Properties of Nonequilibrium Steady States: a Path Integral Approach

Transcription

1 Popeie of Nonequilibiu Seady Sae: a Pah Inegal Appoach E.G.D. Cohen Laboaoy of Saiical Phyic, The Rockefelle Univeiy, 13 Yok Avenue, New Yok, NY 165, USA Abac A nube of popeie of ye in a nonequilibiu eady ae (NESS) ae inveigaed by a genealizaion of he Onage-Machlup (OM) pah inegal appoach fo ye in an equilibiu ae (ES). A heodynaic foally idenical o ha in an ES can be foulaed, bu wih definiion of wok and hea a hoe needed o ainain he NESS. In hi appoach, he hea play a cucial ole and i diecly elaed o he diffeen behavio of a ye fowad and backwad pah in ie in an appopiae funcion pace. Howeve, an abiguiy in he choice of he iebackwad pah coeponding o a given ie-fowad pah peven a unique geneal foal heoy fo ye in a NESS. Unique unabiguou phyically accepable phyical eul fo a ye in a NESS appea o be obainable only afe pecifying he phyical nonequilibiu paaee, which define a ye in a NESS a pa of a lage ye. NESS ye ae heefoe fundaenally diffeen fo hoe in an ES. Fuheoe, an exaple i given fo a paicula ye ha he flucuaion of a ye in a NESS behave in any epec vey diffeenly fo hoe in a ye in an ES. Keywod: Nonequilibiu Seady Sae, pah inegal appoach, nonequilibiu paaee, ie-eveal, heodynaic, flucuaion 1

2 Conen: 1. Inoducion. Illuaion on an exacly oluble odel 3. Flucuaion fo Taniion Pobabiliie 4. Wok diibuion fo dagged Bownian paicle 5. Aypoic Wok Flucuaion Theoe 6. Ineial effec fo finie ie he ciical a 7. Theodynaic fo a NESS 8. Tanien Wok Flucuaion Relaion and Nonequilibiu Deailed Balance 9. Coen and Open Queion 1. Acknowledgeen 11. Refeence 1. Inoducion Thi pape i ean o be an oveview of hee peviou pape [1-3], which apply he OM appoach o udy flucuaion of ye in an ES [4-5], o flucuaion in ye in a NESS. The pape i no inended o be a uvey pape of he lieaue of all he wok done on ye in a NESS; a ciical copaion i inended fo a fuue pape [6]. Fo deail and oe exenive efeence, we efe o he oiginal pape. We will a wih a bief dicuion of oe elevan feaue of he ES. Thi in ode o be able lae o poin ou wha ae, in ou opinion, he fundaenal diffeence in pinciple beween hi ae and he NESS, a lea fo he poin of view adaped in he genealized OM heoy of he NESS, peened hee. By fa he be known and undeood eady ae i he ES. Fo a cloed ye in an ES in conac wih a hea bah a epeaue T, no wok ha o be done on he ye no doe any hea have o be applied o o eoved fo he ye in ode o ainain he ye in i ES.

3 On he ohe hand, icocopically, in e of he paicle ou of which he ye coni, hei (heal) oion caue flucuaion aound he value of he acocopic (i.e.: aveage) popeie of he ye. Thu alhough hee i on aveage no hea exchange wih he envionen hee will be all he ie hea flow ino and ou of he ye o ha hea flucuaion have o be conideed. Howeve, ince no wok i done on he ye, no wok flucuaion have o be conideed. Thi i he cae fo he OM heoy of ye in an ES. Fo he acocopic popeie of ye in an ES, a heodynaic decipion ha been developed culinaing in he fi (enegy conevaion) and he econd (enopy) law, connecing hea, wok, and inenal enegy. Thee law and hei any conequence have been able o decibe all acocopic popeie obeved in an ES, including phae aniion and quanu effec. The ES can be conideed a a pecial cae of he cla of eady ae, which alo include nonequilibiu eady ae (NESS). Up unil vey ecenly hee ha been no heodynaic decipion of ye in a NESS like ha fo an ES. A will be explained lae hi appea o be due in eence becaue a ye in a NESS i no a iple ye like one in an ES coupled o a hea bah o ainain i ES, bu i i conneced o ohe ye needed o ainain i NESS. In fac, if a ye i in a NESS, hee ae nonequilibiu paaee chaaceizing he naue and he engh of he coupling of ha ye (of inee) o ohe ye, which ogehe define he NESS of he ye. One of he eul of hi aicle i an aep o foulae a heodynaic decipion a well a a flucuaion heoy fo ye in a NESS, uing he pah inegal appoach of Onage and Machlup (OM) fo a ye in an ES [4,5]. In doing o, oe new feaue of a NESS eege, which ae fundaenally diffeen fo hoe in an ES. A a conequence, he NESS appea o be in oe ene a new kind of ae, diffeen fo an ES. The udy of he flucuaion in a NESS alo allow u in paicula o ake a connecion wih he flucuaion elaion fo wok and hea deived in he la fifeen yea fo ye wih deeiniic [7,8] a well a ochaic [9,1] dynaic. 3

4 Befoe we ue he OM pah inegal appoach o a heoy of flucuaion fo a ye in an ES fo a ye in a NESS, we vey biefly kech he OM heoy fo flucuaion in a ye in an ES. Fo uch ye, OM conideed he elaxaion back o equilibiu of flucuaion a j ( j = 1,..., n) wih zeo aveage a = of local phyical quaniie A j, away fo equilibiu. Thi elaxaion poce wa aued o be decibed by a j linea Langevin equaion of he fo: n k = 1 [ a&& () + a& () + a () ] = ζ () jk k Hee ζ () i a Gauian ando noie, i a aix epeening he (diipaive) j jk jk jk k j (1) linea law of Ieveible Theodynaic o equivalenly he coniuive equaion beween fluxe (cuen) and foce (gadien) in hydodynaic, while and ae aice epeening he fi non vanihing (econd ode) e in an expeion of he local enopy of he ye in powe of he flucuaion { a& } j, epecively. { a j } and hei ie deivaive We eak ha ou genealizaion of he OM heoy ue he obevaion ha OM do no ue in he developen of hei heoy he pecific naue of he jk jk, a flucuaion of (exenive) local quaniie. Theefoe one can inepe a alo a he poiion x of j a j j a Bownian paicle in a fluid ubjec o ficion, chaaceized by a aix jk and haonic foce, chaaceized by a aix jk. The genealizaion o a NESS poceed a follow [1]. We fi noe ha in a NESS, he aveage a will no longe vanih, i. e., a. Since he a = in he ES, j he { a j } chaaceize he agniude of he deviaion fo equilibiu. Replacing hen in (1) he a k by ( k ak a ), one obain fo ( j 1,..., n) n k = 1 k j = he linea Langevin equaion: [ a& () + a& () + a () a () ] = ζ (), jk & () k () jk whee he conan a incopoae he NESS of he ye. Alhough jk k aheaically he diffeence beween (1) and () i only a conan, i.e. ivial jk k jk k j j 4

5 aheaically, hi diffeence will un ou o be ajo phyically, ince i lead o a nube of phyical popeie of he OM heoy fo ye in a NESS fundaenally diffeen fo hoe in an ES. We will enion hee of he o ineeing of hoe popeie hee. Fi, a heodynaic foally iila o ha fo ye in an ES can be foulaed fo ye in a NESS, by uing appopiae definiion of wok, hea and inenal enegy. The NESS hea and wok ae defined a hoe needed o ainain he NESS: hey vanih in an ES. Second, wok diibuion funcion and a nube of flucuaion elaion can be deived fo ye in a NESS and hei dependence on he iniial ae of uch ye, no eiced o eihe an ES o a NESS iniial condiion, a wa hiheo he cae. Thid, abiguiie peen heelve, howeve, in he choice of he wok, hea and inenal enegy fo a ye in an NESS and which ene alo ino he heodynaic of a NESS. The nube of abiguiie i diecly elaed o he nube of nonequilibiu paaee, enioned above, i.e. o he coplexiy of he ye. Thee uliple poible choice all lead foally o he ae heodynaic law, bu i i no poible o ake, in geneal, a unique choice fo he, valid fo all ye in a NESS, a i, of coue, he cae fo ye in an ES. In fac, only afe one ha defined he ye conceely, a unique choice can be ade. Fo eigh concee odel conideed in [3], he phyically coec definiion of hea, wok and enegy equied one choice fo one half of he, while he ohe half equied a diffeen choice o obain phyically accepable eul. The oigin of hi abiguiy i ooed in an abiguiy in he poible choice of he backwad (ie eveed) pah coeponding o given fowad (in ie) pah in he pah inegal appoach o ye in a NESS. Thi i due o he peence of phyical nonequilibiu paaee, which define he NESS ye conideed and whoe paiy wih epec o ie eveal i elevan o deeine he appopiae backwad pah coeponding o a given fowad pah a i he cae in he odel conideed in [3]. We now give a bief ouline of he conen of hi pape. In ecion, in ode o elucidae vaiou feaue ha diinguih he NESS fo he ES, baic Langevin equaion fo a iple exacly oluble odel of a dagged Bownian paicle in a 5

6 haon ic poenial ae given, a well a a definiion of he wok o ainain he ye in a NESS. Thi odel i ued houghou he pape o illuae vaiou eul. In ecion 3, he pah inegal appoach o obain he popeie of flucuaion via aniion pobabiliie and a (ochaic) Lagangian i dicued. In ecion 4, he copuaion of he wok diibuion fo he dagged Bownian paicle odel i oulined, fo which in ecion 5 an aypoic wok flucuaion heoe i given, valid fo all iniial condiion of he NESS ye. In ecion 6 popeie of he wok flucuaion fo he odel of ecion fo finie ie, which hen depend on he iniial ae, ae keched. The exience of a ciical a i hown o exi, beyond which he flucuaion change fo puely decaying o popagaing a well. In ecion 7 he abiguiy in he choice of he ie eveed backwad pah o a given fowad pah in a ye in a NESS i dicued in a geneal eing. The fi and he econd law of a NESS heodynaic ae foulaed, which eflec hi abiguiy. The neceiy o pecify he NESS ye copleely, befoe a unique choice of he pope phyical quaniie of uch a ye can be ade, i dicued. In ecion 8, anien flucuaion elaion ae deived fo wo nonequilibiu deailed balance elaion fo wo diffeen kind of wok fo he dagged Bownian paicle odel: he echanical wok W, and he ficion wok R, inoduced in equaion (15) and (41), epecively. While he foe obey boh a anien and an aypoic flucuaion elaion, he lae obey only a anien flucuaion elaion. In ecion 9 oe coen and open queion ae dicued.. Illuaion on an exacly oluble odel Ou heoy can be be illuaed by uing a iple ye: ha of a Bownian paicle, confined in i oion by a lae-induced haonic poenial, which i dagged by an exenal foce wih conan velociy υ hough a fluid (hea eevoi) in an ES a epeaue T [1]. Saing he Bownian paicle a he iniial ie = a he boo of he haonic poenial, i will, afe a anien oion in he fluid, be dagged o a aionay poiion in he haonic poenial, whee he ficion foce of he fluid on he paicl e eiing he dagging oion on he one hand and he haonic foce pulling he paicle back o i iniial (lowe enegy) poiion on he ohe hand, balance (cf. fig. 1). Moe peciely, he ye of inee, which coni 6

7 hee of he Bownian paicle in a haonic poenial [11-13], ha hen eached a NESS, in which he Bownian paicle i on aveage a a aionay poiion in he poenial, which ove wih conan velociy υ fowad, ay, in he fluid. The flucuaion conideed in hi ye ae hoe of he poiion of he Bownian paicle aound hi aionay poiion. The nonequilibiu paaee fo he ye i he dagging velociy υ, which vanihe fo an ES, whee he Bownian paicle i a he boo of he haonic poenial. Figue 1 Scheaic illuaion of a Bownian paicle (black do), confined in i oion by a lae induced haonic poen ial (black paabola), which i dagged hough a fluid wih a conan velociy υ. Hee x wih oigin (black) and y wih oigin O (ed) efe o he axe and oigin of he laboaoy Ol (l) and cooving (c) fae, epecively, which ae in he diecion of υ. The paicle poiion i a ( ) y c x in he laboaoy (cooving) fae a ie, epecively, elaed by y = x υ. The figue how he poiion of h e paicle y, afe a chaaceiic elaxaion ie τ = α / κ (cf (3)), a he poiion y = = υτ, when he ye ha eached a NESS and whee he ficion foce f y NESS and he echanical foce p (black), due o he haonic poenial on he paicle, balance. (blue) Maheaically he oion of he paicle can be decibed by a Langevin equaion of he fo: o ( x υ ) ζ & x = α x& κ + (3a) τ && x = x& 1 τ 1 α ( x υ ) + ζ. (3b) 7

8 Hee τ α = andτ = ae, epecively, a elaxaion ie due o he finie α κ a of he Bownian paicle and a chaaceiic elaxaion ie fo he decay of he Bownian paicle o equilibiu when he a of he Bownian paicle i neglece d, i.e. fo he o-called ovedaped Langevin equaion The fi e on he igh hand ide (.h..) of (3) i he ficion foce, whee α i he ficion coefficien beween he paicle and he fluid, he econd e on he.h.. of (3) i he haonic foce exeed on he paicle a ie, which i deeined by he elongaion of he haonic ping x υ and ha engh κ (he haonic ping conan). The hid e on he.h.. of (1) i due o he foce exeed by he heal oion of he fluid paicle on he Bownian paicle, cauing i Bownian oion, chaaceized by a Gauian ando whie noie funcion ζ wih: ζ = and ζ 1 α β ζ = δ ( ) 1 whee β =, wih k B Bolzann conan. k B T 1, (4) The ficion and he noie e ogehe epeen he effec of he fluid on he Bowni an paicle. Changing one will heefoe equie a change in he ohe. Thee wo e alo give he hea poduced in he ye by he oion of he Bownian paicle hough he fluid. We noe ha he nonequilibiu paaee, chaaceizing, he NESS of he ye, i fo hi odel he dagging velociy υ. A fo all ye in a NESS, he coeponding OM eul fo he ae ye in an ES ae obained fo vanihing nonequilibiu paaee i.e. fo υ = hee. The peence of a conan dagging velociy υ lead o he exience hee of wo ineial fae of efeence : a laboaoy and a cooving fae. The wo fae ae illuaed in figue 1. The poiion of he Bownian paicle fae a ie i elaed o ha in he cooving fae y by: x in he laboaoy y = x υ, (5a) o ha 8

9 y & = x& υ and && y = & x. (5b) I i ofen convenien o ue, inead of he Langevin equaion (1) in he laboaoy fae, he coeponding equaion in he cooving fae which ha a (conan ) velociy υ wih epec o he laboaoy fae: o y & & = α y & α υ κ y + ζ, ( 6a) 1 1 τ & y& = y& υ y ζ. (6b) τ α We eak ha he equaion (5a) and (5b) allow o anfo any expeion in e of he coodinae coodinae y x in he laboaoy fae ino one in e of he in he cooving fae. We alo noe ha he equaion (3) and (6) ae no idenical, which iplie a lack of Galilean invaiance of he Lan gevin equaion due o he peence of he fluid. Fuheoe he fluid i accouned fo in he Langevin equaion (3) by wo e: one, cauing he ficion α x& of he Bownian paicle in he fluid wh en i i foced o ove backwad, if he haonic poenial i dagged fowad and anohe, by he incean ando colliion of he fluid olecule wih he Bownian paicle a expeed byζ. Befoe we dicu he wok flucuaion in hi odel fo ohe odel we efe o [1-3] - we will give, fo lae ue, a oe geneal foulaion of he equaion (3), by eplacing he haonic foce in (3) by a geneal echanical foce F. Then equaion (3a) becoe: ( x,; μ) ζ, & x = α x& F + (7) whee μ epeen in geneal a nonequilibiu paaee, chaaceizing he NESS of he ye. We have eiced ouelve in hi pape a well a in [1-3] o foce linea in x, o ha: F ( x, ; μ) F( x, ; μ), = (8) whee he backe L indicae an aveage ove he Gauian pobabiliy diibuion (4) of he ζ. We eak ha hi lineaiy aupion of F i only ued in hi pape and 9

10 in [1-3] fo a deivaion of he econd law of heodynaic, alhough hi law hould, of coue, be valid fo any ( x,;μ) F ; ohewie i i no ued in hi pape. 3. Flucuaion fo Taniion Pobabiliie A aid befoe, in ode o dicu he flucuaion popeie of he NESS we genealize he OM pah inegal appoach fo an ES o a NESS. We a by defining fo he NESS he aniion pobabiliy ( x x ) F ha, he Bownian paicle jup fo an iniial poiion x x a ie o a final poiion x a ie. Then an iniial diibuion of he paicle f x, ) will change o a final diibuion ( x, ) ( f accoding o: f ( x, ) dx F (, ( x x ) f x ) =. (9) Uing he funcional inegal appoach of OM 1, one can wie he aniion pobabiliy F ( x x ) of he Bownian paicle fo he iniial ae o a final x ae x a: Hee we ue in geneal X x& && x,, x, a a ho hand fo he coplee dependence of a Lagangian L ( X ;υ) F x ( x x ) D x exp d L( X ; υ). on ( ) = x x L ( X ;υ) and i fi wo ie deivaive and. can be conideed a a ochaic Lagangian decibing he ochaic dynaic of he Bownian paicle baed on he Langevin equaion, in a oewha iila way a a F( x ), ho fo F { x},whee he funcion [, ] x decibe a paicula pah fo x, o, in funcion pace. F ({ }) i heefoe a 1 Thi pape i baed on he noion of a funcional { } { }, ] [ funcion of a funcion o a funcional. x [, ] x [, ] x,,, x x ( ) i a funcional inegal, indicaing an inegaion ove all pah { } fo o whee he inegal can be defined a a lii of dicee ie ep along he pah { } (cf [1,4]). x D x x x (1) 1

11 dynaical Lagangian decibe he dee iniic dynaic of a paicle in Mechanic. Fo he dagged paicle odel in paicula we have ha L ( ; υ) i given by: L ( X ; υ) 1 1 =. 4 τ & x + x& + x υ (11) D τ τ X In (11), D i he Einein diffuion coefficien aociaed wih he Bownian paicle (ochaic) oion hough he fluid, given by: 1 D =. (1) α β ( x ) We noe ha in he epeenaion (9) of F x he funcional exp i popoional o he pobabiliy funcional P d L( X ; υ) ; he occuence of he pah { } [, ] ({ }) x aociaed wih x in funcion pace. In fac, uing he Langevin equaion (3) a well a he Gauian diibuion of he noie (4), one obain iediaely a diibuion fo he pobabiliy funcional P ({ x }) : ({ };υ )= whee C x i a noalizaion conan and P x C x exp d L( X ; υ), ; L ( X ;υ ) i given by (11). The pah inegal aveage of any funcional F( { }) x ove all poible pah (13) { } [, ], wih he iniial condiion a of x x and x x of x and x& i defined by: & = x& o he final condiion a ( ) x, x& F ({ x} ) dx dx dx & dx&, ( x x& ) D x F ({ x} ) ({ x }) f ( ) P x,. (14) ( x }) Hee, a well a he e of he pape, h e dependen ce of F { and P on υ ha been uppeed a well a ha of L on. The neceiy o inegae ove boh he iniial ae a well a ove he final ae i due o he ochaic naue of he dynaic fo he pah { } [, ] x becaue, unlike fo 11

12 deeiniic dynaical ye, fo ochaic dynaic, he final ae i no uniquely deeined by he iniial ae. wok 4. Wok diibuion fo dagged Bownian paicle Due o he lack of Galilei invaiance of he Langevin equaion, he echanical W l o ainain he NESS in he laboaoy fae (l) diffe fo ha in he cooving fae (c), W c. Fo: W l = d κ x υ υ, (15a) while W c = d [ κ y && y ]υ, (15b) o ha wih (4), W c Wl = d ( && y )υ. (15c) Hee - & i a d Alebe-like foce acing in he cooving fae o keep he Bownian y paicle on aveage a a fixed poiion in hi fae [] (fig. 1). Thi wok diffeence in he wo fae i phyically due o he kineic enegy diffeence 1 1 ΔΚ l = x& x& in he laboaoy fae and ΔΚc cooving fae, epecively, leading wih (5b) o (cf [1,]): 1 1 = y& y& in he ΔΚ c ΔΚ l = d ( y & ) υ = d ( & )υ x. (15d) I i convenien o inoduce a paaeeϑ labeling he fae, o ha ϑ =1() efe o he laboaoy (cooving) fae, epecively. Then fo boh fae one ha in geneal: W ({ x }) = d W ( X ) = o d[ κ ( x υ ) (1 ϑ) & x ] υ. (16) 1

13 whee in (16), a in he e of he pape, he paaee ϑ, ha been uppeed in W ({ x} ). The diibuion funcion fo he dienionle wok β W ({ x }) i.e.: he pobabiliy ha βw ({ x }) ha he value W, i given by: P( W, ) = δ ( W βw ({ P( W, ) can be obained fo (17) by aking a Fouie anfo of P ( W, ) and x }). (17) evaluaing i pah inegal aveage opial { ~ } [, ] L defined in (14). Thi involve finding fi he x pah ha axiize a odified Lagangian []: Lw( X ; υ ) = L( X ; υ) + λβw ({ x}), (18) whee L( X ; υ) i given by (11), W ({ x }) by (16) and λ a Lagangian uliplie expeing he conain (17)[] 3. * The opial pah X fo (18) follow fo he condiion: δ d L w ( X * ; υ) =, (19) wih he bounday condiion ha x & * * = x x, * * = x & and x = x, x& = x&. Thi lead, a in echanic, o an Eule-Lagange equaion fo Lw of (18), which educe fo ou odel o a iple linea fouh-ode diffeenial equaion: whee ~ x * i defined by: τ ~ * τ 1 (1 ) ~ * x & + ~ * x x =, τ τ (a) ~ * * x x ( 1 λ) υ τ. = υ + (b) The einology ued in hi pape confo wih ha ued in he aheaical lieaue (ee e.g. [14]) bu diffe fo ha ued in [1-3]. Theefoe, wha wa called hee he o pobable o o conibuing pah which follow fo he condiion ha he ie inegal ove he Lagangian wih fixed iniial and final ie i a iniu i hee called he opial pah. Wha wa called in [1-3] he aveage poiion which follow fo an aveage linea Langevin equaion, o alo fo he condiion ha he Lagangian ielf i a axiu i hee alo he o pobable poiion. Thi hen hold no only fo ye decibed by a linea Langevin equaion, bu alo by a non-linea Langevin equaion. L w( X ; υ) educe o L( X ; υ) = 3 We noe ha fo λ=,, he Lagangian fo he NESS of he ye; fo he nonequilibiu paaee υ, he NESS educe o an ES, in which, a aid befoe, no wok i done. 13

14 ~* C onideing oluion of () of he fo ~ exp ( ν ), one obain a facoizable equaion fo ν leading o fou poible value of ν : ν = ν, ν, ν, ν whee: x + + τ ν = (1) τ The geneal oluion of () i hen a upepoiion of hee fou pecial oluion in e of ν * { x } [ i a u of he opial pah { }, ] Nex, he eal pah x [ and i, ] * coplee n { x = x x } [, ] Δ and obey he ae bounday condiion a (19). Uing hi, one ha 4 : * 1 1 d Lw( X ; υ ) = w( ; ) [ & && ]. d L X υ 4D d Δx + Δx + τ Δx () τ Hee he la e on he. h.. of (), involving he deviaion Δx of he opial pah { x} * fo he eal pah x }, ake cae of he pope noaliza ion of he diibuion { funcion and can hencefoh be oied []. The fi e on he. h.. of () give, afe oe algeba, an explici Gauian expeion fo any iniial condiion f ( x, x, ) & i i []. P( W, ) in he long ie lii fo 5. Aypoic Wok Flucuaion Theoe Fo he aypoical wok diibuion funcion i of he fo: P( W, ) ~ [ ] π, (3) 1 W α β υ [ ( )] ( ) 4 α β υ exp 4α β υ ( ) fo any f x, x&,. We eak ha P( W, ) in he lii iniial diibuion ( ) i i o depend only on α and β, bu no on, i.e. i hold boh fo he Langevin equaion (1) w ih ineia ( ) and wihou ineia ( = : he ovedaped Langevin 4 We noe ha boh OM [4,5] and Beini e al [15] only conide he opial pah of he Lagangian L i.e. L = w wih λ. 14

15 equaion ). Fo (3) one deive, again fo any iniial diibuion, he aypoic wok flucuaion heoe: li P (W, ) = expw. P ( W, ) dependen of he fae, i.e. valid fo W ({ }) We noe ha (4) i in of equaion (15) fo boh ϑ = andϑ = 1. Lae we will genealize hi kech of he genealizaion of he OM heoy fo ES o NESS fo hi iple odel o oe geneal ye, and hen deeine in a oe geneal way he wok and hea needed o ainain a NESS. x (4) 6. Ineial effec fo finie ie he ciical a Alhough he aypoic flucuaion elaion fo P ( W, ) i independen of ineial effec, hey daaically appea fo finie ie. I i convenien o inoduce in hi conex a flucuaion funcion G( W, ) defined by : G( W, ) = W 5 P ( W, ) ln, (5) P ( W, ) wih, fo boh fae, ϑ = andϑ = 1, a iple aypoic behavio: li G( ) = 1. Since he behavio of G( W, ) fo finie depend on he iniial condiion we chooe in ode o obain concee eul, a NESS iniial condiion: 1 1 f ( x, x& β 1, ) = ( ) β ( x& υ) + κ[ x ] κ exp υ( τ ). (6) π Equaion (6) i a Gauian diibuion fo he iniial x and x& fo he paicle, ( ) aound hei aveage value of υ τ and υ, epecively, whee he aionay ( ) poiion υ τ of he paicle in he laboaoy fae i ued. 5 Thi flucuaion funcion diffe fo hoe ued befoe, whee a diffeen caling wa ued (cf. [16]. The eaon i ha (5) i adaped o he Gauian fo of he diibuion funcion G ( W, ) [] and povide oe infoaive expeion hee. 15

16 Evaluaing P ( W, ) fo he iniial condiion (6), a iple explici fo fo G() can be obained [], which i independen of υ and β, bu doe depend onϑ, i.e., on he fae. In () > fo > addiion, G (cf fig and 3). A ho dicuion of he elaxaion of G () in he laboaoy fae and he cooving fae o i aypoic value 1 follow, which i illuaed in fig and 3. () Figue The flucuaion funcion G a a funcion of ie fo he wok done on he dagged ( = 1) Bownian paicle in he laboaoy fae ϑ fo a NESS iniial condiion, fo [,66]. The / coloed line in he figue coepond o paaee value of he caled a of he Bownian paicle fo / 4. In hi figue we ued α = κ = 1 (o ha τ = 1 i a uni of ie and 1 = ). The one of popagaion fo > i clealy viible. 4 16

17 Figue 3 The flucuaion funcion ( ϑ = ). The depicion of hi figue i he ae a of figue. G() a a funcion of ie fo he wok done in he cooving fae A fi appoxiaion o G() fo lage i []: i.e. fo θ = 1(laboaoy fae) : G () while fo ϑ = 1(cooving fae) : τ τ ϑ G( ) ~ 1+, (7) τ + τ ϑ τ τ ~ 1+ = 1. (7a) τ τ ( ) 1 τ τ 1 ( ) ~ 1+, (7b) τ G Alhough (7) i only a fi appoxiaion o he aypoic fo of τ ( ) G(), i include an ipoan ineial conibuion o G viaτ in he laboaoy fae, which i aben in he cooving fae due o he d Alebe-like foce. Fuheoe, in he laboaoy fae, G( ) τ ~ 1 O, depending on whehe τ = τ τ, epecively. 17

18 α When τ = τ o =, a highe appoxiaion o G() i needed o obain he κ appoach o i aypoic value 1. * * Moe ipoanly, hee i a ciical a, uch ha fo >, G() exhibi an ocillaing behavio in ie when appoaching i aypoic value 1, in he laboaoy a well a in he cooving fae. Thi ocillaing behavio i a diec anifeaion of he geneal fo oluion of () fo exp ( v ) j ~* x a a upepoiion of fou e of he fo, wih given by equaion (1), ince all he v becoe coplex when v j α 4 τ >τ o fo > * * =. I appea indeed nueically ha a = a dynaical 4κ phae aniion ake place (cf fig and 3). * Fo > he poiion ~ * * x = x + (1 λ) υτ, i.e. he poiion of he Bownian paicle, ocillae wih a peiod: T * ) 1 j = π (1 (8) κ 1 coeponding o a fequency ω = ( 4τ τ 1) / τ = I ν wih (1). Clealy he exience of wo ie cale, one of which i τ and τ ae neceay fo he occuence of ie ocillaoy behavio in boh fae. We noe ha in he ovedaped cae, whee τ =, no ocillaion occu. We ephaize ha he finie ie behavio of he wok flucuaion funcion G() dicued hee fo he NESS ye i copleely aben in he ES. In fac, in he NESS he appoach of G () o i aypoic behavio depend daaically on he fae conideed. Thi i anohe anifeaion of he lack of Galilei invaiance of he Langevin equaion (3) and (5). Thee ae any ohe ineeing feaue in figue and 3, illuaing he diffeen behavio of G() in he laboaoy and he cooving fae, which could all be obeved, bu fo a deailed dicuion of which we efe o he lieaue []. value Finally, in view of he analogy of hi odel wih an elecic cicui [3], a ciical * L of he elf-inducance L could play he ae ole in he lae a he ciical a 18

19 in he foe. Thi would lead o a iila behavio of he wok flucuaion funcion G() a dicued in ecion Theodynaic fo a NESS In hi ecion we give a oe geneal peenaion of he genealizaion of he OM heoy o ye in a NESS, han ju fo he dagged Bownian paicle ye. Defining he hea Q and he wok W in a NESS a hoe needed o NESS ainain he NESS, a foal analogy o he fi and econd law of heodynaic in an ES can be foulaed fo he NESS. In he following all ubcip NESS will be dopped o iplify he noaion. The exience of NESS heodynaic law hould no be upiing ince hey eely expe fo he NESS he conevaion of enegy (NESS fi law) and he exience of a poiive enopy poducion (NESS econd law), a hey do in an ES. No aeen i ade abou he enopy of a NESS ielf. In ode o dicu a foulaion of he law of heodynaic in a NESS, we fi need definiion of he NESS hea Q and he NESS wok W fo a ye in uch a ae. Due o he vey naue of he NESS he pope choice of Q and W canno be ade wihou pecifying he coupling of he ye o ohe ye, wih which i ineac i.e i phyical nonequilibiu paaee, which will be collecively indicaed by μ. 6 Thi i an eenial diffeence beween NESS and ES heodynaic, whee in he lae hi abiguiy doe no occu. The oigin of he abiguiy i ha hea and wok in he genealized OM heoy ae defined via pah in a funcion pace [3]. In fac, hey ae defined in e of a fowad and a coeponding backwad pah, whee he lae i obained by a ieeveal pocedue. I i hi pocedue which i unique in an ES, bu no a pioi in a NESS. While in an ES, he only ie eveal i, apa fo, ha of he ine nal paicle velociy x& ino x&, hee i an addiiona l opion in a NESS o 6 A nonequilibiu paaee μ need no be a dynaical quaniy, bu can alo be a epeaue diffeence, a in wo exaple in [3]. We alo noe ha fo μ =, he OM eul fo a ye in an ES ae egained. In geneal, like he coodinae x, he nonequilibiu paaee μ can be veco, whoe coponen indicae he nube of degee of feedo of he ye of inee and of he nube of nonequilibiu paaee, epecively. NESS 19

20 change he ign of he nonequilibiu paaee μ. In he dagged Bownian paicle ye, fo exaple, one can o canno change he ign of he velociy υ. To dicu hi ie eveal abiguiy aiing in a NESS aheaically, i i convenien o inoduce a ie eveal opeao Î defined fo any funcional F( { x }; μ) by: Iˆ F ({ }; μ ) = F( { } μ) [, ] x x + ; wih, (7) whee μ chaaceize he nonequilibiu paaee, which pecify he NESS ye. fowad Thu unde hi ie eveal opeao he oion of he (Bownian) paicle on a pah { } (, ) pah { x } [, ] x i anfoed ino a oion along a backwad (ie-eveed) + wih he ae pah geoey bu iniial and final poiion given by x a i e and x a ie, epecively, while he coeponding poiion on he fowad pah wee x a ie and x a ie, epecively. Fo echnical eaon we eplace fo now on, wihou lo of genealiy, he oigin of ie a ( + ) o ha and (cf fig 4). Noe ha he iniial and final poiion on he backwad pah ae iden ical o he final and iniial poiion, on he fowad pah, epecively. The ie eveal fo he inenal oion of he paicle of inee, x, i indicaed by he ha (^) on he opeao Î. On he ohe hand he ie eveal pocedue aociaed wih he nonequilibiu paaee, indicaed collecively by μ, i indicaed by he ubcip, o ha one ha eihe Î o Î +. Thee opeao do o do no change he ign of he coponen of he nonequilibiu paaee μ unde a ie eveal opeaion, i.e. in he dagged paicle cae fo υ o υ o fo υ o υ, epecively.

21 Figue 4 A fowad pah fo an iniial ae x a ie o a final ae a ie (black line and aow) and i coeponding backwad pah fo an iniial ae a o a pah ae a, wih oppoie ie diecion and velociie (ed line and aow) (cf (8b)). Equivalen backwad pah obained fo he ae fowad pah by viiing he ae in he oppoie ode a o a (blue line and aow) (cf (8a)). x x x x x Î ha wo ipoan popeie: 1. I ˆ = 1. Iˆ d f ( X ; μ) = d f (&& x, x&, x, ; μ), (8a) = d f (&& x, x&, x, ; μ) (8b) = d f ( ; μ) fo any funcion f ( X ; ) wih X (&& x, x&, x, ) and X (&& x, x&, x, ) X μ. We noe ha he poibiliy of a ign change of μ i diecly elaed o he oe coplex ucue of a NESS han an ES. The cenal quaniy in he genealized OM heoy of a NESS i he hea, which i diecly elaed o he ochaic Lagangian on which he heoy i baed. Thi alo allow a diec deivaion of he econd law fo NESS heodynaic. We poceed in hee ep: deivaion of (a) he enopy poducion ae, (b) he econd law and (c) he hea aociaed wih he NESS. 1

22 Ad (a). We epaae he ochaic Lagangian (cf (11)) ino a ie-eveal yeic and a ie-eveal aniyeic pa: wih and [ Φ ( X ; μ) S ( X μ) ] 1 ; μ ) = & ;, (9) k L ( X B Φ [ L( X ; μ) + L( X ; )] ( X ; μ ) = k μ, (3) B [ L( X ; μ) L( X ; )] S & ( X ; μ ) k μ, (31) = B which coepond o he yeic and aniyeic pa wih epec o ie eveal of he Lagangian L. Hee, a eveywhee in hi pape, he + and ign on boh ide of an equaion coepond o each ohe. A ajo ep in he (genealized) OM heoy i now o idenify he phyical enopy poducion ae of he ye of inee wih he aniyeic pa of he & X ; μ. Lagangian L wih epec o ie eveal i.e. wih ( ) Ad (b). The econd law of heodynaic follow now in wo ep. 1. Thi law i expeed in e of he o pobable o aveage poiion x, given by he oluion of he aveage Langevin equaion, uing by: S ζ =. 7. One can hen pove he poiiviy of he enopy poducion ae & ( ; μ) & ( X ; μ) = ( X ; μ) >, (3) S Φ uing ha he diipaion funcion Φ > fo all pah in a NESS and (8). Fo fuhe deail we efe o [3] 8 9. The foulaion of he econd law hee in e of X X S X inead of X i a genealizaion of OM pocedue fo he ES, whee μ=. The 7 Alenaively, by he condiion ( X ; μ) 8 Fo he dagged Bownian paicle odel, Φ ( X α ; υ ) = T x& 1 + κ α T L = axiu. x μ = υ and υ & x&, ince in hi odel Iˆ υ = υ. 9 Thee i, in geneal, an addiional conibuion o Φ o S & which can be ignoed fo all eigh odel, conideed in [3].

23 appeaance of X can pehap be undeood on he bai ha he econd law i a acocopic law, foulaed in e of acocopic, i.e. aveage, quaniie X ahe han icocopic quaniie X. Ad (c). Finally he hea fo he NESS can be defined in e of he enopy poducion ae by: ({ } μ) T Q x ; = d S& ( X ; μ). (33) Thi how ha he hea i diecly elaed o he ie-aniyeic pa of he Lagangian (33) (cf (35)), o: ({ x} ; μ) = Q ({ x} ; μ). (34) I ˆ Q A diec connecion of Q ({ x }, ) Lagangian ({ }, μ) x wih he ie-ieveible pa of he baic L of he genealized OM heoy i wih (34): x = 1 Q ({ }; μ ) β d[ L( X ; μ) L( X ; μ)] = β 1 P ( ln Iˆ P { x }; μ) ( { x }; μ). (35) Hee P ({ }; μ) i popoional o he pobabiliy fo a pah{ x } fo he nonequilibiu x paaee μ (cf (13): P ; ~ exp d L( X ; μ. (36) ({ } μ) x ) Thi how ha i i he diffeen behavio of ({ }; μ) x P on he fowad pah and he backwad pah which i eponible fo he hea o ainain he NESS [4]. In ode o obain he fi law of NESS heodynaic in a concee aheaical fo, one ha o inoduce he inenal enegy and he enegy conevaion law. The inenal enegy E ( x, x& ; μ) i given a a u of a kineic and a poenial 3

24 enegy conibuion and i aued o be ie eveal invaian, leading o an inenal enegy diffeence beween he final and he iniial ie: and Δ E E ( x, x& ; μ) E ( x, x& ; μ), (37a) = Δ = Δ Î E E. (37b) We define he wok W done o ainain he NESS a he u of he hea poduced and ubequenly eoved fo he ye and he change of he inenal enegy of he ye i.e. (cf [3]): W ({ x}; μ ) = Q ({ x}; μ) + ΔE. (38) We noe ha he wok W ({ }; μ) i, like he hea and he inenal enegy, x aniyeic unde ie eveal of he exenal nonequilibiu paaee μ : ˆ W ({ x }; μ ) W ({ }; μ) I x =. (39) Thi i a local foulaion of he fi law of NESS heodynaic fo each pah { x } 1 foally idenical o he fi law of equilibiu he heodynaic. Thee local foulaion of he fi and econd law fo a NESS can be genealized o a global foulaion of he fi law of NESS by caying ou a funcional aveage (cf (14)) ove a egion in funcion pace beween any iniial and final ae a - and, epecively. Thi funcional aveage appea a he analogue fo ye wih a ochaic dynaic of a phae pace aveage fo ye wih a deeiniic dynaic. Becaue of he ie eveal abiguiy, hee will be fo any nube of nonequilibiu paaee, in pinciple, a any foulaion of he fi wo law of NESS heodynaic, a hee ae exenal nonequilibiu paaee, indicaed by he nube of coponen of he veco μ. Thi abiguiy heefoe lead o he fundaenal queion, which of he exenal nonequilibiu paaee μ o μ ha o be choen o obain he coec phyical wok and hea o ainain he NESS. The wo law of NESS heodynaic alone do no give a eoluion of hi abiguiy. 1 We ephaize ha he NESS heodynaic foulaed in (3) and (38) hould be diinguihed fo he well-eablihed field of Ieveible Theodynaic (o Theodynaic of Ieveible Pocee), a dicued e.g. in [17] and which i baed on wo ealie pape by Onage [18] han [4,5]. 4

25 I appea ha one can only ake a unique phyically accepable choice a poeioi on phyical gound afe having pecified a pioi he NESS ye conceely i.e. i nonequilibiu paaee. One can expec, a i bone ou by he exaple we have udied, ha diffeen odel will equie diffeen choice of Î o Î, o obain a phyically accepable heoy. I appea ha a eoluion 11 of hi abiguiy can be obained by aking ino accoun he paiy wih epec o ie eveal of he nonequilibiu paaee, which chaaceize he ye in he NESS. Then i hould follow ha he (pope) phyical wok, hea and inenal enegy obained aify he hee condiion of ie eveibiliy of he inenal enegy, poiiviy of he aveage wok done on and he aveage hea eoved fo he ye in a NESS. Thi ha been illuaed in [3] fo eigh diffeen odel, wih a vaiey of nonequilibiu paaee. One of hee odel i he dagged paicle odel ued a an illuaion in hi pape. Addiional odel include elecical cicui, a diven oion pendulu, and an enegy cuen geneaed by a epeaue diffeence. Thee odel ae devied o illuae diffeen eoluion of he abiguiie fo he choice of he pope phyical wok, hea, and inenal enegy aociaed wih he. In fac, fou equie and fou Î Î+, epecively. We noe ha eul fo he odel conideed hee have been o can be veified expeienally Tanien Wok Flucuaion Relaion and Nonequilibiu Deailed Balance Relaion Tanien flucuaion elaion [19]can be deived fo nonequilibiu deailed balance elaion. We will illuae hi on he dagged Bownian paicle odel and efe fo a oe geneal foulaion and connecion wih he lieaue o [1-3, 19,,]. We fi decibe wo nonequilibiu deailed balance elaion, one fo he echanical (NESS heodynaic) wok W done on he ye o ainain i in a NESS and anohe fo he ficion wok R done by he Bownian paicle o ovecoe he ficion in he fluid. 11 Thi eoluion i no incopoaed in he pape [1-3]. 5

26 Fo he echanical wok W in he laboaoy fae, defined by (15a), he following nonequilibiu deailed balance elaion hold (cf [1]): ({ x ) exp ( β W };υ exp d ( X ;υ ) L eq f ( x& ), x = L( X ; ) exp υ eq &, (4a) f ( x, ) while fo he ficion wok R, in he cooving fae, defined below by (41), one ha: exp β R( y, y ; υ ) d = d exp ( Y ;υ ) exp ( Y ) L eq L eq ;υ x f ( y& ), y &. (4b) f ( y, ) On he lef hand ide (l.h..) of boh equaion i a Bolzann faco, which incopoae he echanical wok W o he ficional wok R, epecively. Since boh vanih in equilibiu, when he nonequilibiu paaeeυ =, hee elaion educe hen o he uual equilibiu deailed balance elaion. While in (4a) he echanical wok i given by an inegal ove he full pah { } (cf (15a)), in (4b) he ficion wok i given by a bounday e: R x [, ] y ( y y ; ) = d ( αy ) υ = αυ ( y y )., υ & (41) The ohe wo faco on he l.h.. and.h.. of (4a,b) give he pobabiliie aociaed wih he appopiae fowad and backwad pah in funcion pace and he coeponding iniial ae fo hee wo pah, epecively. The anien flucuaion elaion fo W and R, which can be deived fo (4a) and (4b), epecively fo an iniial equilibiu diibuion, uing (11) and (15a) and ead [1]: fo W and iilaly fo R: (, ) ( W, ) P W P ( R, ) ( R, ) P P Hee W and R ae boh dienionle wok (cf (17)). = expw, = exp R. (4a) (4b) 6

27 Copaing he aypoic flucuaion heoe (4) fo W and he anien flucuaion heoe fo W and R, (4), we noe ha alhough in boh cae wok i involved, fo he echanical wok W he appopiae opeao i Î (i.e. x & x& and υ υ) y & & y, while fo he ficion wok R, on he ohe hand, he opeao i ( (i.e. and υ υ) ha o be ued in he Lagangian (4a) and (4b), epecively, o obain (41a) and (41b), epecively. We illuae hi fuhe fo R fo he ovedaped cae in he cooving fae, i.e., (6) wih =, fo he dagged Bownian paicle odel by chooing an iniial condiion of he fo: wih a conan paaee ( y ) f eq ( y φ) f, = + υτ, (43a) φ 1 and f eq κβ π Uing he pobabiliy diibuion funcion flucuaion elaion: 1 ( ) y = exp κy β P ( R, ) P ( R, ) = exp 1 φ P( - R, ) Î +. (43b) fo hi iniial condiion [1] give he [( ) R] We noe ha (44) only ha he fo of a anien flucuaion elaion fo φ = (cf (4b)), i.e., fo an iniial equilibiu diibuion funcion, while fo φ =1, i.e. fo an iniial (44) nonequilibiu eady ae diibuion, P ( R, ) i Gauian, wih a peak a R= and P ( R, ) = P( - R, ) o ha hei aio i 1. We ephaize ha he above enioned diffeence in he definiion of W and R - an inegal a oppoed o a bounday e i uliaely eponible fo he fac ha, while hee ae boh a anien (4a) and an aypoic flucuaion elaion (4) fo W, hee i only a anien flucuaion fo R (4b) and no aypoic flucuaion elaion, ince R neve looe i eoy of he iniial ae. One can uaize he diffeence beween he anien and he aypoic flucuaion elaion in ha he foe hold fo all ie, bu only fo an equilibiu 7

28 iniial condiion, while he lae hold fo all iniial condiion, bu only fo (ee alo [1]). 9. Coen and Open Queion 1. The baic aupion of hi pape, a o he pope definiion of hea and wok fo a ye in a NESS, bea oe eblance wih a cae conideed by Landau and Lifhiz []. They conideed a coplex heally iolaed ye, no in heal equilibiu, coniing of a nube of (ub) ye which ineac wih each ohe and can do wok on exenal objec. The appoach of uch a ye o equilibiu, a well a he final equilibiu ae ielf, ae hen non-unique and can only be deeined if one pecifie he coplex ye conceely, i.e. idenifie i nonequilibiu paaee, chaaceizing he coupling beween all he ubye and o he exenal objec. They ay: Hee one i only ineeed in he wok poduced due o he fac ha he ye i no in equilibiu. Thi ean ha we u ignoe he wok done [e.g.] by a geneal expanion of he ye, wok which could alo have been done by he ye in an equilibiu ae. Thi la condiion alo peain o hi pape when defining he hea and he wok in a NESS.. In addiion o he abiguiy elaed o he pope choice of he ign of he nonequilibiu paaee, hee i a econd abiguiy, iila, bu diffeen fo, ha which exi in he heodynaic of a ye in an ES. Thee, a fa a he fi law of heodynaic i concened, one can add o he hea and he wok he ae conan wihou violaing he (enegy conevaion) law. Hee, fo a ye in a NESS, he ae abiguiy exi wih epec o he epaaion of he (NESS) hea ino a u of (NESS) wok and (NESS) inenal enegy diffeence, which can alo be done only up o a coon conan, epecively. Thi can e.g. be illuaed in he dagged Bownian paicle odel, whee he diffeence in he wok in he laboaoy fae and he cooving fae equal he diffeence in he inenal enegy (cf 15c,d), while he hea i he ae in he wo fae. 3. The deivaion of he Second Law of NESS Theodynaic peened hee i incoplee fo wo eaon. Fi, becaue he foce acing on he ye have o be linea. Thi ake a genealizaion of he peen heoy o he non-linea cae 8

29 paiculaly ipoan. In he dagged Bownian paicle odel hi ean a non-linea foce in he paicle poiion (cf (8)) and/o e.g. a non-linea dependence of he ficion of he paicle on he dagging velociy. 4. We have eiced ouelve in hi pape o a dicuion of popeie of he wok in a ye in a NESS. Fo a dicuion of he hea in uch a ye, we efe o he oiginal pape [1,3]. 5. We noe ha he eoluion of he abiguiy of he choice of he pope backwad pah fo a given fowad pah enioned in Secion 7 yield he hea, wok and enegy needed o ainain he ye in a NESS. Thee ae quaniie alo aify he law of NESS heodynaic. Howeve, a he dagged Bownian paicle odel how, one can obain in addiion o he echanical wok, which ha all he above popeie, an ohe phyical quaniy in hi odel: he ficion wok, which i alo a eaueable phyical quaniy bu ha diffeen popeie (e.g. diffeen flucuaion elaion) han he echanical wok. I i unclea, a peen, how geneal hi conequence of he ign abiguiy of he nonequilibiu paaee found in he dagged paicle odel i. 6. Fo a geneal poin of view, one can ay ha he behavio of he ye in a NESS conideed hee, i deeined by ha of i Langagian L ( ; μ) a a funcion of he pah { x } and he nonequilibiu paaee() μ, unde he acion of he ie X eveal opeao Î. When he pobabiliy of a fowad pah diffe fo ha of he coeponding backwad pah in funcion pace, diipaion occu, due o he enopy poducion in he NESS (cf (33) and (35)) 1. On he ohe hand, he behavio of a ye in an ES, whee hee ae no nonequilibiu paaee μ and only he velociy x& (and he ie ) will change ign upon ie eveal. Alhough hee i hen ill no equal pobabiliy fo a fowad and a backwad pah (e.g., cf (11) and (13)) no enopy poducion will occu on aveage. 7. In hi pape we confined ouelve o ye of inee, whoe NESS popeie ae due o he being pa of a lage ye, which i chaaceized by 1 The ae i ue fo a deeiniic ye in phae pace, whee he phae pace conacion epeen he diipaion o enopy poducion [8, 1]. 9

30 phyical nonequilibiu paaee μ. In fac, fo he cla ye of inee conideed in [3], hee paaee had a diinc paiy wih epec o ie eveal. Fo he OM pah inegal appoach hi i elevan fo he unique deeinaion of he pope backwad pah coeponding o a given fowad pah of he ye of inee in a NESS. Howeve, hee ae ohe nonequilibiu paaee a, e.g., he ficion coefficien α in he dagged Bownian paicle odel, o he eiance in an elecic cicui, which epeen he coupling o a hea bah uounding he ye of inee. They wee no included in he nonequilibiu paaee μ. Thi i becaue hee i hee no ie eveal abiguiy, ince he econd law equie ha boh ae >. I i no clea a peen how, in geneal, he choice of unique coeponding fowad and backwad pah fo any ye in a NESS hould be ade, wihou a oe deailed claificaion of hei poible copoiion. 8. In hei 1953 pape [4], Onage and Machlup enion poible genealizaion of hei appoach o ohe ye han hoe in an ES. They ay: The exenion o [ohe ye]... uually involve iply a change of language... Bu again, he exenion o open ye (and eady ae) i puely foal. In ou genealizaion of he OM appoach o ye in a NESS hi appea no o be o. Acknowledgeen The auho i indebed o D. T. Taniguchi and D. H. Touchee fo any helpful dicuion and M. J Alonzo fo aiance in he pepaaion of hi anucip. He alo gaefully acknowledge financial uppo of he Maheaical Phyic poga of he Naional Science Foundaion unde gan PHY Refeence [1] Taniguchi T and Cohen E G D, 7 J.Sa.Phy [] Taniguchi T and Cohen E G D, 8 J.Sa.Phy [3] Taniguchi T and Cohen E G D, 8 J.Sa.Phy [4] Onage L and Machlup S, 1953 Phy. Rev [5] Machlup S and Onage L, 1953 Phy. Rev [6] Touchee H and Cohen E G D, o be publihed 3

31 [7] Evan D J, Cohen E G D and Moi G P, 1993 Phy. Rev. Le [8] Gallavoi G and Cohen E G D, 1995 Phy. Rev. Le [9] Kuchan J, 1998 J. Phy. A: Mah. Gen [1] Lebowiz J L and Spohn H, 1999 J.Sa.Phy [11] Cohen E G D and Van Zon R, 7 C. R. Phyique 8 57 [1] van Zon R and Cohen E G D, 3 Phy. Rev. E [13] Tolan RC, 1938 The Pinciple of Saiical Mecahnic, Oxfod Univ. Pe 4 [14] Touchee H, 8 Pepin axiv [15] Beini L, De Sole A, Gabielli D, Jona-Lainio G and Landi C, 1 Phy. Rev. Le Beini L, De Sole A, Gabielli D, Jona-Lainio G and Landi C, Phy. Rev. Le Beini L, De Sole A, Gabielli D, Jona-Lainio G and Landi C, 6 Phy. Rev. Le [16] van Zon R and Cohen E G D, 3 Phy. Rev. Le [17] De Goo S R and Mazu P, 196 Non-Equilibiu Theodynaic(Aeda: Noh-Holland) [18] Onage L, 1931 Phy. Rev Onage L, 1931 Phy. Rev [19] Evan D J and Seale D J, 1994 Phy. Rev. E Evan D J and Seale D J, Adv. Phy. E [] Cook G E, 1999 Phy. Rev. E Cook G E, Phy. Rev. E [1] Cohen E G D and Gallavoi G, 1999 J.Sa.Phy [] Landau L D and Lifhiz E M, 1958 Saiical Phyic (Addion-Weley) 55 [3] Gallavoi G, 8 Eu. Phy. J. B,