Entropy production in irreversible systems described by a Fokker-Planck equation

Transcription

1 Entropy proucton n rreversble systems escrbe by a Fokker-Planck equaton Tâna Tomé an Máro J. e Olvera Insttuto e Físca Unversae e São Paulo Caxa postal São Paulo- SP, Brazl (Date: July 3, 010) We analyze the rreversblty an the entropy proucton n nonequlbrum nteractng partcle systems escrbe by a Fokker-Planck equaton by the use of a sutable master equaton representaton. The rreversble character s prove ether by nonconservatve forces or by the contact wth heat baths at stnct temperatures. The expresson for the entropy proucton s euce from a general efnton, whch s relate to the probablty of a trajectory n phase space an ts tme reversal, that makes no reference a pror to the sspate power. Our formalsm s apple to calculate the heat conuctance n a smple system consstng of two Brownan partcles each one n contact to a heat reservor. We show also the connecton between the efnton of entropy proucton rate an the Jarzynsk equalty. PACS numbers: Ln, Jc, 0.50.Ga I. INTRODUCTION A thermoynamc system n a statonary state s characterze by havng ts propertes such as energy an entropy nvarant n tme. In ths regme, there can be no flow of a conserve quantty such as energy to or from the outse because t cannot be create. However, there mght be a flow of a nonconserve quantty such as the entropy. The flux of entropy to the outse wll be equal to the entropy spontaneously generate nse the system. Only n thermoynamc equlbrum there wll be no proucton of entropy. A nonequlbrum thermoynamc system n the statonary state s thus characterze by a contnuous proucton of entropy. In a transent state, the change n the entropy s not only ue to the entropy flow but s also ue to the spontaneous generaton of entropy wthn the system so that, n general, the tme ervatve of the entropy S of a system can be splt nto two parts [1 3] t = Π Φ, (1) where Π s the entropy proucton rate, whch s always nonnegatve, an Φ s the entropy flux rate from the system to the envronment. In the statonary regme, the entropy rate /t vanshes an Π = Φ. If n aton the system s out of equlbrum then Π = Φ > 0; f t s n equlbrum Π = Φ = 0. The quantty Φ, efne as the flux rate from nse to outse of the system, wll be postve n the nonequlbrum statonary state. The constructon of a mcroscopc theory of nonequlbrum thermoynamc systems s face wth two major problems relate to entropy. The frst concerns the efnton of nonequlbrum entropy an the secon the efnton of entropy proucton. For systems n equlbrum, the entropy S s relate to the probablty P(η) of fnng the system n a certan state η by the well known Boltzmann-Gbbs expresson S = η P(η)lnP(η). () For nonequlbrum systems, escrbe by a tme epenent probablty strbuton P(η,t), t s natural to exten the Boltzmann-Gbbs expresson to these systems. The out-of-equlbrum tme-epenent entropy S(t) s then efne by S(t) = η P(η,t)ln P(η,t), (3) an therefore vares n tme accorng to the specfc ynamcs that governs the evoluton of the probablty strbuton P(η, t). The secon problem, the efnton of entropy proucton rate Π, s equvalent to the problem of efnng the entropy flux rate Φ snce these two quanttes are relate wth each other by means of (1). These two quanttes shoul necessarly be relate to the tme evoluton of P(η,t) an therefore cannot be efne n terms of P(η,t) alone. We nee to known the ynamcs that governs ts tme evoluton. We assume that the system evolves n tme accorng to a Markovan process on a scretze phase space, efne by a transton rate W(η η) from state η to state η. Wthn ths framework, Φ an Π wll be relate to W. The tme evoluton of the probablty strbuton s assume to be governe by the master equaton [4, 5] where t P(η,t) = η J (η η,t), (4) J (η η,t) = W(η η )P(η,t) W(η η)p(η,t) (5) s the probablty current. Here we wll be concerne manly wth the stuy of the Fokker-Planck equaton [4 6] whch we regar as comng from an approprate contnuous lmt of the master equaton (4), as we shall see.

2 From the probablty current, one etermnes the flux rate Φ E of any state functon E(η), whch s Φ E (t) = η,η J (η η,t)e(η), (6) an clearly vanshes n the statonary state as t shoul. A natural way to efne the entropy flux rate s as follows Φ(t) = η,η J (η η,t)ln W(η η). (7) From ths efnton we see mmeately by means of (1), (3) an (4) that the entropy proucton rate s gven by Π(t) = η,η J (η η,t)ln[w(η η)p(η,t)], (8) an expresson that s always nonnegatve an equvalent to that ntrouce by Schnakenberg [7]. In thermoynamc equlbrum, when mcroscopc reversblty takes place, J vanshes an both Φ an Π vansh as well. It worth mentonng that ths efnton of entropy proucton makes no a pror reference to any thermoynamc quantty such as sspate energy as s usually one. It s a unversal efnton n the same sense as the efnton of entropy (3) s unversal. The proucton of entropy n systems escrbe by a stochastc process or by a master equaton has been the subject of several stues [8 5]. Ths nclues the numercal calculaton of entropy proucton n nonequlbrum lattce gas moels [16]. Here we are concerne wth the proucton of entropy n nonequlbrum nteractng partcle systems escrbe by Langevn equatons or, n an equvalent way, by the assocate Fokker- Planck equaton, whch s the approprate framework to escrbe nonequlbrum system uner temperature graents [10, 5]. Our man purpose here s to use expressons (7) an (8) to etermne the entropy flux rate Φ an the entropy proucton rate Π n rreversble systems escrbe by Langevn equatons. The proucton of entropy n systems escrbe by Langevn equatons n the overampe lmt has been prevously stue [17, 18]. Here we conser the general case. Systems escrbe by a Fokker-Planck [4 6] equaton follows a Markovan process n contnuous tme an contnuous confguraton space. The rreversble character comes from the type of forces enterng the Langevn equatons or from the type of contact of the system wth the envronment. As we shall see, f the system s n contact wth a heat reservor that keeps the temperature T constant but the forces are nonconservatve the resultng entropy proucton rate s strctly postve. We wll show that n ths case the sspate power P s relate to the entropy proucton rate by Π = P/T, whch s a fluctuaton sspaton type relaton. When the forces become conservatve but the system s n contact wth more than one heat reservors at stnct temperatures the resultng entropy proucton rate s also nonzero. We apply the results obtane here to a smple system of ths type consstng of two Brownan partcles connecte wth each other by a harmonc force an each one to heat baths at stnct temperatures [6, 7]. Whenever the temperatures are stnct there wll be a heat flow through the system from one reservor to the other. By calculatng the proucton of entropy we etermne the thermal conuctance. Although the forces are conservatve, the fference n temperatures keeps the system n a nonequlbrum state. The proucton of entropy vanshes only when the forces are conservatve an the system s n contact wth only one heat bath. We use the expressons (7) an (8) for the entropy flux an entropy proucton to etermne an equalty of the Jarzynsk type [8 3]. Ths s carre out by conserng the rato of the probablty of a gven trajectory n phase space an the probablty of the tme reversal trajectory. II. FOKKER-PLANCK EQUATION A. Langevn equatons We conser a system of n nteractng partcles that evolves n tme accorng to the followng set of couple Langevn equatons m v t = F αv + F (t), (9) x t = v, (10) where x an v are the poston an velocty of the - th partcle. We are assumng that the mass m an the coeffcent α are the same for all partcles an that the force F actng on the -th partcle epens only on the postons. The forces F mght not be conservatve. The quantty F (t) s the ranom force, a stochastc varable havng the propertes an F (t) = 0, (11) F (t)f j (t ) = αt δ j δ(t t ), (1) where, T > 0 s a constant that mght be stnct for each partcle. The assocate Fokker-Planck equaton, that gves the tme evoluton of the probablty strbuton P(x, v, t), where x an v enote the vectors whose components are the varables {x } an {v }, respectvely, s gven by t P = 1 (F P αv P) m v + α m x (v P)+ T v P, (13)

3 3 whch we wrte n the form t P = where K an J are gven by an ( K + ) J, (14) v K = F P + v P, (15) m v x J = αv m P αt m v P. (16) The Fokker-Planck equaton shoul be solve nse a certan regon of the space spanne by the jont sets of varables x = {x } an v = {v }. We assume that at the bounary of ths regon the probablty strbuton P(x, v, t) vanshes. The set of Langevn equatons (9) an (10) an the assocate Fokker-Planck equaton (13) are assume to escrbe a system that s n contact wth several heat baths, each one havng a temperature T. The contact s accomplshe by the ranom forces F. If T = T s nepenent of then we may say taht the system s n contact wth just one heat reservor at temperature T. If n aton the forces F are conservatve then n the statonary state the system s n equlbrum. B. Equlbrum contons At the statonary state the probablty strbuton P(x,v) s nepenent of tme an s the soluton of (K + v J ) = 0. (17) When mcroscopc reversblty hols we shall see n secton IV that J = 0, that s, αv m P + αt m whch mples that P must be of the form v P = 0, (18) P(x,v) = χ(x)φ(v), (19) that s x an v are nepenent ranom vector varables. Settng (19) nto the efnton (15) of K, t follows that ( K = v φ F χ + ) χ. (0) T x But, snce J = 0, t follows from (17) that the summaton of K must vansh, that s, K = 0. (1) Takng nto account that the expresson nse the parentheses n equaton (0) epens only on x an that (1) must be hel for any velocty, t follows that each term of the summaton n (1) must vansh, that s, K = 0, so that x lnχ = F T. () From ths equaton t follows mmeately that 1 T F x j = 1 T j F j x, (3) for any par,j, whch s the esre equlbrum conton. That s, mcroscopc reversblty mples that the forces F an the parameters T must be such that they satsfy (3). If the temperatures are all the same, then T = T j, (4) F x j = F j x, (5) that s, the forces F must be conservatve. In ths case the system s n thermoynamc equlbrum an s escrbe by the canoncal Gbbs probablty strbuton P(x,v) = 1 exp{ βh(x,v)}, (6) Z whch follows rectly from (17) an (), where H(x,v) = 1 m v + V(x) (7) an T = 1/β. In aton, the forces are relate to the potental V by F = V x. (8) When the contons (5) an (4) are val, the Langevn equatons an the assocate Fokker-Planck equaton escrbe a system wth conservatve forces n contact wth a heat reservor at temperature T = 1/β. However, our am here s to stuy systems that o not satsfy these contons so that, n the statonary state, they are rreversble. We stngush two types of nonequlbrum stuatons. In the frst, the forces F are conservatve but the temperatures T are not all the same. In the secon, the temperatures are all the same but the forces F are nonconservatve. C. Entropy proucton To etermne an expresson for the entropy flux rate an entropy proucton rate we follow a metho smlar

4 4 to that use by Sefert [17] for the case of overampe moton. We start from the entropy S, efne by S = P lnpxv. (9) Its tme ervatve s t = lnp P xv, (30) t or, usng the Fokker-Planck equaton as gven by (14), t = K lnpxv + lnp J xv. (31) v The ntegral n the frst summaton vanshes entcally as can be seen by replacng K by ts efnton, gven by equaton (15), an by performng an ntegraton by parts. The result s ( ) F P P K lnpxv = + v xv = 0, m v x (3) where the secon equalty s obtane by another ntegraton by parts an by takng nto account that F epens on x but not on v. Usng ths last result an ntegratng the secon ntegral n (31) by parts gves t = But from the efnton of J, so that t = J v lnpxv. (33) lnp = mv m J v T αt P, (34) ( m v J + m J ) xv. (35) T αt P The secon term, whch s always nonnegatve, s entfe as the entropy proucton rate Π = m J xv. (36) αt P A smlar expresson for Π has been obtane for the case of overampe moton [17, 18]. In accorance wth relaton (1), the entropy flux rate shoul be then Φ = m v J xv. (37) T In the statonary state, /t = 0 an Π = Φ. In equlbrum J = 0 an both the entropy flux rate an entropy proucton rate vansh, Π = Φ = 0. In nonequlbrum statonary state, J 0 an Π = Φ 0. Equatons (36) an (37) gve the esre expresson for the entropy proucton rate an entropy flux rate for a system escrbe by the Fokker-Planck equaton (13). In secton IV, we wll show that these two expressons can actually be euce from the general expressons (7) an (8) by an approprate master equaton representaton of the Fokker-Planck equaton (13). D. Entropy flux an energy sspaton Usng the efnton of J, gven by equaton (16), the entropy flux rate may be wrtten as Φ = ( α v P + α ) T m v P xv. (38) v Integratng the secon ntegral by parts, Φ = ( α v P α ) T m P xv, (39) or Φ = 1 ( α v α ) T m T. (40) Let us etermne the average rate of energy sspaton P of each partcle. It has two contrbutons: one s the work sspate per unt tme, v F, an the other s the ecrease n knetc energy per unt tme, (m/)(/t)v. That s, P = v F m t v. (41) Now, from the Fokker-Planck equaton, t s straghtfowar to obtan the result m t v j = v j F j α vj + α m T j, (4) whch follows after some approprate ntegraton by parts. Replacng ths result n equaton (41), we get an equvalent expresson for the sspaton power of each partcle, namely P j = α v j α m T j. (43) From ths result we may wrte the entropy flux rate as Φ = P T. (44) If the temperatures are the same T = T then Φ = P T. (45) Where P = P s the total energy sspate per unt tme. In the statonary state, v s a constant so that v /t = 0 an P = v F. We are then left wth the followng expresson Π = Φ = P T = 1 T v F, (46) val n the statonary regme. Usng the nterpretaton that each partcle s n contact wth a heat reservor

5 5 at temperature T, ths result says that the entropy proucton rate s a sum of terms each one beng the rato between the sspaton of energy per unt tme, that s, the sspate power, an the temperature of the heat bath. Let us conser now the case n whch the forces are conservatve n whch case F = V/x. From the Fokker-Planck equaton an after an approprate ntegraton by parts t s straghtforwar to show that t V = v V x = v F. (47) Therefore, the total sspate power s P = ( P = V + ) m t v, (48) If, n aton, T = T s the same for all stes, then the entropy flux rate s gven by ( Φ = 1 V + ) m T t v. (49) From ths equaton we see that the entropy flux rate s equal to the rato between the ecrease n the nternal energy per unt tme an the temperature T of the heat bath. III. THERMAL CONDUCTION IN A SIMPLE SYSTEM A. Equatons of moton We apply the prevous results to a nonequlbrum smple system consstng of two couple partcles of the same mass m, movng along a straght lne. They nteract wth each other an each one s n contact wth thermal reservors at fferent temperatures. Ther movements are governe by the Langevn equatons an m v 1 t = k(x 1 x ) k x 1 αv 1 + F 1 (t), (50) m v t = k(x x 1 ) k x αv + F (t), (51) where x an v = x /t are the poston an velocty of the -th partcle. The quanttes k an k are sprng constants an α s the frcton constant. The ranom forces F 1 an F are Gaussan whte noses wth the propertes F (t) = 0, (5) F (t)f j (t ) = αt δ j δ(t t ), (53) where T 1 an T are the temperature of the thermal reservors connecte to partcles 1 an, respectvely. If we efne the forces F 1 (x 1,x ) an F (x 1,x ) by an F 1 = k(x 1 x ) k x 1, (54) F = k(x x 1 ) k x, (55) then equatons (50) an (51) have the same structure of (9). The assocate Fokker-Planck equaton for the probablty ensty P(x 1,x,v 1,v,t) s gven by t P = x 1 (v 1 P) (v P) 1 (F 1 P) x m v 1 1 (F P) + λ (v 1 P) + λ (v P) m v v 1 v + Γ 1 v1 P + Γ v P, (56) where λ = α/m an Γ = αt /m. To etermne the entropy proucton rate t s necessary to compute averages of the type x x j, x v j an v v j. Snce the Langevn equatons (50) an (51) are lnear equatons they can be solve exactly an so can the Fokker-Planck equaton. From the soluton P(x 1,x,v 1,v,t) of the Fokker-Planck equaton we etermne the esre averages. Here, however, we follow a stnct proceure. Instea of fnng the probablty P tself we set up equatons for the those averages an solve them. From the Fokker-Planck equatons t s straghtforwar to reach the followng equatons for the averages t x 1 = x 1 v 1, (57) t x = x v, (58) t x 1x = x 1 v + x v 1, (59) t x 1v 1 = v 1 K x 1 + L x 1 x λ x 1 v 1, (60) t x v = v K x + L x 1 x λ x v, (61) t x 1v = v 1 v K x 1 x + L x 1 λ x 1 v, (6) t x v 1 = v v 1 K x 1 x + L x λ x v 1, (63)

6 6 t v 1 = K x 1 v 1 + L x v 1 λ v 1 + Γ 1, (64) t v = K x v + L x 1 v λ v + Γ, (65) t v 1v = K x 1 v + L x v K x v 1 + L x 1 v 1 λ v 1 v, (66) where K = (k + k )/m an L = k/m. B. Entropy proucton n the steay state In the statonary regme, the set of equatons above are reuce to followng set of equatons x 1 v 1 = x v = v 1 v = 0, (67) x v 1 + x 1 v = 0, (68) v 1 K x 1 + L x 1 x = 0, (69) K x 1 x + L x 1 λ x 1 v = 0, (70) K x 1 x + L x λ x v 1 = 0, (71) v K x + L x 1 x = 0, (7) L x v 1 λ v 1 + Γ 1 = 0, (73) L x 1 v λ v + Γ = 0. (74) These equatons are lnear n the averages an can realy be solve wth the results v 1 = Γ 1 + Γ 4λ v = Γ 1 + Γ 4λ + Kλ(Γ 1 Γ ) 4(L + Kλ ), (75) Kλ(Γ 1 Γ ) 4(L + Kλ ), (76) x 1 v = x v 1 = L(Γ 1 Γ ) 4(L + Kλ ), (77) x 1 = K(Γ 1 + Γ ) 4λ(K L ) + λ(γ 1 Γ ) 4(L + Kλ ), (78) x = K(Γ 1 + Γ ) 4λ(K L ) λ(γ 1 Γ ) 4(L + Kλ ), (79) x 1 x = L(Γ 1 + Γ ) 4λ(K L ). (80) In the statonary state Π = Φ an we may use expresson (40) for the entropy flux rate to get the entropy proucton rate, gven by or Π = α T 1 v 1 + α T v α m, (81) Π = λ Γ 1 v 1 + λ Γ v 1 λ. (8) Takng nto account the results above for v 1 an v an after straghtforwar calculatons we arrve at the followng expresson for the entropy proucton rate Π = (Γ 1 Γ ) Γ 1 Γ λl L + Kλ. (83) Makng the substtutons K = (k + k )/m, L = k/m, λ = α/m, an Γ = αt /m, we get Π = (T 1 T ) T 1 T αk [mk + (k + k )α ], (84) From the relaton Φ = κ(t 1 T ) /T 1 T between entropy proucton Π an the thermal conucton κ [33], we get κ = αk [mk + (k + k )α ]. (85) whch agrees wth the result obtane by a stnct metho [7]. IV. PRODUCTION OF ENTROPY IN A MARKOVIAN PROCESS A. Master equaton representaton In ths secton we emonstrate two mportant results that we have use prevously. The frst one s relate to the current J as efne by equaton (16). In the steay state an f mcroscopc reversblty hols then J = 0 for each. The secon result refers to the expressons (36) an (37) for the entropy proucton an entropy flux rates. We show here that these two expressons can be obtane from formulas (7) an (8), val for systems escrbe by a master equaton. The emonstraton begns by scretzng the Fokker-Planck equaton (13) transformng t on a master equaton of the form t P(η) = η {W(η η )P(η ) W(η η)p(η)}, (86)

7 7 where η = (x,v) an η = (x,v ) enote scretze states n phase space an W(η η) s the rate of transton from the state η to the state η. To smplfy the notaton we are omttng the tme epenence of P(η). We use two types of scretzatons. In the frst we assume that the velocty v wll ncrease or ecrease by an amount a. Ths proceure s use to wrte own the followng approxmatons for the ervatves of P wth respect to v an v P = 1 a {P(x,v+ ) P(x,v) + P(x,v )} (87) (v P) = 1 v a {(v + a)p(x,v + ) (v a)p(x,v )}. (88) The notaton v ± stans for the vector whose components are the same as those of the vector v except the -component whch equals v ± a. In the secon type of scretzaton the poston x wll ncrease by an amount bv whereas the velocty v wll ncrease by F b/m. Ths proceure s use to wrte own the approxmaton 1 (F P) (v P) = 1 m v x b {P(x,v ) P(x,v)}. (89) The notaton x ± stans for the vector whose components are the same as those of the vector x except the -component whch equals x ± bv an v ± stans for the vector whose components are the same as those of the vector v except the -component whch equals v ±bf /m. Usng the approxmatons gven by equatons (87), (88) an (89), the Fokker-Planck equaton (13) can be represente n the form of a generalze brth an eath master equaton, t P(x,v) = + {A + (x,v )P(x,v ) A (x,v)p(x,v)} {A (x,v+ )P(x,v + ) A + (x,v)p(x,v)} + {B (x,v )P(x,v ) B (x,v)p(x,v)}, (90) where A ± (x,v) are the transton rates from (x,v) to (x,v ± ) an are gven by an A + (x,v) = αt m a αv ma, (91) A (x,v) = αt m a + αv ma, (9) where a s chosen to be suffcently small so that A ± (x,v) wll be nonnegatve. The quantty B (x,v), the transton rate from (x,v) to (x +,v + ), s B (x,v) = 1 b. (93) In the lmt a 0 an b 0, the master equaton (90) turns nto the Fokker-Planck equaton (13). B. Mcroscopc reversblty The state of thermoynamc equlbrum of a system escrbe by a stochastc process s entfe as the state obeyng mcroscopc reversblty, whch occurs whenever the probablty of any trajectory equals the probablty of ts tme reverse. In a stochastc Markovan process ths conton s fullfel f [34] T(η η )P(η ) = T(η η)p(η), (94) for any to state η an η, where T(η η) s the contonal probablty of the transton η η an P(η) s the statonary probablty strbuton. In the contnuous tme lmt, we use the relaton T(η η) = tw(η η), val for small tme nterval t, to get the mcroscopc reversblty conton for system escrbe by the master equaton (86), W(η η )P(η ) = W(η η)p(η), (95) From equaton (95), we get two nepenent contons, an A + (x,v )P(x,v + ) = A (x,v)p(x,v), (96) B (x,v )P(x,v ) = B (x,v)p(x,v). (97) From ths last conton an usng (89) an (93) we get 1 (F P) + (v P) = 0. (98) m v x Therefore, the quantty K efne by (15) vanshes, whch s one of the equlbrum contons foun earler. The conton gven by equaton (96) proves [αt ma α(v a)]p(x,v ) = [αt + ma αv ]P(x,v). (99) Expanng ths expresson n powers of a, the lnear term n a gves αt v P(x,v) + mαv P(x,v) = 0 (100) from whch follows that J, efne by (16), vanshes, whch s the other equlbrum conton.

8 8 C. Conserve quantty Let us conser an elementary trajectory η η n phase space occurrng urng a small nterval of tme t. Suppose that a quantty L(η η), such as the work one by nonconservatve forces, s efne along ths elementary trajectory. The flux of ths quantty urng ths tme nterval s T(η η)p(η)l(η η), (101) η,η where T(η η) = tw(η η) s the transton probablty from η to η. The flux per unt tme Φ L s the rato of ths quantty an t, that s, Φ L = η,η W(η η)p(η)l(η η). (10) If the forces are conservatve, that s, f L(η η) = E(η) E(η ), whch happens for nstance n the case where L(η η) s the work of conservatve force, we may wrte Φ E = η,η{w(η η )P(η ) W(η η)p(η)}e(η). (103) In ths form, whch s entcal to (6), t s easy to see that Φ E vanshes n the statonary state. Inee, the summaton n η s entcally zero n the statonary state. From the master equaton t follows that the tme ervatve of U = E(η) s U t = Φ E, (104) whch agan shows that Φ E vanshes n the statonary state. The flux s efne from the system to the envronment. D. Entropy proucton We assume that the entropy flux rate Φ n a system escrbe by a Markovan process governe by the master equaton (86) s gven by the expresson (10) n whch L(η η) s replace by ln[w(η η)/w(η η )], that s, Φ = η,η W(η η)p(η)ln W(η η) W(η η ). (105) We remark that ths expresson can be unerstoo as the average of η W(η η)ln W(η η)/w(η η ) an n ths sense t can actually be use n numercal smulatons to calculate the entropy flux rate [16]. Equaton (105) can also be wrtten n the form Φ = η,η{w(η η)p(η) W(η η )P(η )}ln W(η η), (106) whch s entcal to expresson (7). Notce that, ln[w(η η)/w(η η )] cannot, n general, that s, for a rreversble system, be wrtten as a fference of the type E(η) E(η ), an therefore t oes not necessarly vansh, except uner thermoynamc equlbrum, n whch case ths quantty equals lnp(η ) lnp(η), as s event from equaton (95). Now, from the entropy of a nonequlbrum thermoynamc system, assume to be gven by equaton (3), t follows that the rate n whch the entropy of the system vares s ( ) t = t P(η) lnp(η). (107) η Usng the master equaton (86), equaton (107) can be wrtten as t = η,η {W(η η )P(η ) W(η η)p(η)}ln P(η), or, equvalently, (108) t = η,η W(η η)p(η)ln P(η) P(η ). (109) The entropy proucton rate s obtane by nsertng expressons (105) an (109) nto relaton (1). We get the followng expresson for the entropy proucton rate Π = η,η W(η η)p(η)ln W(η η)p(η) W(η η )P(η ), (110) whch can be wrtten n the suggestve form Π = 1 {W(η η)p(η) W(η η )P(η )}ln W(η η)p(η) W(η η )P(η ). η,η (111) In ths form Π s manfestly nonnegatve an can be regare as an extenson of the entropy proucton rate ntrouce by Schnakenberg [7]. Usng the transton rates approprate for the master equaton representaton (90) of the Fokker-Planck equaton, the entropy flux rate s explctly gven by + Φ = + x,v x,v x,v A + (x,v)p(x,v)ln A+ (x,v) A (x,v+ ) A (x,v)p(x,v)ln A (x,v) A + (x,v ) B (x,v) B (x,v)p(x,v)ln B (x,v ). (11) Usng the transtons rates (91), (9) an (93), we get the result Φ = ( α v α )P(x,v), (113) T x,v m

9 9 that s, Φ = ( α v α ), (114) T m whch s entcal to the expresson (40) an therefore equvalent to entropy flux rate gven by equaton (37). The rate of proucton of entropy Π can be etermne analogously, Π = A + (x,v)p(x,v)ln A + (x,v)p(x,v) A (x,v+ )P(x,v + ) + x,v an ts probablty of occurrence P(C R ) = T(η 0 η 1 )T(η 1 η )...T(η l 1 η l )P(η l ), (11) whch can also be wrtten as P(C R ) = ( t) l W(η 0 η 1 )W(η 1 η )...W(η l 1 η l )P(η l ), (1) wth the followng unerstanng: whenever W(η η) n equaton (119) s equal to A + then W(η η ) n equaton (1) wll be equal to A an vce-versa. The mcroscopc reversblty happens when a gven trajectory an ts reverse have the same probablty of occurrence, that s, P(C) = P(C R ), so that + A (x,v)p(x,v)ln A (x,v)p(x,v) A + (x,v )P(x,v ) + x,v or T(η 1 η 0 )P(η 0 ) = T(η 0 η 1 )P(η 1 ), (13) + B (x,v)p(x,v) B (x,v)p(x,v)ln B x,v (x,v )P(x,v ). (115) It s straghtforwar but cumbersome to show that ths expresson leas us to the result Π = m J xv, (116) αt P whch s entcal to the expresson (36) foun earler. It suffces to replace A ± an B by ther efntons, gven by (91), (9) an (93), expan P(x,v ± up to secon orer n a an use relaton (89). After takng the lmt a 0 an usng the efnton of J, gven by (16), we arrve at the above result. E. Jarzynsk equalty Here we follow a metho smlar to that use by Crooks [30, 31] an by Gaveau et al. [3]. We scretze the tme n ntervals t so that tw(η η) = T(η η) wll be the transton probablty from η to η. Let us conser a trajectory n phase space C = (η 0 η 1 η... η l ), (117) occurrng urng an nterval of tme equal to l t. The probablty of occurrence of such a trajectory wll be P(C) = T(η l η l 1 )...T(η η 1 )T(η 1 η 0 )P(η 0 ), (118) whch can also be wrtten as P(C) = ( t) l W(η l η l 1 )...W(η η 1 )W(η 1 η 0 )P(η 0 ). (119) Let us conser now the tme reversal path C R, relate to C an efne by C R = (η l η l 1... η 1 η 0 ), (10) W(η 1 η 0 )P(η 0 ) = W(η 0 η 1 )P(η 1 ), (14) whch we use before n equaton (95). Let us conser the rato R = P(C R) l P(C) = W(η j 1 η j ) P(η l ) W(η j η j 1 ) P(η 0 ). (15) One fns that so that R = C j=1 RP(C) = C P(C R ) = 1, (16) e ln R = 1. (17) Now the rato R can be wrtten n the form R = l j=1 W(η j 1 η j )P(η j ) W(η j η j 1 )P(η j 1 ), (18) where P(η j ) s unerstoo as the probablty strbuton at tme t = j t, soluton of the master equaton wth the ntal conton P(η 0 ) at tme t = 0. From (18), t follows where ln R = l σ(η j,η j 1 ) t, (19) j=1 σ(η,η) = 1 t ln W(η η)p(η) W(η η )P(η ) (130) s the ntrnsc entropy proucton rate, wth the conventon that η s the state occurrng at a gven tme t an η at a later tme t + t. A entty of the Jarzynsk type [8 31] follows then exp{ l σ(η j,η j 1 ) t} = 1, (131) j=1

10 10 where the average s to be taken over the probablty strbuton (119) of the path C. The ntrnsc entropy flux rate s gven by so that φ(η,η) = 1 t ln W(η η) W(η η ), (13) σ(η,η) = φ(η,η) + 1 t [S (η ) S (η)], (133) where S (η) = ln P(η) s the ntrnsc entropy. Takng nto account that the entropy proucton rate Π, as gven by (110), s the average of σ, that s, Π = σ(η,η), an that the entropy flux rate Φ, as gven by (105), s the average of φ, that s, Φ = φ(η,η), we get, n the lmt t 0 Π = Φ + t, (134) where S s the average of S, that s, S = S. In the contnuous tme lmt we may wrte the Jarzynsk entty as exp{ t 0 σt} = 1, (135) where the ntegral extens over a gven trajectory n phase space or, takng nto account (133), exp{ t 0 φ t [S (t) S (0)]} = 1. (136) From expresson (44) for Φ an bearng n mn that Φ = φ we get exp{ 1 T ( m [v (t) v (0)] t 0 ) v F t [S (t) S (0)]} = 1, (137) whch s the Jarzynsk equalty for a nonequlbrum systems of partcles followng a Fokker-Planck equaton escrbng the contact wth several heat baths at stnct temperatures. V. CONCLUSION We have etermne an expresson for the entropy proucton rate an entropy flux rate n rreversble systems escrbe by a Fokker-Planck equaton. The rreversble character s represente ether by nonconservatve forces or by the contact of the system wth heat baths at fferent temperatures. The expresson for the entropy proucton was obtane by usng a master equaton representaton of the Fokker-Planck an through a efnton of entropy proucton rate an entropy flux rate that nvolve the transton rates an n ths sense s relate to the rato between the probabltes of a trajectory n phase space an ts tme reversal. We have shown that, n the statonary state, the entropy proucton, or the entropy flux, s relate to the sspate power. More precsely, we have shown that the entropy proucton n a system n contact wth several heat baths s a sum of terms, each one beng the rato between the sspate power an the temperature of the corresponng heat bath. Usually ths relaton s actually use to efne entropy flux. The efntons of entropy proucton an entropy flux as we use here make no a pror reference to the sspate power. In ths sense they are unversal efntons beng val for general open systems not necessarly n contact wth heat reservors. As an example of our formalsm, we have use the expresson for the entropy proucton rate to etermne the heat conuctance of a smple system consstng of two Brownan partcles, each one n contact to heat reservors at stnct temperatures. Our results agree wth those obtane by other methos. Fnally, we have mae a connecton between the efnton of entropy proucton rate an the Jarzynsk equalty. [1] I. Prgogne, Introucton to Thermoynamcs of Irreversble Processes (New York, Wley, 1961), n. e. [] S. R. e Groot an P. Mazur, Non-Equlbrum Thermoynamcs (Amsteram, North-Hollan, 196). [3] G. Ncols an I. Prgogne, Self-Organzaton n Nonequlbrum Systems (Wley, New York, 1977). [4] N. G. van Kampen, Stochastc Processes n Physcs an Chemstry (North-Hollan, Amsteram, 1981). [5] C. Garner, Hanbook of Stochastc Methos for Physcs, Chemstry an Natural Scences (Sprnger, Berln, 1983). [6] H. Rsken, The Fokker-Planck Equaton: Methos of Soluton an Applcatons (Sprnger, Berln, 1984). [7] J. Schnakenberg, Rev. Mo. Phys. 48, 571 (1976). [8] J.-L. Luo, C. Van en Broeck an G. Ncols, Z. Phys. B 56, 165 (1984). [9] C. Y. Muo, J.-L. Luo an G. Ncols, J. Chem. Phys. 84, 7011 (1986). [10] A. Pérez-Mar, J. M. Rubí an P. Mazur, Physca A 1, 31 (1994). [11] J. Kurchan, J. Phys. A 31, 3719 (1998). [1] C. Maes, J. Stat. Phys. 95, 367 (1999). [13] J. L. Lebowtz an H. Spohn, J. Stat. Phys. 95, 333 (1999). [14] C. Maes, F. Reg, an A. Van Moffaert, J. Math. Phys. 41, 158 (000). [15] C. Maes an K. Netočn J. Stat. Phys. 11, 69 (003). [16] L. Crochk an T. Tomé, Phys. Rev. E 7, (005).

11 11 [17] U. Sefert, Phys. Rev. Lett. 95, (005). [18] T. Tomé, Braz. J. Phys. 36, 185 (006). [19] R. K. P. Za an B. Schmttmann, J. Phys. A 39, L407 (006). [0] R. K. P. Za an B. Schmttmann, J. Stat. Mech. P0701 (007). [1] R. J. Harrs an G. M. Schütz, J. Stat. Mech. P0700 (007). [] D. Anreux an P. Gasparp, J. Stat. Phys. 17, 107 (007). [3] A. Imparato an L. Pelt, J. Stat. Mech. L0001 (007). [4] B. Gaveau, M. Moreau an L. S. Schulman, Phys. Rev. E 79, (009). [5] A. Pérez-Mar an I. Santamaría-Holek, Phys. Rev. E 79, (009). [6] A. Dhar, Av. Phys. 57, 457 (008) [7] W. A. Morgao an D. O. Soares-Pnto, Phys. Rev. E 79, (009). [8] C. Jarzynsk, Phys. Rev. Lett. 78, 690 (1997). [9] C. Jarzynsk, Phys. Rev. E 56, 5018 (1997). [30] G. E. Crooks, Phys. Rev. E 60, 71 (1999). [31] G. E. Crooks, Phys. Rev. E 61, 361 (000). [3] B. Gaveau, M. Moreau an L. S. Schulman, Phys. Lett. 37, 3415 (008). [33] D. Konepu an I. Prgogne, Moern Thermoynamcs (New York, Wley, 1998). [34] A. Kolmogoroff, Math. Ann. 11, 155 (1936).