], he exsence nd fne- W x, y 0, x y, from he se x o he se y ness of he so-clled rnson nenses nd he so-clled ex nenses from he ses x of he sysem: Then, he probbly densy (, ) S w x = W x, y d y > 0. (2.35) p defned by Eqs.(2.19) cn be ssumed n he form of p x = p x where p : T R+ s soluon of he followng verson of he so-clled Kolmogorov equon ( see e.g. [11]): ` 6

7 p x, = w x p x, + W y, x p y, d y, p( x,0) = p x. (2.36) Ths equon hs he physcl menng of nec equon h defnes he probblsc represenon of he Mrov-ype evoluon of merl sysem on he bss of he blnce of he nenses of rechng nd levng he ses of hs sysem. Therefore cn be nmed he Kolmogorov s nec equon. Noe h he quny 1 τ ( x) = < (2.37) w x cn be nerpreed s he men resdence me of he Mrov-ype evoluon process n he se x [4]. Le us denoe by L S he lner operor (n generl unbounded) n he Bnch lgebr L defned s L f x = w x f x + W y, x f y d y. (2.38) S Frequenly, we hve o e no ccoun consrns resrcng he se spce [1] s well s concernng he evoluon of he sysem. For exmple, s esy o observe h f he ex nenses w re commonly bounded : w > 0, x, w x w, (2.39) hen he operor L S s bounded: 0 0 L = sup L f 2 w. (2.40) S f 1 where Eq.(2.5) ws en no ccoun. I follows from Eq.(2.36) h hen he sochsc semgroup U consss of operors of he exponenl form [10]: S s n= n S S S ń= 0 n! 0 n 0 U = exp ( L ) L, L I, (2.41) where If = f for f L. The exsence of he nec equons of he form (2.36) s n mporn fc from he pon of vew of he physcl pplcons of sochsc semgroups o he descrpon of rreversble processes. I follows from he followng form of rnson probbles of he Mrov processes governed by such equons [4, 11]: ( + h = = ) = ( h = 0 = ) = + ( + h = ) = ( h 0 = ) = +, P x y x x P x y x x W x, y h o h, P x y x x P x y x x w x h o h (2.42) where o( h) / h 0 for 0 h, unformly wh respec o x for gven y, x y. Hence n hs cse he descrpon of he rreversble evoluon of he sysem cn be reduced o he nvesgon of s behvor for shor me perods, h s o he formulon of he 7

8 physcl hypohess bou he form of he rnson probbles. For exmple, follows from Eqs.(2.39) nd (2.42) h, ndependenly of he choce of x, should be ( ( h 0 ) 0 ) y, y x P x y x = x w h+ o h. (2.43) Fnlly, we cn consder deermnsc merl sysems wh he dynmcs descrbed by nonsngulr semdynmcl sysems s well s sochsc merl sysems [4, 11] defned by he condon h he correspondng sochsc semgroups re genered by he Kolmogorov s nec equon (2.36). The merl sysems revel hen he Mrov-ype rreversble evoluon. Consequenly, one cn sy h such merl sysems nd such sochsc semgroups re he Mrov-ype. 3. Sonry ses Le us consder merl sysem wh he se spce nd such h every se x hs s own energy E x R+ ndependen of he dynmcs of he sysem ( s he so-clled nernl energy of he sysem n he se x). The dynmcs of he merl sysem s descrbed by nonsngulr semdynmcl sysem cng n he se spce (see remrs prevous o Eq.(2.28)). We cn nroduce now he funconl E D : R h defnes + he men nernl energy of he merl sysem ccordng o he followng rule: p D, E p = e x p x d x, e : R, e x = E for x, + x (3.1) where Eq.(2.12) ws en no ccoun, nd we cn dsngush he clss of probblsc mesures gvng he some vlue E of he men nernl energy: { } D, = p D : E p = E. (3.2) E Snce he se spce cn be counble or of he crdnly of he connuum, he dsrbuon of energy e s dmed o be dscree or connuous funcon of he vrble x, respecvely. Though one cn dp he presened descrpon of he collecve properes of merl sysems o he descrpon of mcroscopc sysems, however frs of ll wll neres us smll sysems (see Secon 1 nd he remrs followng Eq.(2.15)). For exmple, le he sysem consss of he fne number N of dencl prcles, he se spce s counble nd x E p = E p <, x x p = 1, p = p x 0, E 0, x x x x (3.3) where Eqs.(2.4)-(2.6) nd (3.1) were en no ccoun. Le us denoe by n x he (fne) number of prcles beng n he se x endowed wh he nernl energy E x. In he clsscl sscl physcs, he probbly p of Eq.(3.3) h prcle of he sysem hs x he nernl energy E x n he se x, s ssumed, under cern physcl condons [9, 12], n he followng form: nx px =, N nx, N = (3.4) x ` 8

9 nd s pproxmed, n he so-clled hermodynmc lm [9, 12], by p x nx = lm. N N (3.5) The pproxmon s he beer, he greer N s. However, he condon (3.5) cn no be cceped n he cse of smll merl sysems. In he sscl physcs s consdered he so-clled Bolzmnn enropy funconl S : D R defned by: where ( ) p D, S p = s p x d x, (3.6) s ( z) Bz ln z for z> 0, = 0 for z= 0, (3.7) nd B s he Bolzmnn consn. Ths funconl s reed, n he hermodynmc lm, s mesure of he sscl nformon concernng he energec ses of mcroscopc sysems [9, 12]. Neverheless, cn be cceped lso s mesure of uncerny n he sscl descrpon of processes n mcroscopc bodes [9]. Ths mesure of nformon es s mxmum vlue conssen wh he fxed vlue of he men energy: ( π) π D, p D, S p S, (3.8), E, E on he so clled Gbbs dsrbuon π of he followng form: π ( x) 1 β Ex = Z exp, B β Ex Z = exp d ( x) <, B (3.9) where β > 0 s consn. For suffcenly smooh dsrbuons, equly n Eq.(3.8) mples h p= π lmos everywhere on. I should be sressed h he Gbbs dsrbuon s, n he frmewor of clsscl sscl physcs, formul vld only n he hermodynmc lm, p = π x. The probblsc h s should be undersood hen n he sense of Eq.(3.5) wh represenon (, π ) defned n hs wy s clled cnoncl ensemble (for he counble se spce) [9, 12]. Denong we obn he followng relon: 1 ( π) θ θ β F = E ln Z, E =, =, (3.10) B B B ( π) ( π) θ ( π), F = E S (3.11) where, E( π) E s fxed men nernl energy of he sysem, x S π s he mxml enropy bsolue hermodynmc emperure of he sysem nd he quny correspondng o E nd defned by Eqs.(3.6), (3.7) nd (3.9), nd he sclr θ > 0 defnes he chrcersc energy E B of he sysem (Eq.(3.10)). If s mcroscopc nd closed sysem (.e. he merl sysem s energeclly soled), hen he sclr θ cn be denfed wh he F π cn be recognzed s he free energy of he sysem [9]. Moreover, for closed sysem, he se of ss- 9

10 cl equlbrum (.e. he condon h he dsrbuon π s ndependen of me) covers wh he se of hermodynmc equlbrum [9]. If he sysem s no closed, hen s ses cn be dependen on he emperure of he envronmen [9]. Prculrly, cn be he cse of he sysem (mcroscopc or mcroscopc) wh hermlly conducng boundry (clled dherml boundry), dmng hermlly cved processes nd wh s envronmen beng hermos. In hs cse we cn ssume F π es hen he physcl h θ covers wh he emperure of he envronmen [4]. menng of generlzed free energy correspondng o he Gbbs dsrbuon nd hs dsrbuon cn descrbe, n generl, sonry nonequlbrum se of smll merl sysem (see remrs followng Eqs.(3.7) nd (3.11)). We wll cll hs se Gbbs se. If we re delng wh Mrov-ype merl sysem, sy wh he dynmcs defned by nonsngulr semdynmcl (see remrs he very end of Secon 2), hen follows from Eqs.(2.35), (3.9), (3.10) nd he condon = 0, (3.12) p h he Gbbs dsrbuon fulflls he Kolmogorov s nec equon (2.36) f he followng nlogue of he so-clled condon of mcroscopc reversbly (clled lso he condon of deled blnce [12]) s ssfed: ( π π ) x, y, x y W x, y > 0, x W x, y = y W y, x. (3.13) Ths condon s fulflled f he rnson nenses W( x, y ) re of he form: E W( x, y) q( x, y) exp x =, E q( y x) B q x, y =, > 0 for x y. The formul (3.14) cn be wren, whou losng generly, n he form: (3.14) U xy W( x, y) = ν exp, EB where ν > 0 s consn wh he dmenson of frequency, nd ws denoed The ro xy (3.15) U = E E, E = E 0. (3.16) xy xy x xy yx (, ) (, ) ( y) Ex exp B W x y π Ey = = = W y x π x E (3.17) defnes he co-clled equlbrum consns consdered n he cse of mcroscopc equlbrum Gbbs ses [12]. I follows from Eqs.(2.4) nd (3.9) h he cse = 1 for rbrry x, y cn e plce only for he fne se spce wh he unform Gbbs dsrbuon, h s f: 1 1 x, π x = Z =, N = crd. (3.18) N The formul (3.15) hs he form of he well-nown lw descrbng he frequency of he rnson x y n he heory of recon dynmcs nd s ppled, for exmple, o he descrpon of he hermlly cved processes [13, 14]. Bsng on hs observon, we cn nerpre E xy s he energy brrer beween he ses x nd y, wh own energes E x xy ` 10

11 nd E, respecvely, wheres y U xy cn be nerpreed s he cvon energy of he chnge of ses of he sysem ledng from x o y [4]. Then he consn ν of Eq.(3.15) hs he menng of he effecve frequency of effors o overcome he energy brrer [13, 14]. Noe h he men resdence me τ n he Gbbs se s gven by τ = τ π d, (3.19) ( x) ( x) ( x) where Eq.(2.37) ws en no ccoun. Ifτ <, hen we cn ssume h [4] nd, for he suffcenly lrge me τ, sy e.g. for 1 ν = (3.20) τ 1 τ τ0 = (3.21) w n he cse of Eq.(2.39), hs nonequlbrum sonry se cn be consdered s mesble se. For exmple, f we re delng wh smll nnosrucured clusers, cn be ssoced wh he phenomenon of he exsence of opmum sze nd shpe ledng o he mos sble pcng of her oms [1] Gbbs dsrbuon nd exernl prmeers Le us consder merl nonsngulr semdynmcl sysem wh he se spce nd such h every se x hs own energy E x R+ ndependen on he dynmcs of he sysem (Secon 3). These energes cn be dependen on fne se of prmeers beng exernl wh respec o he sysem of prcles under consderon [9]. For exmple f we re delng wh sysem of merl prcles conned n hree-dmensonl convex fg- = V,F,M of ure B (consung e.g. nnocluser [1]), hen we cn consder he rple exernl prmeers, where V s he volume of B, F s he surfce feld of he boundry B of B nd M s he men curvure of hs boundry [1]. If we re delng n wo- = F,M, where dmensonl convex fgure B (e.g. grphene smll cluser), hen he pr F s he re of B, M = ( π / 2) L s he men curvure of B, nd L s he permeer of B [15], cn be en no ccoun. Noe h n clsscl hermodynmcs he prmeer = V s frequenly dscussed [9, 16]. The correspondng Gbbs dsrbuon π (Secon 3) depends hen on hese prmeers, h s we hve [9] (, ) ( ) 1 Ex π(, θ, x) = Z (, θ) exp, EB = Bθ, x, EB e x Z (, θ ) = exp d ( x) <, e(, x) = Ex( ), EB (4.1) where θ R+ s n bsolue emperure nd B denoes he Bolzmnn consn. In sscl physcs re consdered he followng equpoenl ses of energy: 11

12 1 ( ε) ( ε) Σ = e, e : R, x, e x = e, x. + (4.2) I s usully dmed h he mesure nduces such mesures, ε on hese hypersurfces n he se spce h he generlzed volume ( ε) ( ε) Ω, = volσ > 0, (4.3) cn be defned s suffcenly smooh funcon of he prmeers nd ε. I follows from Eqs.(4.1)-(4.3) h he densy funcon f ( ε, θ ) of condonl probbly of he dsrbuon of energy ε (wh nd θ eepng consn) [9]: (, ) (, ) Z (, ) 1 exp ε f ε θ =Ω ε θ, Bθ (4.4) descrbes hen he dsrbuon of vlues of own energes of ses of sysem n hermos F θ, by [9, 16]. Inroducng he generlzed free energy of he sysem ( θ) ( θ) F, = E ln Z,, (4.5) B nd ng no ccoun Eqs.(4.1) nd (4.2), we cn wre he densy funcon f of Eq.(4.4) n he followng form [9]: (, ) F θ ε f ( ε, θ) =Ω( ε, ) exp. EB Noe h n he sscl physcs s consdered lso he condonl free energy F( ε, θ ) of sysem n he hermos defned by he relon [9]: f ( ε θ) (, θ) F(, θ ε) F, = exp, EB or, ccordng o Eq.(4.6), by he followng formul: I follows from Eqs.(4.7) nd (4.8) h [9] B B B (4.6) (4.7) F, θ ε = ε E ln Ω, ε, E = θ. (4.8) ( ε θ) = ( ε θ) ( θ ε ) ( θ ε ) f 1, f 2, exp F, 2 F, 1. (4.9) We see h more probble se corresponds o smller vlue of he condonl free energy., S θ, of he sysem s: Noe h nroducng he men energy E( θ ) nd he enropy ( θ) = π( θ ) E, e, x,, x d x, ( θ) = ( π( θ )) S, s,, x d x, (4.10) ` 12

13 where ws denoed s z Bz ln z for z> 0 = 0 for z= 0, we obn he well-nown hermodynmc relon: ( θ) ( θ) θ ( θ) (4.11) F, = E, S,. (4.12) Consequenly, (nonsngulr) semdynmcl sysem endowed wh dsrbuon of own energes of s ses cn be consdered s hermodynmc sysem defned by Eqs.(4.1) nd (4.10)-(4.12). The sonry ses of such defned hermodynmc sysem re Gbbs ses S, ε of (Secon 3). Eqs.(4.8) nd (4.10)-(4.12) sugges o defne he condonl enropy he sysem s [16] (cf. [9]): ( ε) ( ε) S, = ln Ω,. (4.13) B The fundmenl equon of hermodynmcs of reversble qus-sc rnsformons (of sysem embedded n hermos) s he followng [17]: where ( ; 1,2,... n) de= θds A d, = 1,2,... n, (4.14) = = s he se of exernl prmeers (clled lso generlzed coordnes), A, = 1,2,... n, re he correspondng generlzed hermodynmc forces nd he relon dq= θd S, (4.15) where Q s he so-clled heng beng quny descrbng he herml nfluence of hermos on he sysem, s en no ccoun. Le us rewre Eq.(4.14) n he followng form: I follows from Eq.(4.10) h should be: where ws denoed de= dq d A, da= A d. (4.16) ( θ) = α π θ + e( x), θ ( π( θ x) ) ( x) d E,, x d,, x d x, d,, d, e π π α(, x) = (, x), d, d d. θπ = + θ θ Defnng he generlzed hermodynmc forces A s [9]: ( θ) α π( θ ) (4.17) (4.18) A, =, x,, x d x, (4.19) nd comprng Eq.(4.16) wh Eqs.(4.17)-(4.19), we obn he followng represenon of he chnge of heng: ( θ) ( π( θ )) d Q, = e, x d,, x d x. (4.20), θ 13

14 Therefore, dq s defned by he chnge of Gbbs dsrbuon due o he chnge of hermodynmc vrbles nd θ. Noe h follows from Eqs.(4.1), (4.5), (4.18) nd (4.19) h [9] F A (, θ) = (, θ), = 1,2,... n. (4.21) Moreover cn be shown h he dspersons of generlzed hermodynmc forces re gven by he followng formule: A α αα j A Aj = α A α j Aj = E j j B. (4.22) Therefore, n generl, he dspersons of generlzed hermodynmc forces re no defned by hermodynmc funcons only. However, s no e.g. he cse of he followng nernl energy densy funcon e: e, x = e x A, (4.23) frequenly consdered n he cse of sysems nercng wh exernl felds [9]. In hs cse he vrbles re clled felds nd correspondng generlzed hermodynmc forces A re clled polrzons. In he sysem we cn consder smulneously felds s well s nonfeld vrbles (such s e.g. he volume of he sysem). I follows from Eq.(4.23) h n hs cse: A ( α A)( α A ) = EB. j j j (4.24) Treng he enropy S of Eq.(4.14) s funcon of ndependen vrbles E nd obn h [17] n = 1 E,, we 1 1 S S ds= de+ A d = de+ d, θ θ E (4.25) where he symbol z ndces h for he prl dfferenon one should hold consn he vrbles z, nd hus he followng equons hold: 1 S S =, A = θ. θ E E, (4.26) We see h, from hermodynmc pon of vew, he only requremen for he exsence of posve bsolue emperure θ s h he enropy S should be resrced o monoonclly ncresng funcon of he nernl energy E. Noe h f we re delng wh he condonl enropy of Eq.(4.13), hen S ε Ω = Ω B, ε where Ω (, ε) dε, cons. of N prcles wh her own energes conned n he nervl [ ε, ε dε] (4.27) =, cn be nerpreed s number of dmssble ses of sysem + [16]. Ths number ` 14

15 monoonclly ncreses f ε ncreses,.e. we re delng wh ses of he sysems wh no upper lm o he own energes of hese ses, e.g. for he nec energy of gs molecule [16, 18] or n he cse of hrmonc oscllor [16]. Thus, for θ R+, he enropy of he sysem s well s s condonl enropy re monoonclly ncresng funcons of he nernl energy of he sysem. Neverheless, some very peculr sysems, h hve energec upper lms o her llowed ses, re consdered [16, 18]. The descrpon of such sysems n he frmewor of sscl physcs bsed on he exsence of Gbbs dsrbuon, needs o nroduce negve bsolue emperure [18]. In hs cse, ccordng o Eqs.(4.26) nd (4.27), he enropy of hermodynmc sysem s no monoonclly ncresng funcon of s nernl energy. I ough o be sressed h he ssumpon relng o he sgn of he bsolue emperure s no explcly mde n hermodynmcs. I s becuse such n ssumpon s no necessry n he dervon of mny hermodynmc heorems [18]. I esy o see h f R nd he densy f of condonl probbly of Eq.(4.6) hs θ + n exremum, h s here exss ε = εm such h [9, 16]: f ε, θ ( ε θ) m, = 0, (4.28) or, equvlenly, f F ε ( θ ε ), = 0, hen, ccordng o Eq.(4.8), should be ε ε ( θ) Ω ε nd hus f he generlzed volume (, ε) m m, m = where : 1, m, m, θ ( ε ) = Ω( ε ) B Ω, = cons., ncreses f ε ncreses: (4.29) (4.30) Ω ε R +, (, ε) > 0, ε (4.31) henθ > 0. Consequenly, ccordng o Eq.(4.27), n hs cse he emperure θ s posve sclr ff he condonl enropy S of Eq.(4.13) s monoonc funcon of he own energy ε, h s S ε R +, (, ε) > 0. ε (4.32) The condonl probbly funcon f hs he mxmum for he own energy ε = ε ( θ) of Eq.(4.30) f m, 2 f 2 ε, θ ( θ ε ), < 0, m (4.33) or, ccordng o Eqs.(4.7) nd (4.29), f: 2 F 2 ε, θ ( θ ε ), > 0. m (4.34) 15

16 I follows from Eqs.(4.12) nd (4.30) h he nequly of Eq.(4.34) s equvlen o he followng condon: 2 Ω ε 1, ε < Ω, ε, E = θ. 2 m 2 m B B E B (4.35) Le us reurn o Eq.(4.6) (wh θ beng posve hermodynmc emperure). I follows from Eqs.(4.5) nd (4.6) h he men nernl energy ε of he sysem s gven by [9]: 1 2 ln Z ε = Z (, θ) ε exp ( ε / θ) Ω (, ε) d ε = θ (, θ), θ R B B (4.36) + nd he second momen of he nernl energy s gven by: (, ) 2 2 F θ ε ε = ε exp (, ε) dε R Ω = + Bθ 2 1 ε = Bθ Z (, θ) ε exp (, ε) dε θ Ω = (4.37) θ R+ B ε 2 = Bθ Z (, θ) ( ε Z )(, θ) = Bθ (, θ) + ε (, θ). θ θ Thus, he dsperson of nernl energy s gven by he followng formul [9]: ε ε ε = ( ε ε) = B θ (, θ). θ (4.38) We see h he dsperson of own energes of ses of nonsngulr semdynmcl sysem re defned, n he cse of sonry Gbbs ses of hs sysem, by hermodynmc funcons [9]. Le us consder, s n exmple, he cse when s he volume of sysem conssng of N prcles. Then [9] where = V, (4.39) ( ε ε) 2 θ 2 C ( θ) = (4.40) B V, C V s he so-clled he cpcy he consn volume gven by: Q E CV( θ) = ( V, θ) = ( V, θ). θ θ If he consdered sysem consss of n del gs of morphous prcles, hen [9] nd V V (4.41) ε = N 3 Bθ, CV( θ) = C 3 V = N B, (4.42) δε 2 1 =, δ ε = ε ε. (4.43) ε 3 N ` 16

17 Therefore, for mcroscopc bodes, when N 1, relve flucuons of he energy re very smll nd n he hermodynmc lm: N N, V, = cons., (4.44) V equls zero [9]. Ths concluson s lso vld for rel mcroscopc sysems of muully nercng prcles. Nmely, f he number N of prcles s suffcenly lrge, hen ε N nd ( ε) 2 δ 1, (4.45) ε N for fne bu suffcenly lrge rel sysems of dencl prcles [9]. Thus, n he cse of smll merl sysems, he exsence of fne relve flucuons of he energy ough o be en no ccoun. 5. Irreversble processes Le us consder Mrov-ype merl sysem wh s evoluon governed by nonsngulr semdynmcl sysem S (Secon 2) nd dmng s remen s smll hermodynmc sysem wh dherml boundry (Secon 3). Le be he se spce of hs merl { } rjecory of he Mrov-ype semgroup sysem. A (sonry nd nonequlbrum) Gbbs se (beng perhps mesble se) of hs sysem s descrbed by Eqs.(3.1), (3.2), (3.6)-(3.11) nd by Eqs.(2.35), (2.37), (3.15), (3.19), (3.20). I ough o be sressed h he presened here pproch cn be lso ppled n he cse of sochsc Mrov processes (see Secon 2 nd [4]). We cn now defne he hermodynmclly permed Mrov-ype rreversble process of he evoluon of he merl sysem. Frs of ll, such process should be conssen wh he second lw of hermodynmcs. Ths condon cn be formuled n he followng wy. Le us clcule he men nernl energy nd he Bolzmnn enropy (Secon 3) long U = U : D D, T correspondng o S (Secon 2), h s defned by Eqs.(2.18), (2.19), (2.35),(2.36) nd (3.15): where p D S T, E = E U p, S = S U p, (5.1) s n nl sonry dsrbuon. Snce he envronmen of he consdered sysem s hermos (Secon 3), we cn consder nonequlbrum ses of hs sysem wh s consn emperure θ > 0 defned by hs hermos. Ths mes possble he exenson of he defnon (3.11) of he generlzed free energy (correspondng o he Gbbs se) on he cse nonequlbrum soherml processes: T, F = F U p, EB p D, F( p) = E( p) θ S( p), θ =, B (5.2) where E B s chrcersc energy of he sysem ssoced wh he consdered herml phenomen. Noe h he smlr defnon of he nonequlbrum free energy s formuled n order o descrbe rnslonl Brownn moon n n equlbrum medum reed s hermos [9]. I follows from Eqs.(5.1) nd (5.2) h θ 1 ds = de d F, (5.3) 17

18 where ws denoed dh= h ɺ d nd h ɺ = d h / d for dfferenble funcon h : T R. The Bolzmnn enropy ncremen δ e S due o he nercon of he sysem wh s envronmen s gven by: θ 1 δ = δ Q, (5.4) e S where δq s he he ncremen. So, he Bolzmnn enropy ncremen δ S due o he exsence of rreversble processes n he sysem cn be clculed from Eqs (5.3) nd (5.4): 1 δ S = ds δ S = df+ de δ Q. (5.5) e θ The consdered Mrov-ype process wll be conssen he second lw hermodynmcs f nd only f δs 0. (5.6) The nercon of he sysem wh s envronmen hs only he herml chrcer (Secon 3) f nd only f he frs lw of hermodynmcs es he followng form: de= δ Q. (5.7) I follows from Eqs.(5.2) nd (5.5)-(5.7) h he consdered Mrov-ype rreversble processes cn be reed, n he dherml nd soherml condons, s hermodynmclly permed f nd only f he free energy funconl s non-ncresng long he rjecores of he correspondng Mrov-ype semgroups, h s T, Fɺ 0. (5.8) These rjecores re defned by Eq.(2.19), by he Kolmogorov s nec equon (2.36) wh he rnson nenses gven by Eq.(3.15) (nd perhps ddonlly by Eqs.(3.19) nd (3.20)),nd by he rule: x, p x = p x,, T = R +. (5.9) If he sochsc semgroup s he Chpmn-Kolmogorov semgroup, hen Eq.(2.25) should be ddonlly en no ccoun. I s e.g. he cse of Mrov chns [4, 11]. I seems physclly resonble o dsngush he clss of hermodynmclly permed Mrov-ype rreversble processes conssen wh he exsence of Gbbs ses. I cn be formuled s relxon posule sng h he Mrov-ype rreversble processes relx, ndependenly of he choce of he nl condon, o he unvoclly defned Gbbs se (beng perhps mesble se). Noe h snce he Gbbs dsrbuon π of Eq.(3.9) fulfls W x, y gven by Eq.(3.15) nd p= π, he condon (2.22) s denclly Eqs.(2.36) wh fulflled. Moreover, ccordng o hs posule, he condon (2.23) of sympocl sbly should be ssfed. For exmple, n he cse of he Chpmn-Kolmogorov semgroup (Secon 2), he condon (2.22) mens h he should be: π = π K x, y y d y x, (5.10) for every x nd T, nd ddonlly e.g. he condon (2.27) should be fulflled. Ths relxon posule should be reed s he ddonl hermodynmc posule h defnes he noon of Gbbs ses more precsely (cf. [4] nd [19]). I ough o be sressed h n mny cses s dffcul o prove h he probblsc represenon of process n he se spce fulflls he relxon posule. For exmple, he Hmlonn sysem perms he Gbbs se becuse he Gbbs dsrbuon fulfls he Louvlle equon (2.14). Bu hs sys- ` 18

19 em does no relx o he Gbbs se. The relxon posule s fulflled e.g. n he cse of hermodynmclly permed Mrov chns wh he connuous me nd he fne se spce [4, 11] (see Secon 2). 6. Fnl remrs Le us observe h he noon of hermodynmc rreversbly s bsed on he endowng of he me xs wh he dsngushed forwrd orenon. In he clsscl pproch, bsed on he Hmlonn mcrose dynmcs (see remrs n Secons 2 nd 3), he me s consdered s orened only when he collecve ( mcroscopc or mesoscopc ) properes of merl sysems re descrbed, nd remns non-orened when he merl sysem s nlyzed on he mcrolevel ( s becuse he conservve Hmlonn dynmcs does no dsngush ny drecon of me). Ths duly n he remen of he me s elmned f we re delng wh he Mrov-ype evoluon process of he merl sysem (Secons 2 nd 5). Noe h hs pproch s conssen wh he so-clled Prgogne s selecon rule, ccordng o whch only hese probblsc represenons of dynmcs (sy e.g. Mrov processes) h re dreced forwrd descrbe physclly relzble ses [20]. Prgogne ssumes ddonlly, h hs selecon rule cnno be derved from dynmcs n he sense, h s no reled wh he exsence of ny new nercons, no ye en no ccoun. In hs pproch, he me symmery exhbs self on he mcrolevel n he form of nernl rndom nure of he sysem (h s, ndependenly of ny hdden vrbles). Accordng o he Prgogne s pon of vew, he me symmery should be unversl, h s should e plce n ll dynmcl heores: n he clsscl mechncs s well s n he qunum mechncs (nd n relvsc heores). Ths pon of vew seems mporn n he cse of nnosrucured clusers (e.g. n he cse of nnoclusers dscussed n [1]) h re no so lrge s o be compleely free of qunum effecs nd hus, hey no smply obey he clsscl physcs governng he mcroworld (Secon 1). Consequenly, n hs pproch, he second lw of hermodynmcs ns he sus of he fundmenl lw of dynmcs nd he problem of he exsence of mppng relzng he probblsc represenon of he mcrose dynmcs becomes of fundmenl mpornce [20]. Acnowledgemens Ths pper conns resuls obned whn he frmewor of he reserch projec N N fnnced from Scenfc Reserch Suppor Fund n The uhor s grely ndebed o he Polsh Mnsry of Scence nd Hgher Educon for hs fnncl suppor. References [1] A. Trzesows 2009 Nnohermomechncs J. Tech. Phys (Preprn cond-m/ ) [2] M. Hrmn G. Mhler O. Hees Exsence of emperure on he nnoscle 2004 Phys. Rev. Le [3] T. Bcheles H.-J. Gőnherod R. Schäfer Melng of soled n nnoprcles 2000 Phys. Rev. Le [4] A. Trzęsows S. Pers 1992 Mrovn descrpon of rreversble processes nd he me rndomzon Il Nuovo Cmeno 14D [5] A. Szlen 1982 Inroducon o he heory of smooh dynmcl sysems (PWN: Wrsw; n Polsh) [6] Y. Choque-Bruch C. De W-Morew M. Dllrd-Blec 1977 Anlyss, mnfolds nd physcs (Norh-Hollnd: Amserdm) 19

20 [7] A. Lso 1983 Sscl sbly of deermnsc sysems [n:] Equdff 82 H. W. Knoblch K. Schm [eds] (Sprnger-Verlg: Berln) [8] F. Morrson 1996 The r of modelng dynmc sysems (WNT: Wrsw; rnsled from Englsh) [9] Y. L. Klmonovch 1982 Sscl physcs (Nu: Moscov; n Russn) [10] K. Murn 1971 Anlyss 1 (PWN: Wrsw; n Polsh). [11] N. Kovleno N. J. Kuznezov W. M. Schurenov 1983 Sochsc processes (Nuov Dum: Kyev; n Russn). [12] A. Ishr 1971 Sscl Physcs (Acdemc Press: New Yor). [13] J. W. Chrsn 1975 Theory of rnsformons n mels 1 (Pergmon Press: New Yor) [14] W. L. Indebom A. N. Orlov 1973 Ledng rcle [n:] Thermlly cved processes n crysls A. N. Orlov [ed.] (Mr: Moscov; n Russn) [15] L. A. Snlo 1976 Inegrl geomery nd geomerc probbly (Addson-Wesley: Msscuses). [16] T. J. Qunn 1983 Temperure (Acdemc Press: London) [17]. H. Wojewod 1972 Legendre mehod of rnsformon (ITP Repors Wroclv Techncl Unversy: Wroclv; n Polsh) [18] N. F. Rmsey 1956 Thermodynmcs nd sscl mechncs negve bsolue emperure Physcl Revew [19] J. P. Terlezy 1966 Sscl physcs (Vsšcj Šol:Moscov; n Russn) [20] I. Prgogne 1980 From beng o becomng (W. H. Freemn nd Compny: Sn Frncsco) ` 20