Entropy Production in Nonequilibrium Systems Described by a Fokker-Planck Equation

Transcription

1 Brazlan Journal of Physcs, vol. 36, no. 4A, December, Entropy Proucton n Nonequlbrum Systems Descrbe by a Fokker-Planck Equaton Tâna Tomé Insttuto e Físca Unversae e São Paulo Caxa postal São Paulo- SP, Brazl Receve on 24 September, 2006 We stuy the entropy proucton n nonequlbrum systems escrbe by a Fokker-Planck equaton. We have evse an expresson for the entropy flux n the statonary state. We have foun that the entropy flux can be wrtten as an ensemble average of an expresson contanng the force an ts ervatve. Ths result s smlar to the one use by Lebowtz an Spohn for system followng a Markovan process n screte space. We have also been able to obtan a fluctuaton-sspaton type relaton between the sspate power, whch was wrtten as an ensemble average, an the proucton of entropy for the case of systems n contact wth one heat bath. We have apple the results for a smple moel for partcles subjecte to sspatve forces. Keywors: Entropy proucton; Irreversble systems; Fokker-Planck equaton I. INTRODUCTION In the statonary state, rreversble systems are n a contnuous process of proucton of entropy. A measure of the stance from thermoynamc equlbrum can therefore be gven by the proucton of entropy snce ths quantty vanshes n equlbrum. The rate of change of the entropy S of a system can be properly ecompose nto two contrbutons [1] S = Π Φ, (1) t where Π s the entropy proucton ue to rreversble processes ocurrng nse the system an Φ s the entropy flux from the system to the envronment. The quantty Π s postve efnte whereas Φ can have ether sgn. In the statonary state the rate of change of the entropy vanshes so that Φ = Π. The quantty Φ s efne as the flux from nse to outse the system, so that t wll be postve n the nonequlbrum statonary state. Although n equlbrum the entropy s a well efne quantty, n nonequlbrum systems the entropy as well as the proucton of entropy o not have a unversal efnton. Accorng to Gallavott the problem of efnng entropy n a system out of equlbrum has not been solve yet [2]. In ths sense t s nterestng to see how one can efne those quanttes n systems that evolve n tme accorng to specfe ynamcs. In etermnstc Hamltonan ynamcs t s well known that the Gbbs entropy s nvarant [3]. Ths property s a consequence of the ncompressblty of the flu that represents the system n phase space. Irreversble systems, on the other han, are suppose to be escrbe by non-hamltonan ynamcs, that s, by ynamcs comng from nonconservatve forces. In ths case, the flu n phase space becomes compressble an the proucton of entropy may be relate to the contracton of the phase space [2, 4, 5]. Here we are concerne wth the efnton an calculaton of the entropy proucton n nonequlbrum systems escrbe by a Fokker-Planck equaton or n an equvalent escrpton by a Langevn equaton [6, 7]. Systems escrbe by a Fokker-Planck equaton are systems that follow a Makovan process n contnuous tme an contnuous confguraton space. The rreversble character s etermne by the type of force enterng the Langevn an ts assocate Fokker-Planck equaton. As we shall see, f the forces are nonconservatve the resultng entropy proucton s nonzero an postve. When the force becomes conservatve the proucton of entropy vanshes. The stuy of proucton of entropy n systems escrbe by a Markovan process n contnuous tme but screte confguraton space [7, 8] or, n other wors, escrbe by a master equaton, has been the object of several stues [9 13] nclung the proucton of entropy n the majorty vote moel [13, 14]. In ths stuy we wll relate the entropy proucton Π wth the sspate power P occurrng n nonequlbrum systems subject to nonconservatve forces an n contact wth a heat bath at a temperature T. We wll show that n ths case Π = P /T, whch s a fluctuaton-sspaton type relaton. II. FOKKER-PLANCK EQUATION Let us conser a set of n nteractng partcles that evolve n tme accorng to the followng couple set of Langevn equatons x = f (x)+ζ (t), (2) t where x s the poston of the -th partcle an x stans for the collecton of varables {x }. The quantty f (x) = f ({x }) s the force actng on the -th partcle an ζ (t) s the atve whte nose, that s, a stochastc varable havng the propertes an ζ (t) = 0, (3) ζ (t)ζ j (t ) = 2D δ j δ(t t ), (4) where, for smplcty, we are assumng that D 0 are constants that mght be stnct for each partcle.

2 1286 Tâna Tomé The assocate Fokker-Planck equaton, that gves the tme evoluton of the probablty strbuton P(x,t), sgvenby t P(x,t)= [ f (x)p(x,t)] x + 2 D x 2 P(x,t). (5) It s convenent to wrte own the Fokker-Planck equaton as the followng contnuty equaton t P(x,t)= J (x,t), (6) x where J s the -th component of the current of probablty J = {J }, efne by J (x,t)= f (x)p(x,t) D P(x,t). (7) x The Fokker-Planck equaton has to be solve nse a gven regon R of the space spanne by the set of varables x subject to a prescrbe bounary conton whch concerns the behavor of P(x,t) an J (x,t) at the surface S that elmts the regon R. We wll conser two types of bounary contons: peroc an reflectng. For peroc bounary contons both quanttes are peroc. In ths case, the forces f (x) must be peroc. For the reflectng bounary contons the component of the current of probablty J perpencular to S vanshes. In ths case the component of the force perpencular to S s assume to vansh. All results obtane below wll be val uner these two types of bounary contons. III. REVERSIBILITY We wll be concerne here wth systems that are far from equlbrum, that s systems that are rreversble even n the statonary state. Ths means that the current of probablty oes not vansh n the statonary state. If the current of probablty vanshes n the steay state, the system s foun to be n a state of thermoynamc equlbrum an we say that the system s reversble. In ths case, the entropy proucton Π vanshes an the flux of entropy Φ vanshes as well. The property of reversblty hols when the forces f are conservatve, that s, when f = f j, (8) x j x for any par, j an D = D (9) s the same for any partcle. In ths case the f s the graent of a certan potental V (x), that s, V (x) f (x)=, (10) x an the statonary probablty strbuton P e (x) s the soluton of V (x) P e (x) D P e (x)=0, (11) x x that s, P e (x)= 1 Z e V (x)/d, (12) where Z s a normalzaton constant. Assumng that D s proportonal o the temperature T, ths s the canoncal strbuton. The Langevn equaton an ts assocate Fokker-Planck equatons are capable of escrbng system n thermoynamc equlbrum f f an D satsfy the relatons (8) an (9). In ths case we nterpret the equatons as the ones approprate o escrbe a system n contact wth a heat reservor at temperarature T. In the followng we wll be concerne wth systems for whch the contons gven by Eqs. (8) an (9) o not hol. The stuaton n whch conton (8) hols but the D are not the same s approprate o escrbe, for nstante, the contact of the system wth two or more heat reservors wth stnct temperatures. In ths case the statonary state wll be a nonequlbrum steay state. Another rreversble stuaton s foun when D = D, but there exsts at least one par, j for whch f x j f j x. (13) Ths escrbes, for nstance, the contact of an rreversble system, contanng nonconservatve forces, wth a heat reservor. In these nonequlbrum cases the proucton of entropy Π an the flux of entropy Φ are both nonzero. IV. ENTROPY The Gbbs entropy S(t) of the system at tme t s efne by S(t)= P(x,t)lnP(x,t)x, (14) where x = x 1 x 2...x n. Usng the Fokker-Planck equaton n the form of Eq. (6), ts tme ervatve can be wrtten as t S(t)= Integratng by parts we get t S(t)= [lnp(x,t)+1] J (x,t)x. (15) x J (x,t) lnp(x,t)x. (16) x Now from the efnton of current of probablty gven by Eq. (7) t follows that Therefore, we may wrte D x lnp(x,t)= f (x) J (x,t) P(x,t). (17) t S(t)=

3 Brazlan Journal of Physcs, vol. 36, no. 4A, December, = 1 J (x,t) f (x)x+ D [J (x,t)] 2 x. (18) D P(x,t) The last term s clearly postve efnte. We entfy t wth the entropy proucton, that s, Π = [J (x,t)] 2 x. (19) D P(x,t) Comparng wth Eq. (1), t follows that the entropy flux Φ shoul be gven by Φ = 1 J (x,t) f (x)x. (20) D Multplyng the ensty of current gven by Eq. (7) by f an ntegratng we have J (x,t) f (x)x = = [ f (x)] 2 P(x,t)x D f (x) x P(x,t)x. (21) Integratng the last term by parts, we get the followng expresson for the entropy flux, Φ = { 1 [ f (x)] 2 + f (x)}p(x,t)x, (22) D where f (x)= f (x)/x. The rght-han se can be wrtten as an average over the probablty strbuton P(x,t), that s, Φ = { 1 [ f (x)] 2 + f (x)}. (23) D V. MASTER EQUATION In ths secton we show that a scretze Fokker-Planck equaton can be vewe as a master equaton. To ths en let us conser a Fokker-Planck equaton n one varable t P(x,t)= 2 [ f (x)p(x,t]+d P(x,t). (24) x x2 If we use the scretzaton x = na then ths equaton can be wrtten as t P n(t)= 1 a [ f n+1p n+1 (t) f n P n (t)]+ + D a 2 [P n+1(t) 2P n (t)+p n 1 (t)], (25) where P n (t)=p(x,t) an f n = f (x), whch n turn can be entfe as the master equaton t P n(t)=w + n 1 P n 1(t) w + n P n (t)+ where +w n+1 P n+1(t) w n P n (t) (26) w + n = D a 2 (27) s the transton probablty of jumpng from n to n + 1 an w n = D a 2 f n (28) a s the transton probablty of jumpng from n to n 1. Accorng to Lebowtz an Spohn [10] the flux of entropy n a system governe by a master equaton s gven by Φ = w + n ln w+ n w + w n ln w n n+1 w +, (29) n 1 Now, f we take the lmt a 0 t s straghtforwar to show that ths expresson becomes Φ = 1 D [ f (x)]2 + f (x), (30) whch s the expresson gven by Eq. (23) for the partcular case of one varable. VI. TIME AVERAGE OF A STATE FUNCTION Let us conser the tme average of a state functon E(x) gven by U(t)= E(x)P(x, t)x. (31) We have t U(t)= Integratng by parts, t U(t)= where E (x)=e(x)/x,or t U(t)= E(x) J (x,t)x. (32) x J (x,t)e (x)x, (33) E (x) f (x)p(x,t)x Integratng the last term by parts we get t U(t)= + D E (x) P(x,t)x. (34) x E (x) f (x)p(x,t)x+ D E (x)p(x,t)x, (35)

4 1288 Tâna Tomé where E (x)=e (x)/x an we have assume that E (x) s ether peroc or vanshes at the bounary, epenng on the type of bounary conton. The rght-han se can therefore be wrtten as an average, t U(t)= {E (x) f (x)+d E (x)}. (36) compute each of these terms. To ths en we start by splttng the force f (x) nto two parts f (x)= f C (x)+ f D (x), (42) such that f C (x) s conservatve, that s, f C (x) s the graent of a potencal E(x),or VII. DISSIPATED POWER f C = E x, (43) The sspate energy per unt tme ue to the forces f,or the sspate power of the forces f, s etermne by P = x f. (37) t If we use the Langevn equaton ths can be wrtten as P = [ f ] 2 + f ζ (t). (38) Usng now the Stratonovch prescrpton, t s possble to show that f ζ (t) = D f, where f = f /x, so that P = {[ f ] 2 + D f }. (39) Ths expresson has been obtane before [15] by means of a smlar reasonng. Usng a reasonng smlar to those use before we can wrte ths expresson as P = J (x,t) f (x). (40) It s nterestng to notce that there s a smple relatonshp between the sspate power an the entropy flux when the system s n contact wth a heat reservor. In ths case, as we have seen, D = D where D s proportonal o the temperature of the reservor. Comparng these two expresson wth those for the entropy flux Φ we see that Φ = P D. (41) Wrtng D = kt an efnng Φ = kφ we see that Φ = P /T. We remark that ths relaton s val for systems that are n contact wth a heat reservor but subject to nonconservatve forces. VIII. NONCONSERVATIVE FORCE From now one we wll conser the case such that D = D, whch correpons to the escrpton of a system n contact wth a heat reservor. In the statonary state, expresson (23) for the entropy flux Φ gves also the proucton of entropy Π snce the varaton n entropy S/t vanshes. However, f the system s n a transent state ths expresson wll be a sum of two terms: one s the proucton of entropy Π an the other s mnus the entropy varaton S/t of the system. Let us an f D (x) s a nonconservatve force wth a vanshng vergence (solenoal force), that s, f D x = 0. (44) That such a splttng can always be one can be seen as follows. Ths last equaton mples that or yet f f x = x = f C x, (45) 2 E x 2. (46) Therefore, f we are gven a generc force f (x), ths last equaton can be solve to obtan E(x). From E(x) we get f C (x) by means of Eq. (43) an f D (x) by usng Eq. (42). The flux of entropy s wrtten as a sum of two terms, Φ = Φ C + Φ D, where an Φ C = { 1 D f C (x) f (x)+ f C (x)}. (47) Φ D = 1 D f D (x) f (x). (48) Now, snce f C (x) s the graent of a state functon E(x) we may use the entfcaton f C = E an Eq. (36) to conclue that Φ C = 1 E(x). (49) D t In the statonary state, the rght-han se vanshes so that Φ C vanshes an Π = Φ = Φ D. IX. A SIMPLE MODEL Let us conser a smple moel wth two varables, for whch the force s gven by f = Kr + Aẑ r, (50)

5 Brazlan Journal of Physcs, vol. 36, no. 4A, December, where K an A are two parameters, an r = x 1 ˆx + x 2 ŷ an f = f 1 ˆx + f 2 ŷ. As long as A 0 the force s nonconservatve snce f = 2Aẑ 0. In ths case an The Fokker-Planck equaton s Φ C = 1 D {K2 r 2 2DK} (51) Φ D = 1 D A2 r 2. (52) P t = (fp)+d 2 P, (53) whch can be wrtten as P t = K (rp) Aẑ r P + D 2 P. (54) A soluton of ths equaton can be obtane by assumng that P s a functon of r = r only. The vector prouct term n (54) then vanshes an the equaton becomes a Fokker-Planck equaton for a conservatve force gven by f = Kr. The statonary soluton s then P(r)= K 2πD e Kr2 /2D, (55) from whch we get r 2 = 2D/K, an the results Φ C = 0 an Π = Φ D = 2A2 K. (56) The entropy proucton Π s then proportonal to A 2 an vanshes when A 0, that s, n the equlbrum case. We may also etermne the statonary current of probablty. From ts efnton we obtan the result J(r)J(r) =f(r)p(r) D P(r) (57) J(r)=A(ẑ r)p(r). (58) Hence, the current of probablty s proportonal to A an also vannhes n the equlbrum state, as expecte. X. CONCLUSION We have evse an expresson for the entropy flux n the statonary state for systems that follow a Fokker-Planck equaton. We have foun that the entropy flux can be wrtten as an average value of an expresson contanng the force an ts ervatve. Ths result can be compare wth the one by Lebowtz an Spohn. In fact they were able to show that the entropy flux for markovan processes can be calculate as an average of an state functon at the steay state. We have also shown that for a smple moel for partcles subjecte to sspatve forces the entropy flux, whch equals the entropy proucton, n the steay state s nonzero. [1] G. Ncols an I. Prgogne, Self-Organzaton n Nonequlbrum Systems (Wley, New York, 1977). [2] G. Gallavott, Chaos 14, 680 (2004). [3] J. W. Gbbs, Elementary Prncples n Statstcal Mechancs,Yale Unversty Press, New Haven, [4] G. Gallavott an E. G. D. Cohen, Phys. Rev. E 69, (2004). [5] D. J. Evans, D. J. Searles, an L. Ronon, Phys. Rev. E (2005). [6] N. G. van Kampen, Stochastc Processes n Physcs an Chemstry, North-Hollan Amsteram, [7] T. Tomé e M. J. e Olvera, Dnâmca Estocástca e Irreversblae, Etora a Unversae e São Paulo, São Paulo, [8] T. M. Lggett, Interactng Partcle Systems, Sprnger-Verlag, New York, [9] C. Maes, J. Stat. Phys. 95, 367 (1999). [10] J. L. Lebowtz an H. Spohn, J. Stat. Phys. 95, 333 (1999). [11] C. Maes, F. Reg, an A. Van Moffaert, J. Math. Phys. 41, 1528 (2000). [12] C. Maes an K. Netočný J. Stat. Phys. 11, 269 (2003). [13] L. Crochk an T.Tomé, Phys. Rev. E 72, (2005). [14] M. J. e Olvera, J. Stat. Phys. 66, 273 (1992). [15] T. Tomé an M. J. e Olvera, Braz. J. Phys. 27, 525 (1997).