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

riv: [mh-ph] 0805.0944 ON THE DYNAMICS AND THERMODYNAMICS OF SMALL MARKOW-TYPE MATERIAL SYSTEMS Andrzej Trzęsowski Deprmen of Theory of Coninuous Medi, Insiue of Fundmenl Technologicl Reserch, Polish Acdemy of Sciences, Pwińskiego 5B, 02-106 Wrsw, Polnd e-mil dresses: rzes@ipp.gov.pl, rrzes@gmil.com Absrc. The collecive properies of smll meril sysems considered s semidynmicl sysems reveling he Mrkov-ype irreversible evoluion, re invesiged. I is shown h hese meril sysems dmi heir remen s hermodynmic sysems in diherml nd isoherml condiions. A kineic equion describing sisicl regulriies of he Mrkov-ype meril sysems nd consrined by he compibiliy condiion wih he firs nd second lws of hermodynmics nd wih he relxion posule, is proposed. The influence of exernl prmeers on he sionry ses of smll meril sysems endowed wih heir own energy independen of dynmics is discussed. PACS clssificion: 05.20.-y, 05.70.Ln, 02.50.G 1. Inroducion The problem of comprehensive descripion of collecive properies of meril sysems, observed no on he mcroscopic scle bu on differen mesoscles, ppers in he heory of nnosrucured merils [1, 2, 3]. Priculrly, i is indispensble o clrify he pplicbiliy of he conceps of hermodynmics o meril sysems on he smll lengh scles [2]. For exmple, if we re deling wih he mcroscopic observion level scle, hen he sndrd procedure o show he exisence of hermodynmic limi, nd herefore emperure, is bsed on he ide h, s he spil exension increses, he surfce of region in spce grows slower hn is volume [2]. However, i is no he cse of smll meril sysems [1, 3]. The im of his pper is o show h he mehod of descripion of collecive properies of meril sysems, which hs been formuled in [4] for sochsic sysems wih he counble spce of ses, cn be helpful, fer is modificion nd generlizion (bsed on he heory of semidynmicl sysems nd on he sisicl heory of sionry ses), for beer undersnding of he dynmics nd nonequilibrium hermodynmics of smll meril sysems wih he se spce being counble or hving he crdinliy of coninuum. The considered meril sysems cn be deerminisic (wih he dynmics described by semidynmicl sysems) or sochsic bu dmi he Mrkov-ype evoluion nd dmi heir remen s hermodynmic sysems in diherml nd isoherml condiions (Secions 3-5). In Secion 2 re reviewed briefly differen mehods of formuling probbilisic represenion of dynmicl (or semidynmicl) sysems, kineic equion, nmed he Kolmogorov-ype kineic equion, is inroduced, nd he Mrkov-ype evoluion processes of nonsingulr semidynmicl sysems re defined. In Secion 3 re considered no closed Mrkov-ype smll meril sysems

nd he condiion h he Gibbs disribuion fulfills he Kolmogorov-ype kineic equion is given. In Secion 4 he influence of exernl prmeers on he sionry ses of smll meril sysems endowed wih heir own energy independen of dynmics is discussed. Moreover, in Secion 4, he noion of condiionl enropy is considered nd he relionship beween he exisence of posiive bsolue emperure nd he dependence of enropy on he inernl energy of he sysem is presened. In Secion 5 re formuled condiions of he hermodynmicl dmissibiliy of Mrkov-ype evoluion processes of he considered meril sysems. The proposed mehod of he descripion of collecive properies of hese sysems dmis is consisency wih he so-clled Prigogine s selecion rule for irreversible dynmicl processes (Secion 6). In he pper he reversibiliy of deerminisic or sochsic dynmicl processes (see Secion 2) mens h when he direcion of ime is reversed he behvior of hese processes remins he sme. 2. Probbilisic represenions of dynmicl sysems Le us consider meril sysem h dynmicl behvior is defined by opologicl spce (counble or hving he crdinliy of coninuum) of ll is dmissible ses nd by deerminisic opologicl dynmicl sysem [5] defined s coninuous Abelin semigroup or group G = { S :, T} of coninuous rnsformions, where R (, ) T = = + or R [ 0, ) for every 1 2 T T = + = + (wih he inernl operion 1 ± 2 T ) for he group or semigroup, respecively, nd S S S, S id,,. (2.1) s = + s 0 = s T We cn lso consider semigroup G wih T R (, 0] = = s he semigroup of p- is disinguished se of he meril sysem, clled is iniil rmeers. If x0 se, hen x = x x ; S x (2.2) 0 0 denoes he insnneous se of his sysem he insn T. For exmple, if is differenil mnifold nd x ( x0; ) : T is he generl soluion of he differenil equion ( x) x0 x ɺ = v, x 0 =, (2.3) where xɺ = dx/d nd v is vecor field on ngen o, hen Eq.(2.2) cn be considered s definiion of (deerminisic) smooh dynmicl sysem genered by he differenil equion of Eq.(2.3) [5, 6]. We ssume h he spce is ddiionlly endowed wih disinguished σ-finie, nonnegive mesure : Γ R+ where Γ denoes σ-lgebr of subses of [6]. In priculr, if is counble se ( crd ℵ 0 ) nd s Γ is ken he se of ll subses of, hen is he so-clled couning mesure on defined for A Γ by he rule: ` 2

( A) crd A for crd A < ℵ0, = for crd A = ℵ 0. (2.4) Furher on mesure spce (,, ) The spce is lso clled phse spce of he sysem. By L ( ) = Γ [6] is clled he se spce of sysem. we will denoe he liner Bnch spce of -mesurble funcions f : R such h [6]: d Priculrly, if is he couning mesure, hen Noe h L ( ) f = f x x <. (2.5) x f = f x <. (2.6) is he so-clled Bnch lgebr wih respec o he inernl poinwise muliplicion of funcions belonging o his spce [6]. Now, he semigroup (or group) G of rnsformions should consis of mesurble rnsformions [5], h is he so-clled condiion of double mesuremen of hese rnsformions should be fulfilled [7]: ( 1 ) A Γ, T, S A,S A Γ. (2.7) In he lierure deerminisic (nd opologicl or smooh) dynmicl sysem defined s he semigroup G = { S :, T} fulfilling he condiion of double mesuremen is frequenly clled semidynmicl sysem [7]. The mos prominen exmple of dynmicl sysems genered by differenil equions nd fulfilling he bove condiion of double mesuremen is he Hmilonin dynmicl sysem describing he dynmics of meril sysem consising of N 2 idenicl pricles. In his cse = R n, n = 3N, is he Lebesgue mesure, nd Eq.(2.3) wih v ( x) = J H ( x) x 0 I J =, I = dig(1, 1,...,1) GL ( n,r ), I 0, (2.8) where GL ( n,r) is he group of nonsingulr n n rel mrices, H : R is he Hmilonin of he sysem nd x denoes he grdien operor wih respec o vribles x, is considered [8]. Le G denoe he Hmilonin dynmicl sysem defined by Eqs.(2.1)-(2.3) nd (2.8). I cn be shown h G is he one-prmeer group 2 of rnsformions of he spce = R n, clled cnonicl rnsformions, h preserve he Lebesgue mesure (h is preserve he volume in ) [9, 10]. If he iniil se x 0 of meril sysem is known up o he probbiliy of is loclizion in subse A Γ : 3

( ) = P x0 A P A p x d x, p L, p 0, P = 1, A (2.9) hen he probbiliy h n insnneous se of he sysem is loclized he insn T in he se A cn be defined s: ( 0 ) -1 ( ) = = P x A P A p x d x P x S A. (2.10) Priculrly, i resuls from he definiion of cnonicl rnsformions, h A -1 p = p S D, (2.11) where i ws denoed: D ( ) = p L ( ) : p 0, p( x) d ( x) = 1. (2.12) Eqs.(2.2), (2.3), (2.9)-(2.12), semigroup I follows h he Hmilonin dynmicl sysem (2.8) generes, ccording o U = U : D D, T of rnsformions cing ccording o he rule: { } 1 U p = ps, p = p0, U0 = id D ( ), (2.13) exensible o he liner mppings U : L L, nd such h he sufficienly smooh probbilisic densiies of Eq.(2.11) fulfill he so-clled Liouville equion: p( x, ) = Lp( x, ), = = p x, p x, p x, 0 p x, (2.14) where L is he Liouville operor cing ccording o he rule { } { } k Lf = f, H = H, f, f, H C, k 1, (2.15) nd { }, denoes he Poisson brckes [9]. Noe h since in order o solve he Liouville equion we ough o known, in generl, soluion of he Hmilonin equions, i is he problem unrelizble for mcroscopic sysems. Consequenly, in he sisicl physics re considered pproxime soluions of he Liouville equion h describe sisicl regulriies of he Hmilonin sysem [9]. ` 4

If we re deling wih he oriened in ime evoluion of sochsic meril sysem wih he se spce, hen he rndomness of insnneous ses of he sysem cn be described by fmily x { x : T} T = of mppings [ ] P x : Ω, P, T, P : Ω 0, 1, Ω = 1, (2.16) where ( Ω, P) denoes probbilisic spce of elemenry evens nd T ( = R + or R ) is n one-prmeer, ddiive nd Abelin semigroup. If ( ω ( ω ) ) P x A P Ω : x A = p x d x, (2.17) A hen we cn ssume he exisence of n Abelin semigroup U { U, } = T of mppings 0 U : D D, U = id, (2.18) L where Eq.(2.12) is ken ino ccoun, such h if x 0 is rndom vrible wih probbiliy densiy funcion p of Eq.(2.9), hen T, U p = p, (2.19) nd hese mppings re exensible o he liner operors U : L L. I ough o be sressed h he wy in which he probbilisic represenion of dynmicl sysem is inroduced depends on he kind of rndomness ssocied wih he dynmics. For exmple, in he cse of deerminisic dynmicl sysem of Eq.(2.3), he only wy o inroduce is probbilisic represenion is he rndomness of is iniil condiions. Anoher siuion kes plce e.g. in qunum sysems, in which he rndomness cn be reled wih he Heisenberg unceriny relion. More generlly, we cn consider semigroup U of liner operors in L ( ) fulfilling he condiion (2.18) bu no necessrily genered by he fmily x T of rndom vribles. If hese liner operors re he so-clled Mrkov operors, h is: ( ( )) f L, f 0 T, U f 0 nd U f = f, (2.20) hen U is clled sochsic semigroup [7]. I follows from he condiion (2.20) h should be [7]: f L ( ), T, U f f. (2.21) Thus, he Mrkov operors re conrcions. Le D ( ) π be he so-clled sionry densiy, defined by [7]: 5

T, U π = π. (2.22) A sochsic semigroup is clled sympoiclly sble if here exiss excly one sionry densiy π such h [7] p D ( ), lim U p = 0. (2.23) π For exmple, le us consider fmily K { K : R, T} = of he soclled sochsic kernels sisfying, for every s, T nd lmos everywhere on, he so-clled Chpmn-Kolmogorov equion [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. (2.24) Given K we cn define sochsic semigroup by seing for ny f L ( ) Noe h for every 0, T, > 0, we hve [7]: : U f x = K x, y f y d y. (2.25) 0 0 = p D, U p x h x, h x inf K x, z, x. z 0 (2.26) I cn be shown [7] h if K is fmily of sochsic kernels such h h0 x d x > 0 (2.27) for some 0 T, hen he semigroup defined by Eq.(2.25) is sympoiclly sble. The probbilisic inerpreion of he bove sochsic semigroup is defined by Eqs.(2.16)-(2.19) nd (2.25). For exmple, i is he cse of Mrkov chins (i.e. Mrkov processes wih he counble se spces [11]) wih he coninuous ime which hs been considered in [4]. Frher on sochsic semigroup genered by fmily of sochsic kernels is clled he Chpmn-Kolmogorov semigroup. Sochsic semigroups corresponding o he rndomness of insnneous ses of meril sysems (i.e., defined by Eqs.(2.16)-(2.20)) pper minly in pure probbilisic problems such s rndom wlks, sochsic differenil equions nd mny ohers (e.g. in he problem of Mrkovin descripion of collecive properies of sysems wih he counble se spce [4]). However, hey cn ll be genered lso by deerminisic semidynmicl sysems [7]. A semidynmicl sysem is clled nonsingulr if in ddiion ` 6

( ) 1 ( ) A Γ, A = 0 T, S A = S A = 0. (2.28) For ny nonsingulr semidynmicl sysem G we cn univoclly define he sochsic semigroup U U[ G] = { U : L ( ) L ( ), T} ssuming h for every mesurble se A, 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) The such defined sochsic semigroup fulfills ddiionlly he following condiion: ( f ) S ( f ) where supp f { x : f ( x) 0} if A = supp f, hen U f ( x ) = 0 for x ( A) supp U supp, (2.30) = is he suppor of f [7]. I follows from Eq.(2.30) h S [7]. A nonsingulr semidynmicl sysem G is clled sisiclly sble if he corresponding sochsic semigroup is sympoiclly sble. The behvior of U[G] llows o deermine mny properies of he semidynmicl sysem G. For exmple, le us consider he problem of he exisence of mesure 0 invrin under G, h is such h [7] ( S 1 ( A) ) ( A) = (2.31) 0 0 for every mesurble se A nd T (s e.g. in he cse of Hmilonin dynmicl sysems). Assume now h mesure 0 is normlized ( 0 ( ) = 1) nd invrin under G. The pir, G is clled exc if for every mesurble se A he 0 following condiion is fulfilled: ( A) ( ( A) ) > 0 lim S = 1. (2.32) 0 0 Le G be nonsingulr semidynmicl sysem nd le U[G] denoes he sochsic f L, hen he mesure semigroup ssocied wih is. If is invrin under G if nd only if f A = f x d x, A Γ, (2.33) A T, U f = f, (2.34) where Eq.(2.29) ws ken ino ccoun [7]. Moreover, if he semigroup U[G] is sympoiclly sble, f D ( ) is is unique sionry densiy nd f is he mes- 7

ure given by Eq.(2.33), hen he pir (, G f ) is exc nd f is he unique bsoluely coninuous normlized (nonnegive) mesure invrin under G [7]. The probbilisic inerpreion of nonsingulr semidynmicl sysem G cn be formuled if he corresponding sochsic semigroup U[G] is consisen wih sochsic process defined by Eqs.(2.16)-(2.19). I cn be e.g. he cse of Mrkov processes [11]. We will ssume ddiionlly, generlizing he cse of Mrkov chins [4, 11], he exisence nd finieness of he so-clled rnsiion inensiies W ( x, y) 0, x y, from he se x o he se y nd he so-clled exi inensiies from he ses x of he sysem: w x = W x, y d y > 0. (2.35) Then, he probbiliy densiy p defined by Eqs.(2.19) cn be ssumed in he form of p ( x) = p( x, ) where p : T R + is soluion of he following version of he soclled Kolmogorov equion considered in he heory of Mrkov processes: p x, = w x p x, + W y, x p y, d y, p( x,0) = p x. (2.36) This equion hs he physicl mening of kineic equion h defines he probbilisic represenion of he Mrkov-ype evoluion of meril sysem on he bsis of he blnce of he inensiies of reching nd leving he ses of his sysem. Therefore i cn be nmed he Kolmogorov-ype kineic equion. Noe h he quniy 1 τ ( x) = < (2.37) w x cn be inerpreed s he men residence ime of he Mrkov-ype evoluion process in he se x [4]. Le us denoe by L G he liner operor (in generl unbounded) in he Bnch lgebr L ( ) defined s LG f x = w x f x + W y, x f y d y. (2.38) Frequenly, we hve o ke ino ccoun consrins resricing he se spce [1] s well s concerning he evoluion of he sysem. For exmple, i is esy o observe h if he exi inensiies w re commonly bounded hen he operor LG is bounded: w > 0, x, w x w, (2.39) 0 0 ` 8

L = sup L f 2w, (2.40) G G 0 f 1 where Eq.(2.5) ws ken ino ccoun. I follows from Eq.(2.36) h hen he sochsic semigroup U[G] consiss of operors of he exponenil form [10]: n= n 0 U = exp ( L ) L, L I, (2.41) G G G ń= 0 n! where If = f for ny f L ( ). The exisence of kineic equion of he form (2.36) is n imporn fc from he poin of view of he physicl pplicions of sochsic semigroups o he descripion of irreversible processes. I follows from he following form of rnsiion probbiliies of he Mrkov processes governed by such equion [4, 11]: n ( + 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 h 0, uniformly wih respec o x for given y, x y. Hence in his cse he descripion of he irreversible evoluion of he sysem cn be reduced o he invesigion of is behvior for shor ime periods, h is o he formulion of he physicl hypohesis bou he form of he rnsiion probbiliies. For exmple, i follows from Eqs.(2.39) nd (2.42) h, independenly of he choice of x, should be ( ( h 0 ) 0 ) y, y x P x y x = x w h + o h. (2.43) Furher on he considered meril sysems (deerminisic wih he dynmics described by nonsingulr semidynmicl sysems or sochsic [4, 11]) re consrined by he condiion h he ssocied sochsic semigroups re genered by he Kolmogorov-ype kineic equion (2.36). Consequenly, one cn sy h such meril sysems nd such sochsic semigroups re he Mrkov-ype. Noe h such meril sysems cn dmi he Mrkov-ype irreversible evoluion. 3. Sionry ses Le us consider meril sysem wih he se spce nd such h every se x hs is own energy E x R+ independen of he dynmics of he sysem (nd clled inernl energy of he sysem in he se x). The dynmics of he meril sysem is described by nonsingulr semidynmicl sysem cing in he se spce (see remrks previous o Eq.(2.28)). We cn inroduce now he men inernl energy E : D R cing ccording o he following rule: funcionl + 9

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 ken ino ccoun, nd we cn disinguish he clss of probbilisic mesures giving he some vlue ε of he men inernl energy: { } D, = p D : E p =. (3.2) ε ε Since he se spce cn be counble or cn hs he crdinliy of coninuum, he disribuion of energy e is dmied o be discree or coninuous funcion of he vrible x, respecively. Though one cn dp he presened descripion of collecive properies of meril sysems o he descripion of mcroscopic sysems, however firs of ll will ineres us smll sysems (see Secion 1 nd he remrks following Eq.(2.15)). For exmple, le he sysem consiss of he finie number N of idenicl pricles, he se spce is 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 ken ino ccoun. Le us denoe by n x he (finie) number of pricles being in he se x endowed wih he inernl energy E x. In he clssicl sisicl physics, he probbiliy p x of Eq.(3.3) h pricle of he sysem hs he inernl energy E x in he se x, is ssumed, under cerin physicl condiions [9, 12], in he following form: nx px =, N nx, N = (3.4) x nd i is pproximed, in he so-clled hermodynmic limi [9, 12], by p x nx = lim. N N (3.5) The pproximion is he beer, he greer N is. However, he condiion (3.5) cn no be cceped in he cse of smll meril sysems. In sisicl physics is considered he so-clled Bolzmnn enropy funcionl S : D ( ) R defined by: where ( ) p D, S p = s p x d x, (3.6) ` 10

s ( z) kbz ln z for z > 0, = 0 for z = 0, (3.7) nd k B is he Bolzmnn consn. This funcionl is reed, in he hermodynmic limi, s mesure of he sisicl informion concerning he energeic ses of mcroscopic sysems [9, 12]. Neverheless, i cn be cceped lso s mesure of unceriny in he sisicl descripion of processes in microscopic bodies [9]. This mesure of informion kes is mximum vlue consisen wih he fixed vlue of he men energy, ( π ) π D, p D, S p S, (3.8), ε, ε on he so clled Gibbs disribuion π of he following form: π ( x) 1 β Ex = Z exp, kb β Ex Z = exp d ( x) <, kb (3.9) where β > 0 is consn. For sufficienly smooh disribuions, equliy in Eq.(3.8) implies h p = π lmos everywhere on. I should be sressed h, in he frmework of clssicl sisicl physics, he Gibbs disribuion is pplied only in he hermodynmic limi, h is, i should be undersood hen in he sense of Eq.(3.5) p π x, π defined in his wy is clled wih x =. The probbilisic represenion ( ) cnonicl ensemble (for he counble se spce) [9, 12]. Denoing we obin he following relion: 1 ( π ) θ θ β F = E ln Z, E = k, =, (3.10) B B B ( π ) ( π ) θ ( π ), F = E S (3.11) where E ( π ) ε is fixed men inernl energy of he sysem, S ( π ) is he mximl enropy corresponding o E nd defined by Eqs.(3.6), (3.7) nd (3.9), nd he sclr θ > 0 defines, ccording o Eq.(3.10), he chrcerisic energy E B of he sysem. If i is mcroscopic nd closed sysem (i.e. he meril sysem is energeiclly isoled), hen he sclr θ cn be idenified wih he bsolue hermodynmic emperure of he sysem nd he quniy F ( π ) cn be recognized s he free energy of he sysem [9]. Moreover, for closed sysem, he se of sisicl equilibrium (i.e. he condiion h he disribuion π is independen of ime) covers wih he se of hermodynmic equilibrium [9]. If he sysem is no closed, hen his ses cn be dependen on he mbien emperure [9]. Priculrly, i cn be he cse of sysem (mcroscopic or microscopic) wih hermlly conducing boundry (clled diherml boundry), dmiing 11

hermlly cived processes nd coupled wih is environmen ( hermos). In his cse we cn ssume h θ covers wih he emperure of he environmen [4] or, more generlly, h his emperure is produced by hermos. F ( π ) kes hen he physicl mening of generlized free energy corresponding o he Gibbs disribuion nd his disribuion cn describe, in generl, sionry nonequilibrium se of smll meril sysem (see remrks following Eqs.(3.7) nd (3.11)). We will cll his se Gibbs se. If we re deling wih Mrkov-ype meril sysem, sy wih nonsingulr semidynmicl sysem (see remrks he very end of Secion 2), hen i follows from Eqs.(2.35), (3.9), (3.10) nd he condiion = 0 (3.12) p h he Gibbs disribuion fulfills he Kolmogorov-ype kineic equion (2.36) if he following nlogue of he so-clled condiion of microscopic reversibiliy (clled lso he condiion of deiled blnce [12]) is sisfied: ( π π ) x, y, x y W x, y > 0, x W x, y = y W y, x. (3.13) This condiion is fulfilled if he rnsiion inensiies 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. (3.14) The formul (3.14) cn be wrien, wihou losing generliy, in he form: U xy W ( x, y) = ν exp, EB (3.15) where ν > 0 is consn wih he dimension of frequency, nd i ws denoed The rio U = E E, E = E 0. (3.16) xy xy x xy yx k xy (, ) (, ) ( y) Ex exp B W x y π Ey = = = W y x π x E (3.17) defines he co-clled equilibrium consns considered in he cse of mcroscopic equilibrium Gibbs ses [12]. I follows from Eqs.(2.4) nd (3.9) h he cse k xy = 1 for rbirry x, y cn ke plce only for he finie se spce wih he uniform Gibbs disribuion, h is, if: ` 12

1 1 x, π x = Z =, N = crd. (3.18) N The formul (3.15) hs he form of he well-known lw describing he frequency of he rnsiion x y in he heory of recion dynmics nd is pplied, for exmple, o he descripion of he hermlly cived processes [13, 14]. Bsing on his observion, we cn inerpre E s he energy brrier beween he ses x nd y, wih own energies x xy E nd E, respecively, wheres y U xy cn be inerpreed s he civion energy of he chnge of ses of he sysem leding from x o y [4]. Then he consn ν of Eq.(3.15) hs he mening of he effecive frequency of effors o overcome he energy brrier [13, 14]. Noe h he men residence ime τ in he Gibbs se is given by τ = τ π d, (3.19) ( x) ( x) ( x) where Eq.(2.37) ws ken ino ccoun. Ifτ <, hen we cn ssume h [4] nd, for he sufficienly lrge ime τ, sy e.g. for 1 ν = (3.20) τ 1 τ τ0 = (3.21) w in he cse of Eq.(2.39), his nonequilibrium sionry se cn be considered s mesble se. For exmple, if we re deling wih smll nnosrucured clusers, i cn be ssocied wih he phenomenon of he exisence of opimum size nd shpe leding o he mos sble pcking of heir oms [1]. 0 4. Condiionl enropy nd exernl prmeers Le us consider meril nonsingulr semidynmicl sysem wih he se spce nd such h every se x hs is own energy E x R+ independen of he dynmics of he sysem (Secion 3). These energies cn be dependen on finie se of prmeers being exernl wih respec o he sysem of pricles under considerion [9]. For exmple if we re deling wih sysem of meril pricles conined in hree-dimensionl smll convex figure B (consiuing e.g. nnocluser [1]), hen we cn consider he riple = ( V,F,M) of exernl prmeers, where V is he volume of B, F is he surfce field of he boundry B of B nd M is he men curvure of his boundry [1]. If we re deling in wo-dimensionl smll convex figure = F,M, where F is he re of B, B (e.g. grphene smll cluser), hen he pir M = ( π / 2) L is he men curvure of B ken ino ccoun. Noe h he volume prmeer, nd L is he perimeer of B [15], cn be = V is frequenly discussed in 13

clssicl hermodynmics of mcroscopic sysems [9, 16]. The corresponding Gibbs disribuion π (Secion 3) depends hen on hese prmeers, h is we hve [9] (, ) ( ) 1 Ex π (, θ, x) = Z (, θ ) exp, EB = kbθ, x, EB e x Z (, θ ) = exp d ( x) <, e(, x) = Ex ( ), EB (4.1) where θ R+ is n bsolue emperure nd k B denoes he Bolzmnn consn. In sisicl physics re considered he following equipoenil ses of energy: 1 ( ε ) ( ε ) Σ = e, e : R, x, e x = e, x. + (4.2) I is dmied h he mesure induces such mesures, ε on he hypersurfces Σ ε in he se spce h he generlized volume ( ε ) ( ε ) Ω, = volσ > 0, (4.3) cn be defined s sufficienly smooh funcion of he prmeers nd ε. I follows f ε, θ of he from Eqs.(4.1)-(4.3) h he condiionl probbiliy densiy funcion energy disribuion (wih nd θ keeping consn) [9] (, ) (, ) Z (, ) 1 exp ε f ε θ = Ω ε θ k, Bθ (4.4) describes hen he disribuion of vlues of own energies of ses of sysem in F, θ by hermos [9, 16]. Inroducing he generlized free energy of he sysem ( θ ) ( θ ) F, = E ln Z,, (4.5) nd king ino ccoun Eqs.(4.1) nd (4.2), we cn wrie he densiy funcion f of Eq.(4.4) in he following form [9]: B (, θ ) F ε f ( ε, θ ) = Ω( ε, ) exp. EB (4.6) In sisicl physics is considered lso he condiionl free energy F (, θ ε ) of sysem in he hermos defined by he relion [9]: F (, θ ) F (, θ ε ) f ( ε, θ ) = exp, EB or, ccording o Eq.(4.6), by he following formul: (4.7) ` 14

I follows from Eqs.(4.7) nd (4.8) h [9] F, θ ε = ε E ln Ω, ε, E = θ k. (4.8) B B B ( 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 condiionl free energy., S, θ of he sysem s: Inroducing 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) where i ws denoed s z kbz ln z for z > 0 = 0 for z = 0, (4.11) we obin he well-known hermodynmic relion: ( θ ) ( θ ) θ ( θ ) F, = E, S,. (4.12) Consequenly, (nonsingulr) semidynmicl sysem endowed wih disribuion of own energies of is ses cn be considered s hermodynmic sysem defined by Eqs.(4.1) nd (4.10)-(4.12). The sionry ses of such defined hermodynmic sysem re Gibbs ses (Secion 3). Noice h Eqs.(4.8) nd (4.10)-(4.12) sugges o define he condiionl enropy S, ε of he sysem s [16] (cf. [9]): ( ε ) ( ε ) S, = k ln Ω,. (4.13) B The fundmenl equion of hermodynmics of reversible qusi-sic rnsformions (of sysem embedded in hermos) is he following [17]: where ( k ; k 1,2,... n) k de = θds A d, k = 1,2,... n, (4.14) k = = is he se of exernl prmeers (clled lso generlized coordines), Ak, k = 1,2,... n, re he corresponding generlized hermodynmic forces nd he relion dq = θd S, (4.15) 15

where Q is he so-clled heing being quniy describing he herml influence of hermos on he sysem, is ken ino ccoun. Wriing Eq.(4.14) in he following form: de = dq d A, da = A d k, (4.16) nd king ino ccoun Eq.(4.10), we obin h should be: k i ( θ ) = αi π θ + e( x), θ ( π ( θ x) ) ( x) d E,, x d,, x d x, d,, d, (4.17) where i ws denoed e π i π αi (, x) = (, x), d, d d. i θπ = + θ i θ (4.18) Defining he generlized hermodynmic forces A i s [9]: A, =, x,, x d x, (4.19) i ( θ ) α π ( θ ) i nd compring Eq.(4.16) wih Eqs.(4.17)-(4.19), we obin he following represenion of he chnge of heing: ( θ ) ( π ( θ )) d Q, = e, x d,, x d x. (4.20), θ Therefore, dq is defined by he chnge of Gibbs disribuion due o he chnge of hermodynmic vribles nd θ. Noe h i follows from Eqs.(4.1), (4.5), (4.18) nd (4.19) h [9] F Ai (, θ ) = (, θ ), i = 1,2,... n. i (4.21) Treing he enropy S of Eq.(4.14) s funcion of independen vribles E nd k, we obin h [17] n 1 1 k S S k ds = de + Ak d = de + d, θ θ E (4.22) k k = 1 i k E, where he symbol z indices h for he pril differeniion one should hold consn he vrible z, nd hus he following equions hold: 1 S S =, Ak = θ. θ E k i k E, (4.23) ` 16

We see h, from hermodynmic poin of view, he only requiremen for he exisence of posiive bsolue emperure θ is h he enropy S should be resriced o monooniclly incresing funcion of he inernl energy E. Noe h if we re deling wih he condiionl enropy of Eq.(4.13), hen S ε Ω = k Ω B, ε (4.24) where Ω (, ε ) dε, = cons. sysem of N pricles wih heir own energies conined in he inervl [ ε, ε dε ], cn be inerpreed s number of dmissible ses of + [16]. This number monooniclly increses if ε increses, i.e. we re deling wih ses of he sysems wih no upper limi o he own energies of hese ses, e.g. for he kineic energy of gs molecule [16, 18] or in he cse of hrmonic oscillor [16]. Thus, for θ R+, he enropy of he sysem s well s is condiionl enropy re monooniclly incresing funcions of he inernl energy of he sysem. Neverheless, some very peculir sysems h hve energeic upper limis of heir llowed ses, re considered [16, 18]. The descripion of such sysems in he frmework of sisicl physics bsed on he exisence of Gibbs disribuion, needs o inroduce negive bsolue emperure [18]. In his cse, ccording o Eqs.(4.26) nd (4.27), he enropy of hermodynmic sysem is no monooniclly incresing funcion of is inernl energy. I ough o be sressed h he ssumpion reling o he sign of he bsolue emperure is no explicily mde in hermodynmics. I is becuse such n ssumpion is no necessry in he derivion of mny hermodynmic heorems [18]. I esy o see h if θ R+ nd he condiionl probbiliy densiy funcion f of Eq.(4.6) hs n exremum, h is, here exiss ε = ε such h [9, 16]: m f ε, θ ( ε θ ) m, = 0, (4.25) or, equivlenly, if F ε ( θ ε ), = 0, m (4.26) hen, ccording o Eq.(4.8), should be ε ε ( θ ) = where: m m, Ω ε 1, m, m, k θ ( ε ) = Ω( ε ) B (4.27) nd hus if he generlized volume (, ε ) Ω, = cons., increses if ε increses: Ω ε R +, (, ε ) > 0, ε (4.28) 17

henθ > 0. Consequenly, ccording o Eq.(4.24), in his cse he emperure θ is posiive sclr iff he condiionl enropy S of Eq.(4.13) is monoonic funcion of he own energy ε, h is, S ε R +, (, ε ) > 0. ε (4.29) Noice h he densiy f of he condiionl probbiliy hs he mximum for he own ε = ε θ of Eq.(4.27) if energy m, 2 f 2 ε, θ ( θ ε ), < 0, m (4.30) or, ccording o Eqs.(4.7) nd (4.26), if: 2 F 2 ε, θ ( θ ε ), > 0. m (4.31) 5. Thermodynmiclly dmissible Mrkov-ype processes Le us consider Mrkov-ype meril sysem wih is evoluion governed by nonsingulr semidynmicl sysem G such h he corresponding sochsic semigroup U[G] is genered by he Kolmogorow-ype kineic equion defined by (2.35)-(2.37), (3.15), nd (lernively) by (3.19)-(3.21) (Secion 2). We will ssume lso h he sysem dmis is remen s smll hermodynmic sysem wih diherml boundry (Secions 3 nd 4). Le be he se spce of his meril sysem. A (sionry nd nonequilibrium) Gibbs se (being perhps mesble se) of his sysem is described by Eqs.(3.1), (3.2), (3.6)-(3.11) (nd, for exmple, fulfils he condiions (4.1), (4.10)-(4.12)). I ough o be sressed h he presened here pproch cn be lso pplied in he cse of sochsic Mrkov processes (see Secion 2 nd [4]) nd in he cse of processes governed by he Chpmn- Kolmogorov semigroup (Secion 2). We cn now define he hermodynmiclly dmissible Mrkov-ype process of he evoluion of he meril sysem. Firs of ll, such process should be consisen wih he second lw of hermodynmics. This condiion cn be formuled in he following wy. Le us clcule funcionls of he men inernl energy nd he Bolzmnn enropy (Secion 3) long rjecory of he Mrkov-ype semigroup U G = U : D D, T corresponding o G (Secion 2), h is, defined { } [ ] by Eqs.(2.1), (2.2), (2.9)-(2.12), (2.20), (2.29) (in he cse of deerminisic sysems; sy e.g. defined by Eq.(2.3)) or by Eqs.(2.16)-(2.20) (in he cse of sochsic sysems): T, E = E U p, S = S U p, (5.1) ` 18

where p D ( ) is n iniil sionry disribuion. Since he environmen of he considered sysem is hermos (Secion 3), we cn consider nonequilibrium ses of his sysem wih is consn emperure θ > 0 defined by his hermos. This mkes possible he following exension of he definiion (3.11) of he generlized free energy (corresponding o he Gibbs se) o he cse of nonequilibrium isoherml processes: T, F = F U p, EB p D ( ), F ( p) = E ( p) θ S ( p), θ =, k B (5.2) where E B is chrcerisic energy of he sysem ssocied wih he considered herml phenomen. Noe h he similr definiion of he nonequilibrium free energy is formuled in order o describe rnslionl Brownin moion in n equilibrium medium reed s hermos [9]. I follows from Eqs.(5.1) nd (5.2) h 1 ds = de d F, (5.3) θ where i ws denoed dh = h ɺ d nd h ɺ = d h / d for differenible funcion h : T R. The Bolzmnn enropy incremen δ e S due o he inercion of he sysem wih is environmen is given by: θ 1 δ = δ Q, (5.4) e S where δq is he he incremen. So, he Bolzmnn enropy incremen δ i S due o he exisence of hermodynmiclly irreversible processes in he sysem cn be clculed from Eqs (5.3) nd (5.4): 1 δ S = ds δ S = df + de δ Q. (5.5) i e θ The considered Mrkov-ype process will be consisen wih he second lw hermodynmics if nd only if δ 0. i S (5.6) The inercion of he sysem wih is environmen hs only he herml chrcer (Secion 3) if nd only if he firs lw of hermodynmics kes he following form: de = δ Q. (5.7) I follows from Eqs.(5.2) nd (5.5)-(5.7) h he considered Mrkov-ype processes cn be reed, in he diherml nd isoherml condiions, s hermodynmiclly dmissible if nd only if he generlized free energy funcionl (Secion 3) is non-incresing long he rjecories of he corresponding Mrkov-ype semigroups, h is, 19

T, Fɺ 0. (5.8) These rjecories re defined by Eq.(2.19), by he Kolmogorov-ype kineic equion (2.36) wih he rnsiion inensiies given by Eq.(3.15) (nd, perhps, ddiionlly by Eqs.(3.19) nd (3.20)), nd by he rule: x, p x = p x,, T = R +. (5.9) If he sochsic semigroup is he Chpmn-Kolmogorov semigroup (Secion 2), hen Eq.(2.25) should be ddiionlly ken ino ccoun. I is, for exmple, he cse of Mrkov chins [4, 11]. I seems physiclly resonble o disinguish he clss of hermodynmiclly dmissible irreversible Mrkov-ype processes consisen wih he exisence of Gibbs ses. I cn be formuled s relxion posule sing h he Mrkov-ype irreversible processes relx, independenly of he choice of he iniil condiion, o he univoclly defined Gibbs se (being perhps mesble se). Noe h since he Gibbs disribuion π of Eqs.(3.9) or (4.1) fulfils ideniclly Eqs.(2.36) wih W ( x, y ) given by Eq.(3.15) nd p = π, he condiion (2.22) is fulfilled. Moreover, ccording o his posule, he condiion (2.23) of sympoicl sbiliy should be sisfied. The relxion posule is fulfilled, for exmple, in he cse of hermodynmiclly dmissible Mrkov chins wih he coninuous ime nd he finie se spce [4, 11] (see Secion 2). The relxion posule should be reed s he ddiionl hermodynmic posule h defines he noion of Gibbs ses more precisely (cf. [4] nd [19]). However, i ough o be sressed h in mny cses i is difficul o prove h he probbilisic represenion of process in he se spce fulfills his posule. For exmple, in he cse of he Chpmn-Kolmogorov semigroup (Secion 2), he condiion (2.22) mens h should be: π = π K x, y y d y x, (5.10) for every x nd T. Noe lso h he Gibbs disribuion fulfils he Liouville equion (2.14) bu he Hmilonin sysem does no relx o he Gibbs se. 6. Conclusions nd remrks The noion of irreversibiliy is bsed on he endowing of he ime xis wih he disinguished forwrd orienion (see Secion 1). In he clssicl pproch, bsed on he Hmilonin microse dynmics (see remrks in Secions 2 nd 3), he ime is considered s oriened only when he collecive ( mcroscopic or mesoscopic ) properies of meril sysems re described, nd remins non-oriened when he meril sysem is nlyzed on he micro-level (i is becuse he conservive Hmilonin dynmics does no disinguish ny direcion of ime). This duliy in reing he ime cn be elimined, for exmple, in he cse of meril sysems (deerminisic s well s sochsic) reveling Mrkov-ype irreversible evoluion (Secion 2). These evoluion processes re defined s nonsingulr semidinmicl sysems possessing ` 20

probbilisic represenion described by he Kolmogorov-ype kineic equion (2.36) inroduced in Secion 2. The descripion of he irreversible evoluion of he sysem cn be reduced hen o he formulion of he physicl hypohesis bou he form of he rnsiion probbiliies (Secion 2). In Secion 3 re considered sionry ses of no closed Mrkov-ype smll meril sysems (chrcerized, for exmple, by suibly seleced exernl prmeers see remrks he beginning of Secion 4) nd i is shown h he Gibbs disribuion (Secions 3 nd 4) fulfils he Kolmogorovype kineic equion if n nlogue of he so clled condiion of microscopic reversibiliy (clled lso he condiion of deiled blnce) is sisfied. I urns ou h he proposed heory describes hen hermlly cived collecive processes (Secion 3). I is shown lso h he Mrkov-ype processes cn be reed, in he diherml nd isoherml condiions, s hermodynmiclly dmissible if nd only if he generlized free energy funcionl (Secion 3) is non-incresing long he rjecories of he corresponding Mrkov-ype semigroups (Secion 5). The exisence of he bove-menioned diherml condiions enbles o consider smll meril sysem endowed, due o is conc wih hermos, wih he posiive bsolue emperure (Secions 3 nd 4) nd, consequenly, enbles o inroduce he noion of generlized free energy of nonequilibrium isoherml processes (Secions 3 nd 5). Thus, since he considered dynmics of smll meril sysems kes plce in isoherml condiions, we re deling rher wih he descripion of isoherml collecive properies of hese sysems consisen wih hermodynmicl rules hn wih he sndrd sisicl hermodynmics of mcroscopic meril sysems (see lso he cse of nnohermomechnics [1] nnoscle isoherml counerpr of nlyicl mechnics of ffinely-rigid mcroscopic bodies [21] consisen wih he phenomenologicl hermodynmics). For exmple, in Secion 5, in ddiion o he firs nd second lws of hermodynmics reed s hermodynmic consrins of Mrkov-ype meril sysems, n ddiionl isoherml consrin of hese sysems (clled he relxion posule) is formuled nd discussed. Noe h he presened pproch dmis he consisency wih he so-clled Prigogine s selecion rule, ccording o which only hese probbilisic represenions of dynmics h re direced forwrd describe physiclly relizble ses [20]. Prigogine ssumes ddiionlly h his selecion rule cnno be derived from dynmics in his sense h i is no reled wih he exisence of ny new inercions no ye ken ino ccoun. In his pproch, he ime symmery exhibis iself on he microlevel in he form of inernl rndom nure of he sysem (h is, independenly of ny hidden vribles). According o he Prigogine s poin of view, he ime symmery should be universl, h is, i should ke plce in ll dynmicl heories: in he clssicl mechnics s well s in he qunum mechnics (nd in relivisic heories. This poin of view cn be useful in he cse of he descripion of effecive hermomechnicl properies of bulk nnosrucured clusers (see, for exmple, nnohermomechnics of bulk nnoclusers considered in [1]) s well s in he cse of low-dimensionl meril sysems (see, for exmple, [1] remrks concerning plnr grphene shees, [22], [23], nd [24] where he isoherml geomery of corruged grphene shees is formuled). Nmely, hese meril sysems re sufficienly smll (in he cse of bulk nnoclusers) or re sufficienly hin (in he cse of grphene shees) so hey re no compleely free of qunum effecs nd hus, hey no simply obey he clssicl physics governing he mcroworld (Secion 1). Moreover, in he cse of grphene shees, i ough o be ken ino ccoun h grphene is inrinsiclly no fl nd corruged rndomly ([25] nd references herein). Consequenly, 21

les in he cse of low-dimensionl smll meril sysems (deerminisic or sochsic see Secion 2 nd ssumpions he beginning of Secion 5), he second lw of hermodynmics (Secion 5) ins he sus of he fundmenl lw of dynmics of hese sysems. If so, he problem of he exisence of mpping relizing he probbilisic represenion of he microse dynmics of he bove menioned smll meril sysems becomes of fundmenl impornce. Acknowledgemen This pper conins resuls obined wihin he frmework of he reserch projec N N501 049540 finnced from Scienific Reserch Suppor Fund in 2011-2014 The uhor is grely indebed o he Polish Minisry of Science nd Higher Educion for his finncil suppor. References [1] A. Trzęsowski, Nnohermomechnics, J. Tech. Phys., 2009, 50, 353-374, (Preprin riv: cond-m/0801.2049). [2] M. Hrmn, G. Mhler, O. Hees, Exisence of emperure on he nnoscle, Phys. Rev. Le., 2004, 93, 080402-1 080402 4. [3] T. Bcheles, H.-J. Gűnherod, R. Schäfer, Meling of isoled in nnopricles, Phys. Rev. Le., 2000, 85, 1250 1253. [4] A. Trzęsowski, S. Piekrski, Mrkovin descripion of irreversible processes nd ime rndomizion, Il Nuovo Cimeno, 1992, 14D, 767 787. [5] A. Szlenk, Inroducion o he heory of smooh dynmicl sysems (PWN, Wrsw, 1982, (in Polish). [6] Y. Choque-Bruch, C. De Wi-Morew, M. Dillrd-Bleick, Anlysis, mnifolds nd physics, Norh-Hollnd, Amserdm, 1977 [7] A. Lso, Sisicl sbiliy of deerminisic sysems [in:] Equdiff 82, H. W. Knoblch, K. Schmi [eds], Springer-Verlg, Berlin, 1983. [8] F. Morrison, The r of modeling dynmic sysems, WNT, Wrsw, 1996, (rnsled from English). [9] Y. L. Klimonovich, Sisicl physics, Nuk, Moscov,1982, (in Russin). [10] K. Murin, Anlysis, vol.1, PWN, Wrsw, 1971, (in Polish). [11] N. Kovlenko, N. J. Kuznezov, W. M. Schurenkov, Sochsic processes, Nukov Dumk, Kyev, 1983, (in Russin). [12] A. Isihr, Sisicl Physics, Acdemic Press, New York, 1971. [13] J. W. Chrisin, Theory of rnsformions in mels, vol.1, Pergmon Press, New York, 1975. [14] W. L. Indebom, A. N. Orlov, Leding ricle [in:] Thermlly cived processes in crysls A. N. Orlov [ed.], Mir, Moscov, 1973, (in Russin). [15] L. A. Snlo, Inegrl geomery nd geomeric probbiliy, Addison-Wesley, Msscuses,1976. [16] T. J. Quinn, Temperure, Acdemic Press, London, 1983. [17] H. Wojewod, Legendre mehod of rnsformion, ITP Repors Wroclv Technicl Universiy, Wroclv, 1972, (in Polish). ` 22

[18] N. F. Rmsey, Thermodynmics nd sisicl mechnics negive bsolue emperure, Physicl Review, 1956, 103, 20-28. [19] J. P. Terlezky, Sisicl physics, Vysšcj Škol,Moscov, 1966, (in Russin). [20] I. Prigogine, From being o becoming, W. H. Freemn nd Compny, Sn Frncisco, 1980. [21] J. Słwinowski, Anlyicl mechnics, PWN, Wrsw 1982, (in Polish). [22] Mrens, A.: Hmilonin dynmics of plnr ffinely-rigid body, J. Nonliner Mh. Phys, 11, 145 150, 2004. [23] Kovlchuk, V., Rożko, E. E.: Clssicl models of ffinely-rigid bodies wih hickness in degenere dimension, J. Geom. Symm. Phys., 14, 51 65, 2009. [24] Trzęsowski, A., On he isoherml geomery of corruged grphene shees, J. Geom. Symm. Phys., 36, 1 45, 2014, (Preprin riv: mh-phys/1312.4711). [25] Li, T.: Exrinsic Morphology of grphene, Modelling Simul. Mer. Sci. Eng., 19, 054005-1 054005-15, 2011, (Preprin riv: cond-m.mes-hl/1111.2278). 23