Novel dissipative properties for the master equation

Transcription

1 Novel dssatve roertes for the master equaton Lu Hong 1, Chen Ja, Y Zhu 1, Wen-An Yong 1, 1 Zhou Pe-Yuan Center for Aled Mathematcs, Tsnghua Unversty, Bejng , P.R. Chna Bejng Comutatonal Scence Research Center, Bejng , P.R. Chna Corresondence: wayong@mal.tsnghua.edu.cn arxv: v3 [hyscs.chem-h] 5 Nov 015 Abstract Recent studes have shown that the entroy roducton rate for the master equaton conssts of two nonnegatve terms: the adabatc and non-adabatc arts, where the non-adabatc art s also known as the dssaton rate of a Boltzmann-Shannon relatve entroy. In ths aer, we rovde some nonzero lower bounds for the relatve entroy, the entroy roducton rate, and ts adabatc and non-adabatc arts. These nonzero lower bounds not only reveal some novel dssatve roertes for general nonequlbrum rocesses whch are much stronger than the second law of thermodynamcs, but also mose some new constrants on thermodynamc consttutve relatons. Moreover, we also rovde a mathematcal alcaton of these nonzero lower bounds by studyng the long-tme behavor of the master equaton. Extensons to the Tsalls statstcs are also dscussed, ncludng the nonzero lower bounds for the Tsalls-tye relatve entroy and ts dssaton rate. Keywords: nonequlbrum, entroy roducton rate, free energy, Markov rocess, Tsalls statstcs 1 Introducton The master equaton descrbes the dynamcs of the robablty dstrbuton for a Markov jum rocess and has been wdely aled to hyscs, chemstry, bology, and many other scentfc felds [1]. It rovdes an effectve way to model varous stochastc systems such as random walks, brth-death rocesses, Lndblad equatons [], chemcal reacton systems [3], sngle-molecule enzyme knetcs [4], and so on. In recent years, sgnfcant rogresses have been made n the feld of mesoscoc stochastc thermodynamcs [5 7]. The dynamc foundaton of ths feld turns out to be Markov rocesses and the master equaton lays a fundamental role because any Markov rocess can generally be aroxmated by a Markov jum rocess. In the theory of stochastc thermodynamcs, an equlbrum state s defned as a statonary rocess wth detaled balance and the devaton of a system from equlbrum can be characterzed by the concet of entroy roducton rate [8 10]. Motvated by the classcal theory of nonequlbrum thermodynamcs [11], the entroy roducton rate can generally be reresented as a blnear form of the thermodynamcs fluxes and forces. When an oen system s drven by a sustaned energy suly from the envronment, t can aroach a nonequlbrum steady state (NESS) as ts long-tme behavor, wth a ostve entroy roducton rate. It has been shown recently that the entroy roducton rate e for the master equaton can be decomosed as the sum of two nonnegatve terms: e = e (ad) + e (na), where e (ad) (res.

2 e (na) ) s the adabatc (res. non-adabatc) entroy roducton rate [1, 13]. The adabatc art s also known as housekeeng heat [14, 15] and the non-adabatc art s also referred to as the dssaton rate df/ of the free energy F [1]. Moreover, the free energy F can be reresented as a Boltzmann-Shannon relatve entroy between the robablty dstrbuton = ( ) of the system and the steady-state dstrbutonµ = (µ ): F = k B T log µ, where k B s the Boltzmann constant and T s the temerature. The above decomoston s mortant because t rovdes a strengthened verson of the second law of thermodynamcs. In ths work, we shall further exlore ths strengthened verson of the second law. Insred by the entroy-dssaton rncle n nonequlbrum thermodynamcs roosed n [16], we rovde nonzero lower bounds for the free energy, the entroy roducton rate, and ts adabatc and non-adabatc arts. These nonzero lower bounds reveal new dssatve roertes of general nonequlbrum rocesses whch are even much stronger than the strengthened verson of the second law. They also renforce the revous fndngs that the rreversblty of nonequlbrum rocesses has two dfferent mechansms: the devaton from steady state and the breakng of detaled balance [1, 13]. Moreover, we establsh smlar conclusons for the Tsalls statstcs by rovdng nonzero lower bounds for the Tsalls-tye relatve entroy and ts dssaton rate. Fnally, we elaborate the sgnfcance of these nonzero lower bounds from both the hyscal and mathematcal ersectves. It wll be seen that these nonzero lower bounds not only mose some new restrctons on thermodynamc consttutve relatons, but also rovde a smle way to study the long-tme behavor of the master equaton. Ths aer s organzed as follows. In Secton, we ntroduce some relmnares on the master equaton. Secton 3 contans our man results. In Secton 4, we resent some dscussons on the hyscal sgnfcance of the nonzero lower bounds. Secton 5 s devoted to a new and urely analytc roof about the long-tme behavor of the master equaton. Prelmnares Consder a molecular system modeled by a Markov jum rocess [17] wth a fnte number of states 1,,,N and transton rate matrx Q = (q j ), where q j wth j denotes the transton rate from state to j and q = j q j. Let (t) = ( 1 (t), (t),, N (t)) denote the robablty dstrbuton of the system at tme t. Then the dynamcs of the robablty dstrbuton = (t) s governed by the master equaton [1] d = Q. In comonents, the master equaton can be wrtten as d = j j q j = j ( j q j q j ), = 1,,,N. (1)

3 We assume that the system s rreducble, namely, for each ar of states j, there s a sequence of states 1,,, m, such that 1 =, m = j, and q 1 q 3 q m 1 m > 0. Under ths rreducble conon, t s well-known that the comonents of (t) are all ostve for any t > 0. Furthermore, t can be roved that the system has a unque steady-state dstrbuton µ = (µ 1,µ,,µ N ) satsfyng µq = 0 and the comonents of µ are all ostve [17]. In general, the steady state of the system can be classfed nto equlbrum and nonequlbrum ones. In an equlbrum state, the detaled balance cononµ q j = µ j q j holds for any ar of statesandj. In an NESS, however, the detaled balance conon s broken and the system s externally drven wth concomtant entroy roducton. Throughout ths aer, we set the Boltzmann constant k B = 1 and assume the temerature T = 1 for smlcty. Recall that the total entroy roducton ratee of the system s defned as e = 1 ( q j j q j )log q j, () j q j,j where0/0 s understood to be 1. In recent years, t has been shown thate can be decomosed as the sum of two nonnegatve terms: e = e (ad) +e (na) [1, 13]. Here the adabatc art e (ad), also known as housekeeng heat, can be exressed as e (ad) = 1 ( q j j q j )log µ q j 0 (3) µ j q j and the non-adabatc arte (na) e (na) = 1,j can be exressed as To see the nature ofe (na), we recall the followng defnton.,j ( q j j q j )log µ j j µ 0. (4) Defnton.1. Letu = (u 1,u,,u N ) andv = (v 1,v,,v N ) be two robablty dstrbutons. Then the Boltzmann-Shannon relatve entroy (Kullback-Lebler dvergence) between u and v s defned as D(u v) = u log u v. It was shown n [1] that, even f the system s away from equlbrum, the free energy F can stll be ntroduced and can be exressed as the Boltzmann-Shannon relatve entroy between the robablty dstrbuton and the steady-state dstrbutonµ: F = D( µ) = log µ. (5) It s easy to check that the non-adabatc entroy roducton ratee (na) ratef d = df/ for the free energy F : e (na) = f d = df 3 s exactly the dssaton 0. (6)

4 The entroy roducton ratee s always nonnegatve and ths s an equvalent statement of the second law of thermodynamcs. However, the above dscusson shows thate can be further decomosed as the sum of two nonnegatve terms, e (ad) and e (na). Therefore, ths decomoston can be vewed as a strengthened verson of the second law. In what follows, we shall further exlore ths strengthened verson of the second law by rovdng some nonzero lower bounds for the free energy, the total entroy roducton rate, and ts adabatc and non-adabatc arts. 3 Results 3.1 Nonzero lower bound for the non-adabatc entroy roducton rate We start wth the followng nequalty, whch wll be used frequently n the sequel. Lemma 3.1. Let K be a ostve number. Then for any0 x,y K, we have ylog y x 1 K (y x) +(y x). Proof. For any 0 < x K, set f(x) = xlogx x. Then we have f (x) = logx and f (x) = 1/x 1/K. By the mean-value theorem, for any 0 < x,y K, there exsts θ between x and y such that Moreover, t can be seen that Ths shows that for any0 < x,y K, f(y) f(x) = f (x)(y x)+ 1 f (θ)(y x). f(y) f(x) f (x)(y x) = ylog y (y x). x ylog y x (y x) = 1 f (θ)(y x) 1 K (y x), whch gves the desred result. It s a well-known result that the free energy F s nonnegatve [1]. The followng theorem strengthens ths result by gvng a nonzero lower bound of the free energy F. Theorem 3.. The followng nequalty holds: F 1 ( µ ). Proof. Snce0 µ, 1, t follows from Lemma 3.1 that log µ 1 ( µ ) +( µ ). By the defnton off n (5) we have F [ ] 1 ( µ ) +( µ ) = 1 ( µ ), whch gves the desred result. 4

5 The next theorem gves a nonzero lower bound for the non-adabatc entroy roducton ratee (na), also known as the free energy dssaton ratef d = df/. Theorem 3.3. There exsts a constantc 1 > 0 such that [ e (na) = f d = df c 1 ( j q j q j )], wherec 1 only deends on the transton rate matrxq. Proof. Snce0 < µ, 1, t follows from Lemma 3.1 that By the defnton off d n (6), we have e (na) j µ j log µ j j µ 1 ( µ j j µ ) +( µ j j µ ). =,j q j µ j,j = 1,j = 1 q j log µ j = q j µ j log µ j j µ µ,j j j µ [ 1 ( µ j j µ ) +( µ j j µ ),j q j µ j ( µ j j µ ) +,j q j µ j ( µ j j µ ), q j,j ] j µ j µ q j where the last equalty s due to j q j = 0 and µ q j = 0. SetM = max{q j : j} > 0. Then we have e (na) 1 q j q j µ,j j M ( µ j j µ ) = 1 M µ j j [ ] qj ( µ j j µ ). µ j Recall that the comonents of µ are all ostve. Set r = mn{µ 1,µ,µ N } > 0. It thus follows from the Cauchy-Schwarz nequalty that e (na) r MN = r MN [ j q j µ j ( µ j j µ ) [ ( q j j q j )], j j ] = r MN [ ] q j j whch gves the desred result. Remark 3.4. Ths theorem s nsred by the entroy-dssaton rncle n nonequlbrum thermodynamcs roosed n [16] and has been roved when the steady-steady dstrbuton µ satsfes the detaled balance conon [18, 19]. However, we do not need detaled balance here and thus the above theorem can be aled to general Markov jum rocesses. 5

6 The followng result s a drect corollary of Theorems 3. and 3.3. Corollary 3.5. The followng three statements are equvalent: (a) The system s n a steady state; (b) The free energy F vanshes; (c) The free energy dssaton ratef d vanshes. Proof. If (a) holds, then = µ and thus (b) and (c) follow from (5) and (6) mmedately. If (b) holds, then t follows from Theorem 3. that = µ, whch shows that (a) holds. If (c) holds, then t follows from Theorem 3.3 that for any, q j = j q j. j Ths shows thats a steady-state dstrbuton and thus (a) holds. j Remark 3.6. Ths corollary shows that the free energy dssaton rate vanshes f and only f the system s n a steady state. Therefore, a nonequlbrum system wll never sto dssatng free energy unless t has reached the steady state. Thus the free energy dssaton rate characterzes the rreversblty n the sontaneous relaxaton rocess towards the steady state. Ths s an nterestng echo of the famous H-theorem for the Boltzmann equaton, whch characterzes the relaxaton rocess of a thermodynamc system towards thermodynamc equlbrum. 3. Nonzero lower bound for the adabatc entroy roducton rate We have gven a nonzero lower bound for the non-adabatc entroy roducton rate. The followng theorem rovdes an analogue for the adabatc art. Theorem 3.7. There exsts a constantc > 0 such that c (µ q j µ j q j ), e (ad) wherec only deends on the transton rate matrxq.,j Proof. Set M = max{q j : j} > 0. Snce 0 µ q j M for any j, t follows from Lemma 3.1 that By the defnton ofe (ad) e (ad) µ q j log µ q j µ j q j 1 M (µ q j µ j q j ) +(µ q j µ j q j ). =,j n (3), we have q j log µ q j = µ q j log µ q j µ j q j µ,j µ j q j [ 1 M (µ q j µ j q j ) +(µ q j µ j q j ) µ whch gves the desred result.,j = 1 (µ q j µ j q j ) 1 M µ,j M 6 ] (µ q j µ j q j ),,j

7 The followng result s a drect corollary of the above theorem. Corollary 3.8. Assume that the comonents of the ntal dstrbuton (0) are all ostve. Then there exsts a constantc 3 > 0 such that e (ad) c 3 (µ q j µ j q j ), wherec 3 deends on the transton rate matrxqand the ntal dstrbuton(0).,j Proof. By Theorem 3.7, there exsts a constant c > 0 deendng on the transton rate matrx Q such that Let r be a constant defned by e (ad) c (µ q j µ j q j ). (7),j r = mn mn t 0 (t). Snce the comonents of (0) are all ostve, t s easy to see that the comonents of (t) are all ostve for any t 0. Snce the system s rreducble, the comonents of µ are all ostve and (t) µ as t [17]. The above two facts mly that r > 0. It thus follows from (7) that whch gves the desred result. e (ad) c r,j (µ q j µ j q j ), Remark 3.9. We know from [17] that even f some comonents of the ntal dstrbuton(0) are zero, all comonents wll become ostve after an arbtrarly small tme. Therefore, the assumton of the above corollary does not mose too much restrcton on the system. In adon, t has been shown that f the steady state of the system s an NESS, then the adabatc entroy roducton rate e (ad) > 0 [1, 13]. The above corollary further strengthens ths result by rovdng a ostve lower bound for e (ad), whch s also tme-ndeendent. Ths reveals an ntrnsc strong dssatve roerty for the master equaton. The next result follows drectly from Corollary 3.8. Corollary Assume that the comonents of the ntal dstrbuton (0) are all ostve. Then the followng three statements are equvalent: (a) The steady state of the system s an equlbrum state; (b) The steady state of the system satsfes the detaled balance conon; (c) The adabatc entroy roducton ratee (ad) vanshes. Proof. The equvalence of (a) and (b) follows from the defnton of the equlbrum state. We next rove the equvalence of (b) and (c). If (b) s true, then the detaled balance conon 7

8 µ q j = µ j q j holds for any ar of statesandj. It thus follows from (3) thate (ad) = 0. On the other hand, f (c) holds, then t follows from Theorem 3.7 that (µ q j µ j q j ) = 0.,j Snce the comonents of the ntal dstrbuton (0) are all ostve, the comonents of (t) are all ostve for any t 0. Ths shows that µ q j = µ j q j for any ar of states and j, whch mles that (b) holds. Remark Ths corollary shows that adabatc entroy roducton rate vanshes f and only f the steady state of the system s an equlbrum state. Thus the adabatc art reflects the rreversblty n an NESS caused by the breakng of detaled balance. In fact, even f the system has reaches the steady state, some knd of crcular motons may stll exst to mantan an NESS and gve rse to a ostvee (ad) [8]. Ths henomenon s tycal n many bochemcal rocesses and consttutes a major dfference between an NESS and an equlbrum state [10]. 3.3 Nonzero lower bound for the total entroy roducton rate We have gven nonzero lower bounds for the adabatc and non-adabatc entroy roducton rates, whch automatcally gve rse to a nonzero lower bound for the total entroy roducton ratee. The followng theorem rovdes another nonzero lower bound fore. Theorem 3.1. There exsts constantsc 4 > 0 such that e c 4 ( q j j q j ), wherec 4 only deends on the transton rate matrxq.,j Proof. Set M = max{q j : j} > 0. Snce 0 q j M for any j, t follows from Lemma 3.1 that By the defnton ofe n (), we have q j log q j j q j 1 M ( q j j q j ) +( q j j q j ). e = q j log q j,j j q j,j = 1 ( q j j q j ) + M,j,j whch gves the desred result. [ ] 1 M ( q j j q j ) +( q j j q j ) ( q j j q j ) = 1 M ( q j j q j ), Remark It s nterestng to note that, n the above theorem, the lower bound for the entroy roducton ratee s gven by the sum of squares of the local fluxes J j = q j j q j between each ar of states and j. In contrast, t s shown n Theorem 3.3 that the free energy dssaton rate f d s bounded from below by the sum of squares of the total fluxes J = j ( jq j q j ) through each state.,j 8

9 3.4 Results for the Tsalls-tye relatve entroy Theorems 3. and 3.3 gve the nonzero lower bounds for the free energy and ts dssaton rate, where the free energy can be exressed as the Boltzmann-Shannon relatve entroy betweenandµ. Here we try to extend these results to the Tsalls-tye relatve entroy, whch was ntroduced by Tsalls n [0] as a generalzaton of the classcal Boltzmann-Gbbs statstcs. The resultng theory s consdered to be mortant for the non-extensve thermodynamcs. It has also been found alcatons n a wde range of natural, artfcal, and socal comlex systems [1]. Secfcally, we recall the followng defnton. Defnton Let u = (u 1,u,,u N ) and v = (v 1,v,,v N ) be two robablty dstrbutons where the comonents of v are all ostve. Then, for any real number α 0,1, the Tsalls-tye relatve entroy of orderαbetween u and v s defned as [ 1 ( ) α 1 u D α (u v) = u 1]. α(α 1) v By L Hostal s rule, t s easy to see that lm D α(u v) = lm α 1 α 1 u ( u v ) α 1 log u v = D(u v). Ths shows that the Tsalls-tye relatve entroy converges to the Boltzmann-Shannon one as α 1. In analogy to the defnton of the free energy F, we defne the Tsalls-tye free energy F α as [ 1 ( ) α 1 F α = D α ( µ) = 1]. α(α 1) µ It has been roved n [] that F α has the followng nce roertes smlar to those of the free energy F : F α 0, df α 0. Next we shall strengthen these results by rovdng nonzero lower bounds for the Tsalls-tye free energy F α and ts dssaton rate df α /. To ths end, we need the followng two lemmas. Lemma (See [3]) Let α and β be two real numbers satsfyng α <,α 0,1 and β 0. Then for any x [ 1, β], the followng nequalty holds: 1 α(α 1) [(x+1)α αx 1] 1 (β +1)α x, where the case ofx = 1 s understood n the lmt sense. Lemma (See [4]) For any α, there exsts a constant u α > 0 deendng on α such that for any comlex numberz, the followng nequalty holds: 1 α(α 1) [ z +1 α 1 αre(z)] u α z α. 9

10 The followng theorem gves a nonzero lower bound for the Tsalls-tye free energy. Theorem For any α 0, 1, 1 F α u α whereu α s the constant n Lemma µ 1 α ( µ ), f α <, µ 1 α µ α, f α, Proof. Let x = /µ 1 and β = 1/µ 1 0. Then x [ 1,β] and t follows from Lemma 3.15 that for anyα < and α 0,1, 1 α(α 1) [(x +1) α αx 1] 1 µ α x = 1 µ α ( µ ). Thus we have [ ] 1 F α = µ (x +1) α 1 = α(α 1) 1 µ 1 α ( µ ). 1 α(α 1) µ [(x +1) α αx 1] Ths shows that the theorem holds when α <. On the other hand, t follows from Lemma 3.16 that for anyα, [ ] 1 F α = µ (x +1) α 1 = α(α 1) u α µ x α = u α µ 1 α µ α. Ths shows that the theorem also holds when α. 1 α(α 1) µ [(x +1) α αx 1] The followng theorem gves a nonzero lower bound for the dssaton rate df α / of the Tsalls-tye free energy. Theorem For any α 0,1, the Tsalls-tye free energy has the followng dssatve roerty: df α c α [ j j q j )] j (q, f α <, c α j (q j j q j ) α, f α, wherec α > 0 s a constant only deendng on α and the transton rate matrxq. 10

11 Proof. Let y = /µ. It follows from the master equaton (1) that df α = 1 α 1 = 1 α 1 = 1 α 1 =,j 1 α(α 1) µ y α 1 dy y α 1 = 1 α 1 ( q j j q j ) = 1 α 1 j µ q j (y α y y α 1 j )+ 1 α,j Let z j = y /y j 1. It s easy to see that df α = y α 1 d,j q j (y α 1 µ q j (yj α y α ),j µ q j [y α αy y α 1 j +(α 1)y α j ]. y α 1 j ) 1 µ q j y α(α 1) j[(z α j +1) α αz j 1]. (8),j Let β = 1/µ j 1 0. Then z j [ 1,β] and t follows from (8) and Lemma 3.15 that for anyα < and α 0,1, df α 1 µ q j yj α (µ j ) α zj = 1 µ q j y α j (µ j ) α (y y j ),j,j = 1 µ q j (µ µ j ) α (y y j ).,j Recall that the comonents of µ are all ostve. Set r = mn{µ 1,µ,µ N } > 0 and M = max{q j : j} > 0. It thus follows from the Cauchy-Schwarz nequalty that df α r4 α r4 α MN = r4 α MN µ q j (y y j ) r4 α,j,j [ ] [ ] µ q j (y y j ) = r4 α q j MN j j [ ( q j j q j )]. j j µ q j µ q j M (y y j ) Ths shows that the theorem holds when α <. On the other hand, t follows from (8) and Lemma 3.16 that for anyα, df α u α µ q j yj α z j α = u α µ q j y y j α,j ( µ q ) α 1 y j u α µ q j y j α M,j = u α (µ M α 1 q j y y j ) α.,j,j 11

12 It thus follows from the Hölder nequalty that [ df α α u α u α µ (MN) α 1 q j y y j ] µ (MN) α 1 q j (y y j ) j j u α α u α α = (MN) α 1 q j = ( (MN) α 1 q j j q j ). Ths shows that the theorem also holds when α. j We conclude ths secton wth the followng observaton. Remark It s nterestng to note that by takngα 1 n Theorem 3.17, we obtan that F = lmf α 1 ( µ ). α 1 Ths s exactly the nequalty stated n Theorem 3. whch gves the nonzero lower bound for the free energy F. Smlarly, by takng α 1 formally n Theorem 3.18, we can recover the nonzero lower bound for the free energy dssaton rate f d roved n Theorem Physcal sgnfcance of the nonzero lower bounds In ths secton, we shall resent some hyscal nterretatons of the nonzero lower bounds obtaned n the revous secton. It s well-known that the entroy roducton rate of a system s always nonnegatve and ths s actually the second law of thermodynamcs, whch characterzes the rreversblty of nonequlbrum rocesses. Recently, t has been shown n [1, 13] that the rreversblty has two dfferent mechansms: the devaton from steady state and the breakng of detaled balance. The former s quantfed by the non-adabatc entroy roducton ratee (na), whle the latter by the adabatc entroy roducton rate e (ad). Therefore, the decomoston of the total entroy roducton ratee = e (na) +e (ad) rovdes a dee nsght nto the rreversblty. Ths mortant ont of vew s by no means obvous wthout the ad of the nonzero lower bounds obtaned above. For a long tme, t s known that the entroy roducton ratee of a system can generally be wrtten as a blnear form of the thermodynamc fluxes and forces [11]. For the master equaton, we have e =,j J j X j = <j j j ( q j j q j )log q j j q j, wherej j = q j j q j s the local thermodynamc flux between states and j and X j = log q j j q j α s the corresondng local thermodynamc force. In fact, the non-adabatc art e (na) wrtten n a smlar way as e (na) = df = F d = J X, can be 1

13 wherej = d / = j ( jq j q j ) s the total thermodynamc flux through stateand X = F [ = log ] +1 µ s the corresondng total thermodynamc force. Thus both the entroy roducton rate e and ts non-adabatc art e (na) can be cast nto a smlar blnear form. The former concerns wth local thermodynamc fluxes and forces and the latter only deals wth the total ones. Wth Theorems 3.1 and 3.3, we show that there exst constants c 1,c 4 > 0 only deendng on the transton rate matrxqsuch that and e =,j e (na) = J j X j c 4,j J j J X c 1 J. These nequaltes reveal two strong dssatve roertes for the master equaton. They suggest that a blnear form of the thermodynamc fluxes and forces generally has a nonzero lower bound whch s roortonal to the sum of squares of the thermodynamc fluxes. We beleve that ths roerty s ossessed by qute general thermodynamc systems [18], not necessarly modeled by the master equaton. Recall that a central task of modern nonequlbrum thermodynamcs s to rovde gudng rncles for modelng varous rreversble rocesses, where the second law of thermodynamcs lays a fundamental role. For examle, classcal rreversble thermodynamcs (CIT), ratonal extended thermodynamcs (RET), and extended rreversble thermodynamcs (EIT) are all such theores that have been develoed and aled wth great success for ths task [11, 5, 6]. Therefore, the strong dssatve roertes obtaned above may rovde such a gudng rncle for the constructon of thermodynamc consttutve relatons. Let J = (J k ) and X = (X k ) denote two column vectors comosed of all thermodynamc fluxes and forces of a molecular system, resectvely. In nonequlbrum thermodynamcs, we hoe to fnd a consttute relaton J = J(X) between the thermodynamc fluxes and forces. In ths way, we can obtan a closed and solvable mathematcal model together wth balance equatons [7]. In the regon not far from equlbrum, the theory of CIT clams a lnear consttutve relaton: J(X) = MX, where M s a constant matrx whch should be nonnegatve defnte n order to guarantee e 0. The celebrated Onsager s recrocal relaton further requres M to be symmetrc [1]. However, when the system s far from equlbrum, the matrx M cannot be vewed as constant. A ossble modfcaton s the nonlnear consttutve relaton of J(X) = M(X)X, where M(X) s a nonnegatve defnte matrx deendng on X. Ths knd of relatons s allowed by the second law of thermodynamcs. In ractce, we are often concerned wth the asymtotc behavor ofj(x) asx tends to zero or nfnty. To ths end, we assume thatj(x) behaves lke 13

14 X α asx tends to zero or nfnty, namely, there exst constantsγ 1 > 0 and γ > 0 such that γ 1 X α J(X) γ X α. (9) Remarkably, the nonzero lower bounds obtaned above mose adonal constrants on the exonentα. In fact, t s easy to see from (9) that e = J X = J(X) X X J(X) γ X α+1. (10) On the other hand, f the entroy roducton rate e has a nonzero lower bound as n Theorem 3.3, t s easy to check that Combnng (10) and (1), we obtan that e c 4 J = c 4 J(X) c 4 γ 1 X α. (11) X α 1 γ. (1) c 4 γ1 Ths mles thatαmust satsfyα 1 asx 0 andα 1 asx. Therefore, the nonzero lower bounds obtaned above ndeed rovde gudance to the constructon of thermodynamc consttutve relatons and such gudance does not seem to be reorted n the lterature before. 5 A mathematcal alcaton of the nonzero lower bounds The nonzero lower bounds obtaned n ths aer not only rovde hyscal nsghts nto the dssatve roertes of nonequlbrum rocesses, but also mly some mathematcal consequences such as the long-tme dynamcs of the master equaton. It s a classcal result that the robablty dstrbuton (t) of an rreducble Markov jum rocess wll converge to the steady-state dstrbutonµast. Ths fact can be roved ether by the algebrac method based on the Perron-Frobenus theorem [8] or by the robablstc method based on the coulng of Markov chans [17]. However, both the two methods turn out to be rather nvolved. In ths secton, we shall gve a smle and urely analytc roof of ths classcal result by usng the nonzero lower bounds obtaned n ths aer. To ths end, we recall the followng elementary fact whch wll be roved for comleteness. Lemma 5.1. Letf be a Lschtz contnuous and ntegrable functon on[0, ). Then we have lm f(t) = 0. (13) t Proof. Sncef s Lschtz contnuous, there exsts a constantk > 0 such that f(x) f(y) < K x y for any x,y 0. Assume that f(t) does not converge to 0 as t. Then there exst 0 < ǫ < K and a sequence t n such that t n+1 t n > 1 and f(t n ) > ǫ for any n 1. For any n 1 and h < ǫ/k, t s easy to see that f(t n +h) f(t n ) Kh < ǫ. 14

15 Ths shows that f(t n +h) > ǫ for anyn 1 and h < ǫ/k. Thus we obtan that tn+ǫ/k ǫ f(t) f(t) K =. 0 n=1 t n Ths contradcts the ntegrablty off on[0, ) and hence we obtan the desred result. We are now n a oston to study the long-tme dynamcs of Markov jum rocesses based on the nonzero lower bound for the free energy dssaton rate. Theorem 5.. Assume that the system s rreducble. Then we have lm (t) = µ. t Proof. Defne a functonf = f(t) fort [0, ) as f(t) = [ ] ( j (t)q j (t)q j ) = j n=1 [ ] j (t)q j. Integratng the nequalty n Theorem 3.3 and notng that F(t) 0, we obtan that Ths ndcates thatf s ntegrable on [0, ). t c 1 f(s)ds F(0) F(t) F(0). 0 Moreover, set M = max{ q j : 1,j N}. It follows from the master equaton (1) that for any, Drect comutaton shows that f(t) = (t) j j j (t)q j MN. (14) [ ][ j (t)q j j j ṗ j (t)q j ]. (15) From (14) and (15), t s easy to see that f(t) s unformly bounded and thus f s Lschtz contnuous on [0, ). It thus follows from Theorem 5.1 that lm f(t) = 0. (16) t Assume that(t) does not converge toµast. Snce(t) s unformly bounded, there exst a stateµ and a sequencet n such that lm (t n) = µ µ. n Owng to the unqueness of the steady-state dstrbutonµ, t s obvous that r := [ ] µ jq j > 0. Thus when n s suffcently large, we have f(t n ) = [ ] j (t n )q j r. j Ths contradcts (16) and hence we obtan the desred result. j 15

16 Acknowledgment Ths work s suorted by the Natonal Natural Scence Foundaton of Chna (Grants and ) and the Intatve Scentfc Research Program of Tsnghua Unversty (Grant ). References [1] Rechl LE (1980) A Modern Course n Statstcal Physcs (Unversty of Texas Press, Austn). [] Van Kamen NG (199) Stochastc Processes n Physcs and Chemstry (Elsever, Sngaore). [3] Kurtz TG (197) The relatonsh between stochastc and determnstc models for chemcal reactons. The Journal of Chemcal Physcs 57: [4] Ge H, Qan M, Qan H (01) Stochastc theory of nonequlbrum steady states. Part II: Alcatons n chemcal bohyscs. Phys Re 510: [5] Jarzynsk C (011) Equaltes and nequaltes: rreversblty and the second law of thermodynamcs at the nanoscale. Annu Rev Condens Matter Phys : [6] Sefert U (01) Stochastc thermodynamcs, fluctuaton theorems and molecular machnes. Re Prog Phys 75: [7] Van den Broeck C, Esosto M (015) Ensemble and trajectory thermodynamcs: a bref ntroducton. Physca A: Statstcal Mechancs and ts Alcatons 418:6 16. [8] Jang DQ, Qan M, Qan MP (004) Mathematcal Theory of Nonequlbrum Steady States: On the Fronter of Probablty and Dynamcal Systems (Srnger). [9] Kjelstru S, Bedeaux D (008) Non-equlbrum Thermodynamcs of Heterogeneous Systems (World Scentfc, Sngaore). [10] Zhang XJ, Qan H, Qan M (01) Stochastc theory of nonequlbrum steady states and ts alcatons. Part I. Phys Re 510:1 86. [11] De Groot SR, Mazur P (196) Non-equlbrum Thermodynamcs (North-Holland Publshng Comany, Amsterdam). [1] Ge H, Qan H (010) Physcal orgns of entroy roducton, free energy dssaton, and ther mathematcal reresentatons. Phys Rev E 81: [13] Esosto M, Van den Broeck C (010) Three faces of the second law. I. Master equaton formulaton. Phys Rev E 8: [14] Oono Y, Pancon M (1998) Steady state thermodynamcs. Prog Theor Phys Su 130:9 44. [15] Hatano T, Sasa S (001) Steady-state thermodynamcs of langevn systems. Physcal revew letters 86:3463. [16] Yong WA (004) Entroy and global exstence for hyerbolc balance laws. Arch Raton Mech An 17: [17] Norrs JR (1998) Markov Chans (Cambrdge Unversty Press). [18] Yong WA (008) An nterestng class of artal dfferental equatons. J Math Phys 49: [19] Yong WA (01) Conservaton-dssaton structure of chemcal reacton systems. Phys Rev E 86: [0] Tsalls C (1988) Possble generalzaton of Boltzmann-Gbbs statstcs. J Stat Phys 5: [1] Tsalls C (011) The nonadve entroy S q and ts alcatons n hyscs and elsewhere: some remarks. Entroy 13: [] Shno M (1998) H-theorem wth generalzed relatve entroes and the Tsalls statstcs. J Phys Soc Jn 67: [3] Mtrnovc DS, Pecarc J, Fnk AM (1993) Classcal and New Inequaltes n Analyss (Kluwer Academc, Dordrecht). [4] Lendler L (197) On a generalzaton of Bernoull s nequalty. Acta Sc Math 33:

17 [5] Müller I, Rugger T (1998) Ratonal extended thermodynamcs (Srnger-Verlag, New York). [6] Jou D, Casas-Vázquez J, Lebon G (010) Extended rreversble thermodynamcs (Srnger, Netherlands), 4th eon. [7] Zhu Y, Hong L, Yang Z, Yong WA (014) Conservaton-dssaton formalsm of rreversble thermodynamcs. J Non-equl Thermody 40: [8] Berman A, Plemmons RJ (1979) Nonnegatve Matrces n the Mathematcal Scences (Academc Press). 17