Waiting for rare entropic fluctuations
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Waiting for rare entropic fluctuations
Keiji Saito and Abhishek Dhar Department of Physics, Keio University, Yokohama 223-8522, Japan International centre for theoretical sciences, TIFR, IISC campus, Bangalore 560012 (Dated: September 7, 2018)Non-equilibrium fluctuations of various stochastic variables, such as work and entropy production,have been widely discussed recently in the context of large deviations, cumulants and fluctuationrelations. Typically, one looks at the distribution of these observables, at large fixed time. Tocharacterize the precise stochastic nature of the process, we here address the distribution in thetime domain. In particular, we focus on the first passage time distribution (FPTD) of entropyproduction, in several realistic models. We find that the fluctuation relation symmetry plays a crucialrole in getting the typical asymptotic behavior. Similarities and differences to the simple randomwalk picture are discussed. For a driven particle in the ring geometry, the mean residence time isconnected to the particle current and the steady state distribution, and it leads to a fluctuationrelation-like symmetry in terms of the FPTD.
PACS numbers: 05.40.-a,05.40.Jc,05.70.Ln
Introduction.—
The past two decades have witnessedsignificant development in nonequilibrium thermody-namics [1–4]. The fluctuation relations are remarkablediscoveries which have quantitatively refined the conceptof the second law [5–9] as applied to small systems. Oneof the central issues in nonequilibrium statistical physicshas been in characterizing the universal nature of fluctua-tions of thermodynamic variables, such as heat and workthat quantify entropy generated in non-equilibrium pro-cesses. Usually, one measures the accumulated entropicvariable, say X , over a fixed time interval τ , and its fluc-tuations are then characterized through a distribution P ( X ). Defining, for example, X as the stochastic totalentropy, one can then prove the detailed and integral typeof fluctuation relation for any fixed time interval τ , in var-ious Markov processes [9]. For large observation times,one finds the large deviation form P ( X ) ∼ e τh ( X/τ ) [10],where h ( x ) is the large deviation function. The corre-sponding cumulant generating function (CGF) is definedby µ ( ξ ) = log h e ξX i /τ where h ... i is an average over thesteady state, and this generates the n th order of cumu-lant I n as I n = ∂ n µ ( ξ ) /∂ξ n | ξ =0 . (1)For physical quantities related to entropy production, itis well known that the CGF shows the fluctuation re-lation symmetry [11, 12]. This symmetry is not onlymathematically beautiful but also physically importantsince it reproduces linear response results and also givesnontrivial relationships on nonlinear responses [13–15].Large deviations and the CGF have been crucial towardsconstructing universal thermodynamic structure of thenonequilibrium steady state [16–18].The large deviation function gives us the probabilityof observing rare events in some fixed observation timewindow. An interesting and natural question to ask is asto how long would one have to wait to see a rare event of a specified size? . This is just the question of the first pas-sage problem for the stochastic variable X . Although thephysics of fluctuation at fixed time has been intensivelystudied and a lot of discoveries have been made, sur-prisingly, only a little is known on the stochastic natureof its time evolution itself. One expects that the time-evolution of stochastic thermodynamic variables shouldbehave like a biased random walk in some configurationspace, but the details of the temporal aspects have notbeen investigated.The main aim of this Letter is to investigate this as-pect, which is clearly necessary for a deeper understand-ing of stochastic thermodynamics. In particular we con-sider the problem of the first passage time distribution(FPTD) of the desired stochastic variable, which is anexperimentally measurable quantity. The FPTD here isthe distribution of waiting time at which a stochasticvariable first reaches some target value. We consider thetypical properties of the FPTD of entropy-related vari-ables within the broad and well-established paradigms ofstochastic thermodynamics. Three examples of nonequi-librium processes are considered: (a) an over-dampeddriven particle in a ring geometry, (b) classical chargetransfer via a quantum-dot, and (c) heat transfer acrossa coupled oscillator system [see Fig. (1) for a schematicdescription]. Note that due to recent development intime-resolved measurement techniques, there are a num-ber of relevant experiments for these setups that look atnonequilibrium fluctuations [19–24].Using these models, we address the following questions.Is there a typical functional form for the FPTD and espe-cially its tail ? How does it depend on the sign of entropyproduced ? What are the differences as compared to thetime-evolution of a simple biased random walk ? Con-cerning this last question, consider the case where X isthe position of a biased diffusing particle on the openline. Then, defining F rw ( t, X ) dt as the probability that f ǫ µ L µ R L (a) (b) (c) L β L β R ν ¯ α + ¯ α − FIG. 1: (color online) Schematic picture of setups. (a): Theover-damped driven particle in the ring geometry. The rightpicture shows the potential landscape in an infinite line pic-ture. (b): Classical charge transfer via a quantum-dot. (c):Heat transfer via a coupled oscillator system. the particle hits X for the first time between times t and t + dt , one easily finds [25] F rw ( t, X ) = | X | e − ( X − I t ) / I t √ πI t → e − I I t − log t . (2)We will use this as a reference form, and aim to figureout the similarities and dissimilarities between this sim-ple random walk picture, and the real stochastic time-evolution of entropic variables. Intriguingly with use ofthe fluctuation relation symmetry, one can derive theasymptotic form of the FPTD for these models, anddissimilarities to F rw can be argued. In addition, wederive the exact expression for mean residence time forthe driven particle in the ring geometry which leads tofluctuation-relation-like equality in terms of the FPTD. Driven particle in the ring geometry.—
We consider acolloidal particle driven by a constant force and confinedto move on a ring, as depicted in Fig. (1a). The dynamicsis well-described by the over-damped Langevin equationswith temperature β − . The Boltzmann constant is set tounity and let us also set f >
0. To proceed, we discretizethe space into L sites on the ring separated by a smallspacing a . Let P ν ( t ) be the probability to find the parti-cle on the ν th site at time t . Its evolution is given by ∂P ν ( t ) ∂t = W ν,ν − P ν − ( t ) + W ν,ν +1 P ν +1 ( t ) − W ν,ν P ν ( t ) , (3)where W ν,α is the transition rate matrix elementwhich satisfies the local detailed balance condition W ν +1 ,ν /W ν,ν +1 = e − β ( U ν +1 − U ν )+ βaf and U ν is the po-tential energy at the ν th site. It is useful to introducethe winding number, N , which, for any given particle tra-jectory, is obtained by counting the number of times theparticle makes the transition from site L to the first site(reverse transitions from the first site to the site L arecounted with a negative sign). The particle’s state can belabeled by the duplet ( ν, N ). Suppose that in any given realization of the stochastic process, the particle makesthe transition ( ν, → ( α, N ) in time t . Then the workdone is w = f ( N L + α − ν ) while the heat dissipated intothe bath is q = w − U α + U ν . Thus, at sufficiently largetimes, entropy production rate is proportional to the av-erage rate of the winding number. Due to the positiveforce f , the particle on average moves in the positive di-rection and the average winding number rate is positive.However there is a finite probability to observe the parti-cle moving in the opposite direction. The ratio of proba-bilities between positive and negative winding number atany finite time is quantitatively given by the fluctuationrelation. We address the FPTD for the winding number.Let T α,ν ( N, t ) be the transition probability from ( ν, α, N ) and let F α,ν ( N, t ) be the probability that firstpassage between ( ν,
0) to ( α, N ) occurs between time t to t + dt . We note the relation for N = 0 T α,ν ( N, t ) = Z t du T α,α (0 , t − u ) F α,ν ( N, u ) . (4)Taking the Laplace transformation T α,ν ( N, s ) = R ∞ dt e − st T α,ν ( N, t ) and similarly for the FPTD we get F α,ν ( N, t ) = 12 πi Z c + i ∞ c − i ∞ ds e st T α,ν ( N, s ) T α,α (0 , s ) . (5)We consider the asymptotic behavior of the FPTD atsufficiently large waiting time. To this end, one can writethe time-evolution equation of the joint probability ofthe variables ν, N . Solving this through Fourier-Laplacetransformation, one can get the formal expression for thetransition probability matrix [26] T ( N, s ) = 12 πi I dzz N +1 A ( z, s )det [ s − W z ] , (6)where the matrix W z is given by the matrix W replacing(1 , L ) and ( L,
1) elements by zW ,L and z − W L, respec-tively, and A ( z, s ) is the co-factor matrix for the matrixof denominator. There are two singular values z ± ( s ) fromthe denominator, which are connected to each other bythe fluctuation relation symmetry [26] z + ( s ) z − ( s ) = e − βfLa , (7)where βf La is the entropy produced in the reservoir fora single winding around the ring. Setting α = ν , usingthe above symmetry and Eq. (5), one can express thedistribution in terms of only one singular point [26] F ν,ν ( N, t ) = C ν ( N )2 πi Z ds e t [ s + b log z + ( s )] , (8)where b = N/t . The steadty state FPTD of windingis then given by F ( N, t ) = P ν F ν,ν ( N, t ) p SSν , with thesteady state distribution p SSν . A further careful exami-nation reveals that the singular value z + ( s ) is connectedto the CGF, µ ( ξ ), for the winding number s − µ ( ξ ) = 0 , where ξ = log z + ( s ) . (9)Based on these relations, a saddle point analysis leads tothe following exact asymptotic expression of the FPTD ∝ F asym ( t ) for sufficiently large waiting time [26] F asym ( t ) = exp[ − Γ t − (3 /
2) log t ] , Γ = ∞ X n =0 ( − I ) n +2 ( n + 2)! q n ( ξ ) | ξ =0 (10)= I I + I I I + (3 I − I I ) I I + · · · , where the function q n ( ξ ) is connected to the CGF; q n ( ξ ) = h ( d µ ( ξ ) dξ ) − ddξ i n ( d µ ( ξ ) dξ ) − .Some important observations on Eq. (10) are now inorder. The asymptotic temporal decay form dependsneither on the sign nor on the amplitude of the wind-ing number, although the actual probability of negativeand positive winding numbers differ by exponential fac-tor (in entropy produced). Thus, even extremely rareevents follow the same asymptotic form. In the linear re-sponse regime with small first cumulant, the asymptoticbehavior is well-explained by the simple random walkpicture F rw . In the far-from-equilibrium regime, how-ever, critical deviation from this picture reveals itself inthe higher order terms with nontrivial expressions. Thisdeviation will be significant in small systems where thedegree of nonequilibrium is easily increased. We notethat the asymptotic form is given by the general form,in terms of cumulants, irrespective of detailed potentialforms. This indicates that it might be applicable to widerclasses of physical situations. As we see below, it turnsout that the expression is valid for many other situationswhen the cumulants are calculated for appropriate phys-ical quantities. Two other examples.—
We now show that the asymp-totic form (10) also appears for open nonequilibriumsystems such as (b) classical charge transport via aquantum-dot and (c) heat transfer via coupled oscilla-tors [See the Figs. (1b,1c) for schematic pictures].Case (b): Let µ L and µ R be respectively the chemicalpotential for the left and right leads and consider spin-lesselectrons transmitted via a quantum-dot with an onsiteenergy ǫ . We measure transmitted electron at the rightcontact to the reservoir, and let the accumulated electrontransfer till time t be n . Charge transfer produces Jouleheating and is directly connected to entropy productionrate as h ˙ Si = β ( µ L − µ R ) h ˙ n i .We now consider the FPTD of the accumulation ofelectron number, an experimentally measurable quantity.Let 1 and 2 respectively denote the unoccupied and occu-pied states of the quantum-dot. Then the time-evolutionof the two states is described by the same type of dy-namics as in Eq. (3). The transition probability W i,j iscomposed of two contributions from the left and rightreservoirs W = W L + W R , where W r , = γ [1 − f r ( ǫ )]and W r , = γf r ( ǫ ) where f r is the Fermi distribution of the r th lead ( r = L, R ). Hence these elements satisfy thedetailed balance W r , /W r , = e β ( ǫ − µ r ) . The modifiedtransition probability matrix W z in (6) is given by thematrix W on replacing W R , and W R , in the off-diagonalmatrix elements by W R , z and W R , z − respectively. Thesingular points in the denominator are z ± ( s ) which areagain connected by the fluctuation relation symmetry z − ( s ) z + ( s ) = e − β ( µ L − µ R ) . (11)In the present example it is easy to see that the firstpassage from the initial state ( i, n = 0) to any fixeddesired value of n , also fixes the final configuration j .Using the renewal equation, we can obtain FPTD from( i, → ( j, n ), using the same argument as for the drivenparticle in the ring geometry, and find that the FPTDis proportional to Eq. (10) where now the cumulants arefor charge transfer and known exactly (see [26]). Forthe case of many sites with strong onsite-Coulomb inter-action, one may employ the symmetric simple exclusionprocess [28]. We can demonstrate that the same expres-sion is obtained analytically for this system of coupledquantum-dots.Case(c): We consider the example of the coupled os-cillator system, exchanging heat with two heat reservoirsat temperatures T L , T R , whose dynamics is described bythe overdamped Langevin equation γ ˙ x = − kx + η L ( t ) , γ ˙ x = kx + η R ( t ) , (12)where x , are the positions of the first and second par-ticles which are coupled via spring constant k , and x = x − x . The noise terms η r satisfy the fluctuation dis-sipation relations h η r ( t ) η r ′ ( t ′ ) i = 2 δ r,r ′ γβ − r δ ( t − t ′ ). Inthis case we consider the heat transfer into the right bathin time t and this is given by Q = R t dt ′ kx ( t ′ )[ kx ( t ′ ) + η R ( t ′ )] /γ and are interested in the FPTD for transitionfrom an initial state ( x, Q = 0) to a state with Q amountof heat transferred. As in the previous examples, we canthink of our system executing biased diffusion in ( x, Q )space. However in this case fixing Q an the initial x doesnot fix the final position and so an extension of the for-mulation is required. A heuristic derivation is given in[26] but the final result for the tail of the FPTD for Q turns out to be the same as given by Eq. (10) where nowthe cumulants for heat transfer are known exactly from[29, 30]. Numerical demonstration of the asymptotic formula forseveral cases.—
We numerically verify the asymptotic be-havior (10) for the three examples discussed and shownin Fig. (1). In the case (a), we consider the dynamics incontinuous space given by γ ˙ x = − dU ( x ) /dx + f + η ( t ),where we impose the periodic boundary condition withthe periodic length 1 and we employ the potential U ( x ) =sin(4 πx ) / (4 π ). The variable η ( t ) is the Langevin noisesatisfying h η ( t ) η ( t ′ ) i = 2 γβ − δ ( t − t ′ ). In case (b), we nu-merically update the state using a Monte-Carlo approach − − F ( Q , t ) t (cid:2) Γ − (cid:3) Q = − . Q = 0 . F asym F rw . F ( n , t ) n = − n = 1 F asym F rw . F ( N , t ) N = − N = 1 F asym F rw . . N = − N =1 t (a)(b)(c) FIG. 2: (color online) The FPTDs for the three models areshown with unit normalization. In the inset in (a), unnormal-ized data are also shown, which shows that the FPTD withthe negative entropy production is very small. The black solidlines are fitting by the theory (10), while the dotted line isthe asymptotic curve of the random walk (2). Parameterssets are (a): β = 5 .
0, (b): ( µ L , µ R ) = (6 . , .
0) and (c):( β L , β R ) = (5 . , . with the specified transition rates. For (c) the systemevolves through the Langevin dynamics in Eq. (12). Inall cases we sample the initial state from the steady stateand then measure the FPTD for specified values of wind-ing number [in case (a)], the charge transfer into rightreservoir [in case (b)] and the heat transfer [in case (c)].For events with a negative entropy production there isa finite probability of the event not occurring at all ina given realization. Hence for negative values, we plotthe distribution, conditioned on the probability that itoccurs. The results are shown in the Fig. (2). In shorttime scale, non-universal behavior is observed. However,all three cases clearly show that the asymptotic behavioris well-described by the theory (10), irrespective of thefixed values including even very rare events. At finitetimes the logarithmic correction is important. The devi-ation from the simple random walk picture (which givesΓ = I / (2 I )) is also clear. Basic equation and integral fluctuation relation interms of first passage.—
Let us consider the entropy pro-duced in the thermal reservoir for case (a). Let T α,ν ( S , t )be the transition probability from the site ν to α in time t during which the entropy of the heat reservoir increasesby the amount S . The entropy is determined by the po-tential at the sites α and ν and the work done by exter-nal force, i.e., S = β [ U ν − U α + f a ( α − ν + LN )]. Notethat fixing ν, S does not necessarily fix α, N . Let us de- fine F ¯ α,ν ( S , t ) to be the FPTD only for S [not for ( α, S )],while reaching ¯ α from ν . For fixed S , the site ¯ α dependson ν . It is uniquely determined if |S| is sufficiently large,while for small |S| , there can be at most two choices of¯ α , respectively on the two sides of ν . As example, see theright figure in Fig.(1a) of a case where two ¯ α (denoted by¯ α ± ) can be reached for a fixed negative S , starting fromthe site ν . Then we note the following basic equation inthe Laplace representation T ¯ α,ν ( S , s ) = X ¯ α ′ T ¯ α, ¯ α ′ (0 , s ) F ¯ α ′ ,ν ( S , s ) . (13)This type of equations provides in general, a basis forconsidering the FPTD for entropic variables.We now establish several relations. The first is an exactrelation for the mean residence time at a given latticepoint for given entropy production, given by R ∞ dt T α,ν ( −S , t ) = e −S p SSν /J , R ∞ dt T ν,α ( S , t ) = p SSν /J , (14)where p SSα is the steady state distribution at the site α and J is the steady state particle current. In the firstrelation, it is assumed that the process ( ν, → ( α, −S )is opposite to the direction of current, while in the sec-ond relation ( α, → ( ν, S ) is in the same direction ascurrent. These are connected by the detailed fluctuationrelation [27]. Note that the sign of S is not specified.The proof for this is presented in [26]. Eqs.(13) and (14)are crucial for deriving other relations as we now show.We now employ the usual definition of total entropy S tot α,ν = ln( p SSν /p SSα )+ S . Then for fixed negative entropy S <
0, using relations (13) and (14) leads to the equality P ¯ α e −S tot¯ α,ν F ¯ α,ν ( S , s = 0) = 1 . Multiplying both sides ofthis equation by p SSν and summing over ν immediatelyleads to the integral type of fluctuation relation hh e −S tot ii S = 1 , (15)where the average hh ... ii S implies taking all possible firstpassage paths producing the negative entropy S , and thatstart from the steady state. Numerical demonstration ispresented in [26]. Summary.—
In general, it is difficult to characterizegeneral temporal aspects in stochastic time-evolution ofthermally fluctuating objects. As a first step in this di-rection, we consider the first passage time distributionof entropy-production in several models that are relevantto recent experimental setups. We find the asymptoticbehavior of Eq. (10), which seems to be the typical func-tional form, valid in many situations. For the paradig-matic example of a particle driven round a periodic po-tential, we find further properties, given by (14) and (15),that characterize the FPTD. It is proposed that Eq.(13)is in general the basic equation needed while consideringthe FPTD for entropic variables.
Acknowledgment
K.S was supported by MEXT (25103003) and JSPS(90312983). AD thanks DST for support through theSwarnajayanti fellowship. [1] U. Seifert, Rep. Prog. Phys. , 126001 (2012).[2] M. Campisi, P. H¨anggi, and P. Talkner, Rev. Mod. Phys. , 771 (2011).[3] M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod.Phys. , 1665 (2009).[4] D. J. Evans and D. J. Searles, Adv. Phys. , 1529(2002).[5] D. J. Evans, E G. D. Cohen, and G. P. Morriss, Phys.Rev. Lett. , 2401 (1993).[6] J. Kurchan, J. Phys. A: Math. Gen. , 3719 (1998).[7] C. Maes, J. Stat. Phys.
367 (1999).[8] G. E. Crooks, Phys. Rev. E , 040602 (2005).[10] H. Touchette, Phys. Rep. , 1 2009.[11] G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. ,2694 (1995).[12] J. L. Lebowitz and H. Spohn, J. Stat. Phys. , 333(1999).[13] G. Gallavotti, Phys. Rev. Lett. , 4334 (1996),[14] D. Andrieux and P. Gaspard, J. Stat. Mech. P02006(2007),[15] K. Saito and Y. Utsumi, Phys. Rev. B , 115429 (2008).[16] L. Bertini, A De Sole, D. Gabrieli, G. Jona-Lasinioand C. Landim, Phys. Rev. Lett. , 040601 (2001);arXiv:1404.6466.[17] T. Bodineau and B. Derrida, Phys. Rev. Lett. , 180601(2004)[18] B. Derrida, J. Stat. Mech. P07023 (2007).[19] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki and M.Sano, Nature Physics , 988 (2010).[20] V. Blickle, T. Speck, C. Lutz, U. Seifert and C.Bechinger, Phys. Rev. Lett. , 210601 (2007).[21] Y. Utsumi, DS. Golubev, M.Marthaler, K.Saito, T. Fu-jisawa and G. Sch¨on, Phys. Rev. B , 125331 (2010).[22] B. K¨ung, C. R¨ossler, M. Beck, M. Marthaler, D.S. Gol-ubev, Y. Utsumi, T. Ihn, K. Ensslin, Phys. Rev. X ,011001 (2012)[23] J. R. Gomez-Solano, L. Bellon, A. Petrosyan and S. Cilib-erto, Europhys Lett. , 60003 (2010).[24] S. Ciliberto, A. Imparato, A. Naert and M. Tanase, Phys.Rev. Lett. , 180601 (2013).[25] S. Redner, A Guide to First-Passage Processes , Cam-bridge University Press (2001).[26] In the supplementary material, the formal expression forthe transition matrix, detailed derivation of Eq.(10) forcase (a)-(c), derivation of the mean residence time (14),and numerical demonstration of the relation (15) are pre-sented.[27] C. Jarzynski, J. Stat. Phys. , 77 (2000).[28] P. -E. Roche, B. Derrida and B. Doucot, Eur. Phys. J. B , 529 (2005).[29] P. Visco, J. Stat. Mech. P06006 (2006).[30] A. Kundu, S. Sabhapandit and A. Dhar, J. Stat. Mech.P03007 (2011). Supplementary Material for“Waiting for rare entropic fluctuations”
Keiji Saito and Abhishek Dhar Department of Physics, Keio University, Yokohama 223-8522, Japan International centre for theoretical sciences, TIFR, IISC campus, Bangalore 560012
Derivation of Eq. (6) in the main text.—
Let us definethe winding number N of a typical trajectory of the dif-fusing particle by counting the number of times it makesthe transition from the site L to the first site, L → → L . Let P ν ( N, t ) be the joint probability that theparticle is at the ν th site with winding number N at time t . We are interested in finding the probability vector P ( N, t ) = { P ( N, t ) , P ( N, t ) , · · · , P L ( N, t ) } . (16)Then it is easy to see that this joint probability satisfiesthe following equation ∂ P ( N, t ) ∂t = W − P ( N − , t ) + W + P ( N + 1 , t ) + W P ( N, t ) . (17)Here W − is a L × L matrix whose only non-vanishingelement is [ W − ] ,L = W ,L , W + is a L × L matrixwhose only non-vanishing element is [ W + ] L, = W L, ,and W = W − W − − W + .We define the generating function P ( z, s ) = ∞ X N = −∞ Z dt z N e − st P ( N, t ) . (18)Then one readily find that this satisfies the equality P ( z, s ) = 1 s − W z p (19)where W z = W − z + W + z − + W and p is the initialcondition p = P ( N = 0 , t = 0). From this, one gets P ( N, t ) = 12 πi Z c + i ∞ c − i ∞ dse ts P ( N, s ) , (20) P ( N, s ) = 12 πi I dzz N +1 s − W z p . (21)Hence the transition matrix T ( N, s ) is given by T ( N, s ) = 12 πi I dzz N +1 s − W z = 12 πi I dzz N +1 A ( z, s )det [ s − W z ] , (22)where the matrix A ( z, s ) is the cofactor matrix of s − W z . Derivation of the FPTD, Eq. (10), in the main text.—
Overall structure and fluctuation relation symmetry
In the expression (22), crucial roles are played by thesingular points in the denominator. We first note thatthe determinant has the functional formdet [ s − W z ] = − Q Lk =1 W k +1 ,k z [ z − z + ( s )] [ z − z − ( s )] , (23)where z + ( s ) and z − ( s ) are the singular points. By look-ing at the tridigonal matrix W z , one finds the relationbetween these singular points z + ( s ) z − ( s ) = L Y k =1 W k,k +1 W k +1 ,k = e − βfLa . (24)Hence we can take these points as | z + ( s ) | ≥ | z − ( s ) | ≤
1. In the limit of s → z + ( s ) →
1. Therelation (24) corresponds to the fluctuation relation interms of winding number which stands for the entropy βf La generated for every increase of the winding num-ber.We consider the matrix element of the transition ma-trix T ( N, s ). We first note that the cofactor matrix hasdependence on either z or z or z − . Hence a matrixelement is given by the following type of integration C πi I dz z − n [ z − z + ( s )] [ z − z − ( s )] , (25)where n = N − , N, N + 1 and C is a constant depen-dent on the matrix element. We here note I dz πi z − n ( s )[ z − z + ( s )] [ z − z − ( s )] = z − n + ( s ) z − ( s ) − z + ( s ) , n ≥ z − n − ( s ) z − ( s ) − z + ( s ) , n ≤ . (26)This implies that depending the sign of the winding num-ber N ( | N | > z + or z − . However, using the symmetry (24) we can inunified way express those only with the singular point z + : T α,ν ( N, s ) = C α,ν ( s ) z −| n | + ( s ) z − ( s ) − z + ( s ) , (27)where the prefactor C α,ν ( s ) accounts for the amplitudeof transition. For instance, between positive an negativewinding number there is exponentially large difference inthe amplitude of the prefactor. Saddle point analysis
Using the relation [Eq (5) in main text] between theFPTD and the transition probability, we then get F ν,ν ( N, t ) = 12 πi Z dse st T ν,ν ( N, s ) T ν,ν (0 , s ) · · · N = 0 . (28)Using Eq.(27), we discuss the asymptotic behavior of theFPTD. At large t , the function F ν,ν ( N, t ) is given by F ν,ν ( N, t ) ∝ πi Z dse − tg ( s,b ) , (29) g ( s, b ) = − s + b log z + ( s ) , (30)where b = | n | /t . We make the saddle point analysis,where the saddle point s ∗ satisfies − b d log z + ( s ) ds (cid:12)(cid:12)(cid:12) s = s ∗ = 0 , (31)where z ′ + ( s ) = dz + ( s ) /ds . As shown in the next subsec-tion g ′′ ( s ∗ ) ∝ /b . From this, one gets F ν,ν ( N, t ) → e − th ( b =0) − (3 /
2) log t , (32) h ( b ) = g ( s ∗ ( b ) , b ) = − s ∗ ( b ) + b log z + ( s ∗ ( b )) , (33)where we introduced the function h ( b ) to emphasize thatthe saddle point s ∗ is a function of b . We noted in Eq.(32)that asymptotic behavior implies b → h ( b ) is in fact precisely the large deviationfunction (LDF). To see this, we first note that the cumu-lant generating function (CGF) is given by the largest eigenvalue of the matrix W z . Thus if λ k ( z ) is the k th eigenvalue of W z and λ k =0 is, say, the largest eigenvaluethen we have µ ( ξ ) = λ ( z ) , with ξ = log z . (34)The eigenvalues λ k are given by the roots of the de-terminantal eqation det [ λ − W z ] = 0. This is identi-cal to the equation for finding the roots z + , z − , namelydet [ s − W z ] = 0 if we replacing λ by s . We also notethat λ ( z ) → z →
1. Using this continuity in termsof the variable s around s = 0, we see that the singularvalue z + is related to the CGF via the relation s − µ ( ξ ) = 0 , where ξ = log z + ( s ∗ ( b )) . (35)Thus the function h ( b ) is completely specified by the fol-lowing equations h ( b ) = bξ ( s ∗ ) − s ∗ , (36) s ∗ = µ ( ξ ) , (37) b dξ ( s ) ds (cid:12)(cid:12) s ∗ − . (38)From the last two equations it is easy to see that b = dµ/dξ and hence it is clear that h ( b ) is the LDF corre-sponding to the CGF µ ( ξ ).We now express the value Γ = h ( b = 0) in terms ofphysical quantities. To this end, we expand the func-tion in a Taylor series around its maximum, b m , whichsatisfies dh ( b ) /db (cid:12)(cid:12)(cid:12) b = b m = log z + ( s ∗ ( b m )) = 0 . (39)This implies z + ( s ( b m )) = 1, hence s = 0 and therefore h ( b m ) = 0. Also ξ = 0 and b = dµ/dξ implies that b m = I , the first cumulant of the winding number. Thuswe get Γ = h ( b = 0) = ∞ X k =2 ( − I ) k h ( k ) ( I ) /k ! . (40)The final task is to express the derivatives h ( k ) ( I ) interms of cumulants of the winding number. To derivethe expression of the second derivative h (2) ( I ), we startwith the expression h (2) ( b m ) = b − ds ∗ /db | b = b m . Usingthe relation b = dµ/dξ one gets ds ∗ /db | b = b m = I /I .Hence h (2) ( I ) = 1 I . (41)Higher order terms are systematically derived in a similarmanner and we get h (2) ( b m ) = 1 /I (42) h (3) ( b m ) = − I /I (43) h (4) ( b m ) = (cid:2) I − I I (cid:3) /I (44) h (5) ( b m ) = − (cid:2) I − I I I + I I (cid:3) /I (45) h (6) ( b m ) = (cid:2) I − I I I + 10 I I + 15 I I I − I I (cid:3) /I (46)...In conclusion we get the asymptotic behavior of theFPTD F asym ( t ) F asym ( t ) = e − Γ t − (3 /
2) log t , (47)Γ = ∞ X n =0 ( − I ) n +2 ( n + 2)! q n ( ξ ) | ξ =0 , (48)where q n ( ξ ) = h ( d µ ( ξ ) dξ ) − ddξ i n ( d µ ( ξ ) dξ ) − . In the maintext, we wrote the expression with up to h (4) . Logarithmic correction term: g ′′ ( s ∗ ) ∝ /b We consider the equation (23) for determining z + ( s ).We consider the structure of the quadratic equation z + c ( s ) z + d = 0 , (49)where d = Q Lk =1 W k,k +1 /W k +1 ,k = e − βfLa . Then thesolution is z + ( s ) = h − c ( s ) + p c ( s ) − d i / . (50) Hence the function g ( s, b ) is given by g ( s, b ) = s + b log nh − c ( s ) + p c ( s ) − d i / o . (51)The first derivative is then given by ∂g ( s, b ) ∂s = 1 + b dc ( s ) ds (cid:20) − c ( s ) √ c ( s ) − d (cid:21) z + ( s ) . (52)Now we consider the case of b ≪
1. For the above tobe zero, the term p c ( s ) − d must be extremely small.Hence, we make the rough estimate p c ( s ∗ ) − d ∝ b . (53)The second derivative is then estimated to be ∂ g ( s, b ) ∂s (cid:12)(cid:12)(cid:12) s = s ∗ ∝ b (cid:2) c ( s ∗ ) − d (cid:3) − / ∝ /b . (54) Derivation of Eq. (14) in the main text.—
We firstnote that the steady state distribution and current can beexactly solved for the driven particle in the ring geometry. p SSα = h W α,α − W α − ,α − + W α,α − W α − ,α − W α − ,α − W α − ,α − + · · · + W α,α − · · · W α − L +2 ,α − L +1 W α − ,α − · · · W α − L,α − L +1 i Q Lk =1 W k,k +1 W α − ,α . Z (55) J = h L Y α =1 W α +1 ,α − L Y α =1 W α,α +1 i / Z , (56)where we used the notation W i,j = W i + L,j + L and Z is the normalization factor. There are two approachestowards getting the expression of mean residence time in terms of steady state and currents.In the Laplace representation for the time-domain, P ( N, s ), the Eq.(17) is reduced to s P ( N, s ) = W − P ( N − , s ) + W + P ( N + 1 , s ) + W P ( N, s ) + δ N, p . (57)For N = 0, let us of the form P ( N, s ) = z − N V , (58) where V is a constant vector. Plugging this into (57) for N = 0 gives the following equation for determining z and V [ s − W z ] V = 0 . (59)A careful look at these equations reveals that there aretwo sets of solutions to these equations. To see this, wewrite the above equation in the following form: (cid:18) s + [ W L, + W , ] − Z + − Z − U (cid:19) (cid:18) V ′ (cid:19) = 0 . (60)where Z + = ( W , , , · · · , zW ,L ) , (61) Z − = ( W , , , · · · , z − W L, ) T , (62) U = s I L − − W L , (63) V ′ = ( V , V , · · · , V L ) T . (64)Here the matrix W L denotes ( L − × ( L −
1) sub-matrixof W excluding the first row and column, while I L − isunit matrix of dimension ( L − V to one. Then we get the following equationsfor V ′ and zs + W L, + W , − W , V − zW ,L V L = 0 , (65) V ′ = U − Z − . (66) The second equation leads to the relation V α +1 = (cid:2) U − α W , + U − αL − z − W L, (cid:3) , for α = 1 , · · · , L −
1. Since U does not depend on z we see that V α are linearfunctions of z − . Hence putting back V , V L into thefirst equation above, we get a quadratic equation for z . For the two solutions we get two corresponding ex-plicit forms for the vectors V . We denote the two solu-tions by { z + ( s ) , V + ( s ) } and { z − ( s ) , V − ( s ) } . From theequation for z , we see the fluctuation relation symmetry z + ( s ) z − ( s ) = Q Lk =1 W k,k +1 /W k +1 ,k = e − βfLa ( < ν with N = 0.A possible solution of Eq.(57) is P ( N, s ) = A + z − N + V + for N > A − z − N − V − for N < V for N = 0 . (67)The unknown constants A + , A − and V can fixed by re-quiring that our above solution satisfies Eq.(57) at thesites corresponding to ν and its two nearest neighbors.Clearly then the vector V must have the following struc-ture V = (cid:0) A − V − , A − V − , · · · , A − V − ν − , A , A + V + ν +1 , · · · , A + V + N (cid:1) . (68)There are three constants ( A − , A + , A ) to be determinedand these will follow by writing the three special equa-tions at the site ν and its neighbors. Let us assume, for the moment, that none of these three sites is a boundarysite on the cell (i.e., ν − > , ν + 1 < L ). Then we getthe following equations by looking at the block of N = 0 (cid:8) [ s + W ν − ,ν − + W ν,ν − ] V − ν − − W ν − ,ν − V − ν − (cid:9) A − = W ν − ,ν A , (69)[ s + W ν − ,ν + W ν +1 ,ν ] A − W ν,ν − V − ν − A − − W ν,ν +1 V + ν +1 A + = 1 , (70) (cid:8) [ s + W ν,ν +1 + W ν +2 ,ν +1 ] V + ν +1 − W ν +1 ,ν +2 V + ν +2 (cid:9) A + = W ν +1 ,ν A , (71)Using the equation satisfied by V ± which is given fromthe block of N = 0, we find that the first and thirdequations yield A − = A /V − ν , A + = A /V + ν . (72) Plugging these into the middle equation, one gets A : A = h s + W ν − ,ν + W ν +1 ,ν − W ν,ν − V − ν − V − ν − W ν,ν +1 V + ν +1 V + ν i − = W − ν,ν − h V − ν +1 V − ν + V + ν +1 V + ν i − . (73)0Special case s = 0. For this case the equation W z V = 0has two solutions. One cleary is for z + = 1 and this isthe steady state solution so we can choose V + = p SS . (74)The other solution for z − = Q Lk =1 W k,k +1 /W k +1 ,k isgiven by V − = (cid:18) , W , W , , W , W , W , W , , · · · , W , · · · W L,L − W , · · · W L − ,L (cid:19) T , (75)as can be easily verified. From (73) we then get A = W − ν,ν +1 h W ν +1 ,ν W ν,ν +1 − V + ν +1 V + ν i − (76)Using the fact that J = (cid:2) W ν +1 ,ν V + ν − W ν,ν +1 V + ν +1 (cid:3) wethen get A = V + ν /J . (77)From Eq.(72) we get A − = V + ν /JV − ν , A + = 1 /J , (78)From this we finally obtain, for the case s = 0, the fol-lowing transition matrices for any states α, ν where ν isthe one “down the hill” (i.e. current is in the direction α → ν ). T α,α ( N = 0 , s = 0) = p SSα
J , (79) T α,ν ( N = 0 , s = 0) = p SSα J k = α Y ν (cid:18) W k − ,k W k,k − (cid:19) , (80) T ν,α ( N = 0 , s = 0) = p SSν
J . (81)We finally explain how to obtain Eq. (14), in the maintext, using Eq. (79). We note that the entropy producedin the thermal reservoir for the process α → ν is given by S = β [ U α − U ν + f a ( ν − α + LN )]. This implies thatthe process ( α, N = 0) → ( α, N = 0) is equivalent to the process ( α, S = 0) → ( α, S = 0). Hence Eq.(79) isequivalent to T α,α ( S = 0 , s = 0) = p SSα
J . (82)Now consider the process ( α, S = 0) → ( ν, S ) whosedirection is the same as the average current. In this casewe note F ν,α ( S , s = 0) = 1 which means that the processwill occur with probaility one. This gives T ν,α ( S , s = 0) = F ν,α ( S , s = 0) T ν,ν ( S = 0 , s = 0)= p SSν
J . (83)In the backward process ( ν, S = 0) → ( α, −S ) which isopposite to the direction of average current, the detailedfluctuation relation immediately gives T α,ν ( −S , s = 0) = e −S p SSν
J . (84)These give Eq. (14) in the main text.
The FPTD of charge transfer via a quantum-dot.—
The dynamics of the charge transfer in classical transportis described by ∂ P ( t ) /∂t = W P ( t ) , (85) W = X r = L,R W r , (86)where both W and W r are 2 × P ( t ) and P ( t ) are respectively stand for the probability for un-occupied and occupied state inside the quantum-dot.Standard setup takes the transition matrix element as W r , = ¯ γ [1 − f r ( ǫ )] and W r , = ¯ γf r ( ǫ ) where f r ( ǫ ) isthe Fermi-distribution for the r th reservoir and ¯ γ is ahopping rate. Hence it satisfies the detailed balance W r , /W r , = e β ( ǫ − µ r ) . Without loss of generality, wecan impose µ L > µ R .Let P i ( Q, t ) be the joint probability for the i (= 1 , Q measured at the right reservoirs till time t . Thedynamics is given by ∂ t P ( Q, t ) = W , P ( Q, t ) + W L , P ( Q, t ) + W R , P ( Q − , t ) , (87) ∂ t P ( Q, t ) = W , P ( Q, t ) + W L , P ( Q, t ) + W R , P ( Q + 1 , t ) . (88)We define the generating function P ( z, s ) = ∞ X Q = −∞ Z ∞ dt z Q e − st P ( Q, t ) , (89) From the dynamics for the joint probabilities, this is for-mally given by the expression P ( z, s ) = 1 s − W z p , (90)1where W z is given by W z = (cid:18) W W L , + W R , zW L , + W R , z − W (cid:19) . (91)The transition matrix is hence given by T ( Q, s ) = 12 πi I dzz Q +1 s − W z , (92)= 12 πi I dzz Q +1 A ( z, s )det [ s − W z ] . (93)Now one can see the same structure to the case of ringgeometry. From this, we find the singularities z ± ( s ) bysolving the equation det [ s − W z ] = 0, which are con-nected by the fluctuation relation symmetry z − ( s ) z + ( s ) = W L , W R , W R , W L , = e − β ( µ L − µ R ) . (94)In the same way as in the driven particle in the ringgeometry, any matrix elements of the FPTD have thefollowing dependence F α,ν ( Q, t ) = 12 πi Z c − i ∞ c + i ∞ ds e st T α,ν ( Q, s ) /T α,α (0 , s ) ∝ πi Z ds e − tg ( s,b ) , (95) g ( s, b ) = − s + b log z + ( s ) , (96)where b = | Q | /t and C is time-independent matrix. Theargument from this point follows the calculations fromEqs.(29) and (30) in the driven particle in the ring ge-ometry. Hence, we can reach the same expression as inEqs.(47) and (48). Heuristic derivation of the FPTD asymptotic.—
Con-sider a general process with discrete configuration space X and let us look at the joint distribution of X andsome quantity Q (like heat or charge). This distribution, P ( X , Q, t ), will satisfy the equation of motion ∂P ( X , Q, t ) ∂t = L Q P ( X , Q, t ) , (97)while the generating function Z ( X , ξ ) = R dQe ξQ P ( X , Q ) will satisfy the equation ∂Z ( X , ξ, t ) ∂t = L ξ Z ( X , ξ, t ) . (98)The large time solution of this equation with the initialcondition X = Y , Q = 0 at t = 0 is given by T X , Y ( ξ, t ) ∼ e µ ( ξ ) t φ ( X , ξ ) χ ( Y , ξ ) , (99)where µ, φ, χ are respectively the largest eigenvalue of L ξ ,and the corresponding right and left eigenvectors. Takinga time-Laplace transformation we get T X , Y ( ξ, s ) ∼ φ ( X , ξ ) χ ( Y , ξ )[ s − µ ( ξ )] , (100) Taking an inverse Laplace transformation in the variable Q , we get T X , Y ( Q, s ) ∼ e − ξ + ( s ) | Q | φ ( X , ξ + ( s )) χ ( Y , ξ + ( s )) , (101)where we assume, based on empirical observations, thatonly the singularity, ξ + , which satisfies the following re-lation, contributes: µ ( ξ + ) − s = 0 . (102)We now consider first passage only of the variable Q with-out caring for the configuration coordinates X . Let us de-fine F X , Y ( Q, t ) as the probability that the system startsfrom Y at time t = 0 with Q = 0, first reaches Q inthe time interval ( t, t + dt ) and is at X during that timeinterval. Then we have, for Q = 0, F X , Y ( Q, s ) = X X ′ T − X , X ′ ( Q = 0 , s ) T X ′ , Y ( Q, s ) . (103)Now using Eq. (101) and assuming that the wave-functions do not contribute to the asymptotic behaviourwe get F X , Y ( Q, s ) ∼ e − ξ + ( s ) | Q | . (104)Finally, transforming back to the time domain, and doinga saddle-point analysis, we get F X , Y ( Q, t ) ∼ e − g ( b ) t , where g ( b ) = ξ + ( s ∗ ) b − s ∗ , b = Q/t , (105)and s ∗ is determined by s ∗ = µ ( ξ ) , dξ + ds b − . (106)Thus we recover the required Eqs. (36,37,38) .We here note that in all models (a)-(c) the equation(102) yields the quadratic equation of the following type z ( s ) + c ( s ) z ( s ) + d = 0 , (107)which gives two solutions z ± ( s ) and z ( s ) is connectedto ξ ( s ) by the relation ξ ( s ) = log z ( s ), and the constantterm d comes from the fluctuation relation symmetry (asin Eq.(49) for the case of driven particle in the ring ge-ometry). Then from the same argument as in Sec.II-C,the same logarithmic correction is obtained. Hence, weget asymptotic behavior (47) and (48). Integral fluctuation relation in terms of first passageand numerical verification.—
We consider the entropyproduced in the thermal reservoir S for the driven par-ticle in the ring geometry. Let F ¯ α,ν ( S , t ) be the FPTDonly for S [not for ( α, S )], while reaching ¯ α from ν . De-pending on ( ν, S ), there are two possible situations; in thefirst case ¯ α is unique, and in the second case there are2two values ¯ α , as depicted in the right figure in Fig. (1 a )in the main text. In both these cases we note the relation T ¯ α,ν ( S , s ) = X ¯ α ′ T ¯ α, ¯ α ′ (0 , s ) F ¯ α ′ ,ν ( S , s ) . (108)Then we consider P ¯ α e −S tot¯ α,ν F ¯ α,ν ( S , s = 0) for negative S . We first consider the case where ¯ α is unique. Notethat in this case ( ν, S = 0) → (¯ α, S ) is opposite to thedirection of the average current. Then e −S tot¯ α,ν F ¯ α,ν ( S , s = 0) = e −S p SS ¯ α p SSν T ¯ α,ν ( S , s = 0) T ¯ α, ¯ α (0 , s = 0)= e −S p SS ¯ α p SSν e S p SSν p SS ¯ α = 1 , (109)where we used Eqs. (82)-(84) to get the final result.We next consider the case where ¯ α has two choices.We call ¯ α ± these two points, located respectively on thepositive and negative side of ν [See the right figure inFig. (1a) in the main text]. Using Eq. (108), we have thefollowing relation (cid:18) T ¯ α + ,ν ( S , T ¯ α − ,ν ( S , (cid:19) = M (cid:18) F ¯ α + ,ν ( S , F ¯ α − ,ν ( S , (cid:19) , (110)where the matrix M is given by M = (cid:18) T ¯ α + , ¯ α + (0 , T ¯ α + , ¯ α − (0 , T ¯ α − , ¯ α + (0 , T ¯ α − , ¯ α − (0 , (cid:19) (111)We note the following expressions in terms of the steadystate distribution and steady state current (cid:18) T ¯ α + ,ν ( S , T ¯ α − ,ν ( S , (cid:19) = 1 J (cid:18) p SS ¯ α + p SSν e S (cid:19) , (112) M = 1 J (cid:18) p SS ¯ α + p SS ¯ α + p SS ¯ α + p SS ¯ α − (cid:19) . (113)Using these expressions, straight forward calculationleads to X ¯ α =¯ α + , ¯ α − e −S tot¯ α,ν F ¯ α,ν ( S ,
0) = 1 . (114) Hence, in any cases, we have the identity X ¯ α e −S tot¯ α,ν F ¯ α,ν ( S ,
0) = 1 . (115)This immediately leads to hh e −S tot ii S = 1 , ( S < . (116)We finally present the numerical demonstration of(116) as well as the relation Eq. (82). The Langevinequation was numerically solved with the same parame-ters set as in Fig. (2a) in the main text. . .
01 0 0 .
25 0 . .