Universal bounds on current fluctuations
UUniversal bounds on current fluctuations
Patrick Pietzonka, Andre C. Barato, and Udo Seifert II. Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, 70550 Stuttgart, Germany Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Straße 38, 01187 Dresden, Germany
For current fluctuations in non-equilibrium steady states of Markovian processes, we derive fourdifferent universal bounds valid beyond the Gaussian regime. Different variants of these boundsapply to either the entropy change or any individual current, e.g., the rate of substrate consumptionin a chemical reaction or the electron current in an electronic device. The bounds vary with respectto their degree of universality and tightness. A universal parabolic bound on the generating functionof an arbitrary current depends solely on the average entropy production. A second, stronger boundrequires knowledge both of the thermodynamic forces that drive the system and of the topologyof the network of states. These two bounds are conjectures based on extensive numerics. Anexponential bound that depends only on the average entropy production and the average number oftransitions per time is rigorously proved. This bound has no obvious relation to the parabolic boundbut it is typically tighter further away from equilibrium. An asymptotic bound that depends on thespecific transition rates and becomes tight for large fluctuations is also derived. This bound allowsfor the prediction of the asymptotic growth of the generating function. Even though our results arerestricted to networks with a finite number of states, we show that the parabolic bound is also validfor three paradigmatic examples of driven diffusive systems for which the generating function canbe calculated using the additivity principle. Our bounds provide a new general class of constraintsfor nonequilibrium systems.
PACS numbers: 05.70.Ln, 05.40.-a
I. INTRODUCTION
Equilibrium statistical physics is governed by a uni-versal principle stating that in an isolated system eachmicrostate is equally likely. For a system in contact witha heat bath, thus the famous Gibbs-Boltzmann distribu-tion arises that involves only the Hamiltonian of the sys-tem and the temperature of the bath. In non-equilibrium,a similarly universal principle is not known and may noteven exist. One characteristic feature of non-equilibriumsystems is that they necessarily come with dissipation,i.e., entropy production. Non-equilibrium steady states,generated by time-independent driving have a constantaverage entropy production. Observed for a finite time,the entropy change exhibits fluctuations that are univer-sally constrained by the fluctuation theorem [1–6], whichis arguably the most universal principle discovered fornon-equilibrium systems so far.The fluctuation theorem relates the probability to ob-serve a negative entropy change to the one for observ-ing the corresponding positive value. In this sense, itconstrains “half” of the distribution. Experiments haveillustrated and tested this symmetry, inter alia , for col-loidal particles [7–9], energy exchange between two con-ductors [10], small electronic systems at low temperature[11], molecular motors [12], and shaken granular mat-ter [13–15]. In a refined version, the fluctuation theo-rem holds not only for entropy change but also for thejoint probability of all currents contributing to the en-tropy change [5, 16], which involves the correspondingaffinities like non-conservative forces for colloids, chem-ical potential differences for bio-molecular reactions or voltage drops for electronic circuits. Generally, these in-dividual currents in a multi-cyclic network, however, arenot restricted by the fluctuation theorem or any otheruniversal result.In this paper, we introduce a complementary classof constraints, not only on the distribution of entropychange, but of any individual current in a network. Theseconstraints universally bound the fluctuations over thefull range of positive and negative values, in particularthe extreme fluctuations. The crucial parameters char-acterizing these bounds are the average entropy produc-tion, the affinities, topological features like the number ofstates in a cycle and the activity, i.e., the average num-ber of transitions per unit time. If one knows such pa-rameters, current fluctuations can be bounded indepen-dently of the specific transition rates. Correspondingly,a measurement of such current fluctuations will make itpossible to infer constraints on these parameters whichin an experiment may not be known or not be directlyaccessible. This study substantially extends and gener-alizes work in which we have recently explored universalrelations between dissipation and dispersion of currentsleading to a general thermodynamic uncertainty relation[17] and allowing the inference of topological propertiesof enzymatic networks [18, 19].We employ the formalism of large deviations [20, 21] inwhich for large times the exponential decay of the tailsof the distribution function is characterized by a ratefunction. This rate function can be obtained from theLegendre transformation of the scaled cumulant generat-ing function. For the latter, we derive a series of lowerbounds that can be divided into four classes: a parabolic a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y bound, a hyperbolic cosine bound, an exponential bound,and an asymptotic bound relevant for large values of z ,where z is the real variable in the scaled cumulant gen-erating function. The last two bounds can be provedexactly while the first two are conjectures based on ex-tensive numerics. These universal bounds are valid forany nonequilibrium system described by a Markov pro-cess with a finite number of states.The hydrodynamic fluctuation theory for driven dif-fusive systems in contact with two reservoirs by Bertini et al. [22–24] has been another major development innonequilibrium statistical physics. This theory leads toa (typically hard) variational problem that, if solved,leads to the exact rate function of the current of parti-cles or heat between reservoirs. The additivity principlederived in [25] is a more direct method that allows forthe calculation of the scaled cumulant generating func-tion related to the current in driven diffusive systems.For example, this method has been used to calculatethis function for the symmetric simple exclusion process(SSEP) [25, 26], the Kipnis-Marchioro-Pressuti (KMP)model [27, 28], and the weakly asymmetric simple exclu-sion process (WASEP) [29]. These results are valid inthe limit of large system size, for which the number ofstates diverges. Even though our bounds are restrictedto the case of a finite number of states, we show that thescaled cumulant generating functions obtained from theadditivity principle for these three models lies inside ourparabolic bound.The paper is divided as follows. In Sec. II we definethe entropy, the currents and their generating functions.Our main results are summarized in Sec. III. Secs. IV,V, VI, and VII contain the parabolic, hyperbolic cosine,exponential, and asymptotic bounds, respectively. InSec. VIII, the parabolic bound is compared with exactresults for the SSEP, the KMP model, and the WASEP.We conclude in Sec. IX. The appendices contain variousproofs and details on the numerics. II. LARGE DEVIATIONS IN MARKOVIANNETWORKS
We consider a Markovian network consisting of N discrete states { i } and allow for transitions with rates k ij ≥ i to j . All transitions are taken tobe reversible, i.e., k ij > k ji >
0. The time de-pendent probability distribution p i ( t ) of state i at time t evolves according to the master equation ∂ t p i ( t ) = (cid:88) j L ij p j ( t ) ≡ (cid:88) j [ k ji − r i δ ij ] p j ( t ) , (1)where the exit rate from state i is defined as r i ≡ (cid:88) (cid:96) k i(cid:96) . (2) (a) FIG. 1. (a) Unicyclic network with affinity A and five states.(b) Multicyclic network with two fundamental cycles, onewith three states and affinity A and the other with four statesand affinity A . The red dashed lines indicate a cycle withaffinity A + A and five states. (c) Multicyclic network withthree fundamental cycles with three states each. The affini-ties of these cycles are A , A , and A . The red dashed linesindicate a cycle with affinity A + A and four states. For large times t , p i ( t ) tends to the stationary distribu-tion p s i , which satisfies (cid:80) j L ij p s i = 0.Following Schnakenberg, we identify a complete set offundamental cycles { β } within the network [30]. Eachcycle is associated with an affinity A β , a fluctuating cur-rent X β ( t ) that counts cycle completions after time t (theso-called integrated current ) and an average current J β ≡ (cid:104) X β ( t ) (cid:105) /t, (3)where the brackets indicate an average over stochastictrajectories. The average is independent of t for initialconditions drawn from the steady state distribution.Upon a transition i → j , X β increases by the gener-alized distance d βij = − d βji . These increments are con-strained to add up to one for every closed loop that com-pletes the cycle once in forward direction. For example, X β could be the scaled displacement of a molecular mo-tor. The affinity A β would then be given by the exter-nal force times the length of a full motor step, while d βij denotes the relative length of a sub-step related to theconformational change i → j . The ratio of forward andbackward transition rates fulfill the local detailed balancerelation ln( k ij /k ji ) = (cid:88) β d βij A β + E i − E j , (4)where E i denotes the equilibrium free energy associatedwith state i , and, in general, the sum over β is a sumover all fundamental cycles in the network of states [30](see also [16, 31] for a precise definition of a fundamen-tal cycle). For notational convenience we have set Boltz-mann’s constant k B and the temperature T to unity, thusenergies and affinities are given in units of the thermalenergy k B T . The (fluctuating) entropy change in thesurrounding medium s m ( t ) is given by the increments d s ij = ln( k ij /k ji ). The average entropy production reads σ ≡ (cid:104) s m ( t ) (cid:105) /t = (cid:88) β A β J β . (5)Adopting a vector notation X for the set of all cyclecurrents X β the scaled cumulant generating function isdefined as λ ( z ) ≡ lim t →∞ t ln (cid:104) exp[ z · X ( t )] (cid:105) , (6)where z is a real vector. As an abbreviation we will referto λ ( z ) simply as the “generating function”. It can beshown that λ ( z ) is the largest eigenvalue of the modifiedMarkov generator L ij ( z ), which is defined as [5, 32] L ij ( z ) ≡ L ij exp( z · d ji ) , (7)where d ji is a vector with components d βji . The variable X β /t , or more conveniently the scaled variable ξ β ≡ X β / ( tJ β ) , (8)satisfies a large deviation principle [21, 33] of the formProb( X , t ) ∼ exp[ − th ( ξ )] (9)with a rate function h ( ξ ) that is given by the Legendre-Fenchel transform h ( ξ ) = max z (cid:88) β z β J β ξ β − λ ( z ) . (10)The fluctuation theorem is a symmetry, known asGallavotti-Cohen symmetry, on the generating functionof the form [5, 16] λ ( z ) = λ ( − A − z ) . (11)In terms of the rate function this symmetry reads − h ( ξ ) + h ( − ξ ) = (cid:88) β ξ β J β A β . (12)The generating function related to a single fluctuatingcurrent X α is λ α ( z ) ≡ λ ( z e α ) = lim t →∞ t ln (cid:104) exp[ zX α ( t )] (cid:105) , (13)where e α is the unit vector associated with the currentin cycle α . Generally, this function does not exhibit asymmetry of the form (11) as extensively discussed in [31] (see also [34, 35]). In contrast, the evaluation of λ ( z ) along the vector A yields λ s ( z ) ≡ λ ( z A ) = lim t →∞ t ln (cid:104) exp[ zs m ( t )] (cid:105) (14)as the generating function of the entropy change. It issymmetric with respect to z = − /
2, which expresses thefluctuation theorem that holds for this observable. Therate functions associated with the probability distribu-tions of these variables read h α ( ξ α ) = max z [ zJ α ξ α − λ α ( z )] (15)and, introducing the scaled entropy change s ≡ s m ( t ) / ( σt ) in analogy to Eq. (8), h s ( s ) = max z [ zσs − λ s ( z )] , (16)respectively.An important distinction in this paper is the one be-tween unicyclic and multicyclic networks of states, illus-trated in Fig. 1. For unicyclic networks, where there isonly a single affinity A ≡ A α and a single fluctuatingcurrent X ≡ X α , we no longer have to distinguish be-tween the different types of generating functions and cansimply write λ ( z ) ≡ λ α ( z ) = λ s ( z/ A ) . (17)In the following, we will be interested in functions b ( z )that bound the generating function λ ( z ) from below, i.e., b ( z ) ≤ λ ( z ) (18)for all z . As special cases, the relation (18) can beused to extract bounds for individual fluctuating cur-rents, λ α ( z ) ≥ b α ( z ) ≡ b ( z e α ), and the entropy change, λ s ( z ) ≥ b s ( z ) ≡ b ( z A ). Such bounds immediately implyupper bounds on the rate functions h α ( ξ α ) ≤ max z [ zJ α ξ α − b α ( z )] . (19)and h s ( s ) ≤ max z [ zσs − b s ( z )] . (20)For any generating function the coefficients of the Tay-lor expansion around z = 0 correspond to the cumulants.The Fano factor that quantifies the dispersion of the dis-tribution is defined as F ≡ lim t →∞ (cid:10) X ( t ) (cid:11) − (cid:104) X ( t ) (cid:105) (cid:104) X ( t ) (cid:105) = λ (cid:48)(cid:48) (0) λ (cid:48) (0) , (21)where X is a random variable. We denote the Fano factorassociated with an individual current by F α and the oneassociated with the entropy change in the medium by F s .Since global lower bounds b ( z ) with b (0) = 0 must share -20246-20 -15 -10 -5 0 5 10 λ α ( z ) z parabolicexponential FIG. 2. Illustration of our two main results for the generatingfunction of an individual cycle current (thick black curve).The parabolic bound that depends on the entropy productionis shown as a green dotted line and the exponential bound thatdepends on the entropy production and the activity is shownas a blue dashed line. The generating function refers to thecycle α = 1 of the “house-shaped” network shown in Fig. 1b.The affinities are A = 8 and A = 6, all transition rateswere set to 1 except for k = exp( A / k = exp( −A / k = exp( A / k = exp( −A / J α (cid:39) . σ (cid:39) .
6, and the activity R (cid:39) . a tangent with λ ( z ) at z = 0 while having a strongercurvature, every such bound implies with F ≥ b (cid:48)(cid:48) (0) b (cid:48) (0) (22)a bound on the Fano factor.Since λ (0) = 0 holds trivially for all networks, we usu-ally require that our bounds are saturated for z = 0. Thisrequirement will only be lifted for a bound that capturesthe asymptotic behavior for large | z | in Sec. VII. Hence,if λ ( z ) is analytic, b ( z ) must have the same gradient as λ ( z ). III. SUMMARY OF MAIN RESULTS
The two main bounds on the generating function ob-tained in this paper are illustrated in Fig. 2. First, thegenerating function λ ( z ) for any network is bounded bya parabola according to λ ( z ) ≥ z · J (1 + z · J /σ ) . (23)This parabolic bound depends only on the average en-tropy production. Second, λ ( z ) is also bounded frombelow by an exponential function of the form λ ( z ) ≥ R (cid:104) e ( | σ/ z · J |− σ/ /R − (cid:105) . (24)This second bound depends on the average entropy pro-duction and on the activity R ≡ (cid:80) i p s i r i , which is the average number of transitions per time in the whole net-work.Choosing a specific direction for the vector z , bothbounds are valid both for any individual fluctuating cur-rent and for the entropy change. It is quite remarkablethat the fluctuations of any current in an arbitrary mul-ticyclic network can be bounded by a function involv-ing only the average entropy production in the case ofthe parabolic bound and the average entropy productionand activity in the case of the exponential bound. Eventhough there is no obvious relation between the parabolicbound and the exponential bound, typically, the expo-nential bound becomes tighter than the parabolic boundboth for far from equilibrium conditions and for large | z | .We derive two further relevant lower bounds on λ ( z )in this paper. (1) A hyperbolic cosine bound, which is anextension of the parabolic bound that is tighter and re-quires further knowledge of the affinities and the topologyof the network. (2) An asymptotic bound that becomestight for large values of | z | and requires knowledge of alltransition rates.These bounds are complementary to the fluctuationtheorem. Whereas they establish the minimal value that λ ( z ) can take, the fluctuation theorem constrains λ ( z )to have the symmetry (11). IV. PARABOLIC BOUNDA. Linear response regime
In the limit of small affinities A β , the average current J α depends linearly on the affinities, J α = (cid:88) β L αβ A β (25)with the symmetric and positive definite Onsager matrix L αβ ≡ ∂J α /∂ A β | A =0 . In the region z < ∼ O ( A ), thegenerating function λ ( z ) can be expanded as a quadraticform around its center of symmetry, which is, due toEq. (11), located at z = − A /
2. The requirement λ (0) =0 and ∇ λ (0) = J fixes this expansion to λ ( z ) = (cid:88) β,γ ( z β + A β / L βγ ( z γ + A γ / − σ/ , (26)where the entropy production σ is given in Eq. (5). Eval-uating this function for z = z e α yields as generatingfunction related to the individual current λ α ( z ) = zJ α + z L αα . (27)The positive definiteness of the matrix G βγ ≡ ( L αα L βγ − L αβ L αγ ) [36] (with α fixed) yields (cid:88) β,γ G βγ A β A γ = L αα σ − J α ≥ . (28)Hence λ α ( z ) is bounded from below by λ α ( z ) ≥ zJ α (1 + zJ α /σ ) . (29)Using the Legendre transform (19), this bound can betransformed into a bound for the rate function h α ( ξ α ) = L αα J α ( ξ α − ≤ σ ξ α − . (30)Since the direction e α can be chosen arbitrarily, thebound (29) can be stated in a multidimensional formula-tion as λ ( z ) ≥ z · J (1 + z · J /σ ) . (31)Equality holds along the line z ∝ A , which correspondsto the generating function λ s ( z ) = λ ( A z ) associated withentropy change. Within linear response, the rate functionfor the scaled entropy change s is thus given by h s ( s ) = σ s − . (32)Eq. (30) shows that the knowledge of the average en-tropy production is sufficient to bound the whole rangeof fluctuations of any individual current in the linearresponse regime. Surprisingly, as we show next, thisparabolic bound is also valid beyond the linear responseregime.The parabolic bound has also an important conse-quence for fluctuations in systems at equilibrium. Tostudy this case it is more convenient to scale the fluc-tuating currents as x β ≡ X β /t = J β ξ β . For the corre-sponding rate function ˜ h α ( x α ) = h α ( x α J α ), the bound(30) then reads˜ h α ( x α ) ≤ (cid:80) γδ L γδ A γ A δ (cid:16)(cid:80) β L αβ A β (cid:17) x α + O ( A ) , (33)for small A β with fixed x α . Here, we have represented theaverage currents using Eq. (25). This bound is supportedby our numerics presented in appendix B as, where wehave checked (29) also for z > ∼ O ( A ). For multicyclicnetworks at equilibrium, the prefactor in (33) dependson the direction in which the limit A → A ∝ e α yields˜ h α ( x α ) ≤ x α / (4 D α ) . (34)Thus, the equilibrium fluctuations of any current X α canbe bounded by the parabola that is defined as the con-tinuation of the quadratic expansion of the rate functionaround x α = 0. In other words, the Gaussian approxi-mation for typical fluctuations always underestimates theprobability of extreme fluctuations in equilibrium sys-tems. Since this bound is exact for small x α , performingthe limit A → e α cannotyield a stronger bound. λ A R W ( z , A , ) z / AAր h A R W ( ξ , A , ) / A ξ Aր FIG. 3. The generating function λ ( z ) and the rate function h ( ξ ) of the asymmetric random walk for selected affinities A (2 , , ,
50) and N = 1. Black arrows indicate the direction ofincreasing A . The parabolic bound for the generating func-tion (40) and for the rate function (41) are shown as blackdashed curves. B. Beyond linear response: Unicyclic case
The parabolic shape of the generating function for z < ∼ O ( A ) and of the rate function for ξ < ∼ O (1) canbe regarded as a signature of linear response. It arisesonly for nearly vanishing affinities or for freely diffusingparticles, where the linearity between affinity and cur-rent persists even for high affinities. Beyond this regime,one universally observes two characteristic changes in therate function [37–39]. First, the tails for large values of | ξ α | grow no longer quadratically but with a scaling some-where between linear and quadratic. Second, there is aformation of a “kink” around the value ξ α = 0. For fi-nite numbers of states, the rate function is still analyticin this region, but it exhibits a significantly enhancedcurvature. In the Legendre transformed picture of thegenerating function λ α ( z ), these two effects show up as afaster than quadratic growth for large z and a pronouncedplateau around the minimum of λ α ( z ).This behavior of the generating function is best il-lustrated with an asymmetric random walk (ARW), asshown in Fig. 3. Consider a network consisting of a singlecycle with N vertices and affinity A , as shown in Fig. 1a.The hopping rates in forward and backward directions k + and k − are uniform withln k + k − = A /N. (35)The average current in this model is J = ( k + − k − ) /N and the entropy production is σ = J A . It can be shownthat the generating function is given by [5] λ ( z ) = k + (cid:104) e z/N + e − ( z + A ) /N − − e −A /N (cid:105) (36)= Jλ ARW ( z, A , N ) , where λ ARW ( z, A , N ) ≡ cosh[( z + A / /N ] − cosh[ A / (2 N )](1 /N ) sinh[ A / (2 N )] . (37)Similarly, the rate function corresponding to the gener-ating function (36) is given by [5] h ( ξ ) = J h
ARW ( ξ, A , N ) , (38)where h ARW ( ξ, A , N ) ≡ N sinh[ A / (2 N )] (cid:104) aξ arsinh( aξ ) − aξ A N − (cid:112) aξ ) + (cid:112) a (cid:105) (39)and a ≡ sinh[ A / (2 N )]. As shown in Fig. 3, the generat-ing function (36) is bounded from below by the parabola λ ARW ( z, A , N ) ≥ z J (1 + z/ A ) , (40)and the rate function is bounded from above by theparabola h ARW ( z, A , N ) ≤ A ( ξ − / . (41)In Sec. V, we will show in the context of an evenstronger, affinity-dependent bound, that the bound (40)holds also for arbitrary unicyclic networks with non-uniform transition rates. C. Beyond linear response: multicyclic case
Based on numerical evidence we conjecture that λ ( z ) ≥ z · J (1 + z · J /σ ) (42)holds globally for all vectors z and for all types of Marko-vian networks. In terms of the individual current in acycle α , this conjecture can be formulated as λ α ( z ) ≥ zJ α (1 + zJ α /σ ) (43)whereas for the entropy change λ s ( z ) ≥ zσ (1 + z ) . (44)The bound (42) becomes the same as (31) in the lin-ear response regime. However, (42) is also valid beyondthis regime where the currents J are the actual average -0.3-0.2-0.100.10.20.30.4-1.5 -1 -0.5 0 0.5 λ α ( z ) / σ zJ α /σ N = 4-0.3-0.2-0.100.10.20.30.4-1.5 -1 -0.5 0 0.5 λ α ( z ) / σ zJ α /σ N = 6 FIG. 4. Generating functions λ α ( z ) for an individual currentin fully connected networks with random transition rates. Theblack curves in the upper and lower panels correspond to net-works with N = 4 and N = 6 vertices, respectively. Theparabolic bound (43) is shown as a dashed curve. currents in the steady state, as determined from ∇ λ (0),which are different from the linear response currents (25).The numerical evidence for this bound is illustrated inFig. 4. We generated a large set of networks of stateswith random transition rates, drawn according to theprocedure described in appendix B. As the affinity in-creases, generating functions globally deviate in a pos-itive direction from the parabolic shape. Only at thetrivial points z = 0 and z = − A does the generatingfunction in Eq. (42) acquire with zero the same valuefor all networks. For larger networks ( N = 6) the lefthand side of the plot becomes less populated, since theprobability of the vectors e α and A being nearly paral-lel becomes smaller in higher dimensions. The full nu-merical evidence for this parabolic bound is explained inappendix B.The local evaluation (22) of the parabolic bound (43)for an individual current yields the relation F α σ/J α ≥ X α . This“thermodynamic uncertainty relation”, which imposes aminimal energetic cost that must be paid for small uncer-tainty in the output of an enzymatic reaction, has beenderived in [17]. Hence, the parabolic bound is a general-ization of this relation. From relation (45), measurementsof the dispersion and average of an individual current canprovide a lower bound on the average entropy production -10-5051015 -100 -80 -60 -40 -20 0 λ ( z ) / J z SD = 210.50.20.10.050.01
FIG. 5. Generating function λ ( z ) scaled by the steady statecurrent J in unicyclic networks with N = 10 states and fixedaffinity A = 100. The gray-scale (color-scale) encodes thestandard deviation (SD) used for sampling the transition rates(see appendix B). Dark gray (blue) corresponds to a nearlyuniform distribution of transition rates and light gray (yellow)to a broad distribution of transition rates. The lower andupper bounds (50) and (51), respectively, are shown as (red)dashed lines. σ ≥ J α /F α . This bound makes it possible to estimatethe entropy production by measuring a single individualcurrent [40, 41]. Applied to the entropy production inthe medium, the parabolic bound (44) leads to F s ≥ . (46) V. HYPERBOLIC COSINE BOUNDA. Unicyclic networks
For a unicyclic network, the parabolic bound is satu-rated in the linear response regime. As shown in Fig. 3for the asymmetric random walk, the generating functiondeviates more from the parabolic bound as the affinity A increases. We now discuss an affinity dependent boundthat is stronger than the parabolic bound. This affinitydependent bound is also less universal as it requires theknowledge of A . For example, in a biochemical network afixed affinity means that the chemical potential differencedriving a chemical reaction is known.The transition rates for an arbitrary unicyclic modelwith N states and periodic boundary conditions are de-noted by k i,i +1 = k + i and k i,i − = k − i , (47)where i = 1 , , . . . , N . A fixed affinity A implies theconstraint (cid:81) Ni =1 k + i (cid:81) Ni =1 k − i = e A (48) on the transition rates. Different choices of the transi-tion rates that fulfill this restriction can lead to differentgenerating functions, as shown in Fig. 5. In particular,if the transition rates are uniform, i.e., k + i = k + and k − i = k − the generating function divided by the averagecurrent λ ( z ) /J becomes λ ARW ( z, A , N ), which is given inEq. (37). The opposite extreme choice for the transitionrates is the case where the network behaves effectivelylike there was only one link between states ( N = 1)and all the affinity is concentrated in this single link.In this case λ ( z ) /J becomes λ ARW ( z, A , λ ARW ( z, A , ≥ λ ARW ( z, A , N ).From these considerations we conjecture that for uni-cyclic networks λ ARW ( z, A , ≥ λ ( z ) /J ≥ λ ARW ( z, A , N ) . (49)Hence, using the definition (37), Eq. (49) implies thelower bound λ ( z ) ≥ J cosh[( z + A / /N ] − cosh[ A / (2 N )](1 /N ) sinh[ A / (2 N )] , (50)which we call the hyperbolic cosine bound for a unicyclicnetwork. This conjectured bound is supported by thenumerical evidence shown in Fig. 5. Eq. (49) also leadsto an upper bound λ ( z ) ≤ J cosh( z + A / − cosh( A / A / . (51)A rigorous proof for this upper bound is provided in ap-pendix C.In Fig. 6, we show the bounds (50) and (51) for differ-ent values of the affinity. For smaller A the bounds arecloser to each other. In the linear response regime, up toquadratic order in z , they become the same and equal tothe parabolic bound in Eq. (40).The bounds in Eqs. (50) and (51) lead to the boundscoth (cid:18) A (cid:19) ≥ F ≥ N coth (cid:18) A N (cid:19) (52)on the Fano factor defined in Eq. (22). The lower boundis an affinity dependent bound on the Fano factor thathas been obtained in [17]. In the formal limit A → ∞ it becomes F ≥ /N . This bound for formally divergentaffinity is a key result in statistical kinetics [42] as itallows for an estimate on the number of states N frommeasurements of the Fano factor. The upper bound on F in Eq. (52) is a new result. It is a generalization ofthe known result F ≤
1, which is also valid in the limit
A → ∞ [42].For systems at equilibrium, the bounds (49) become2 D [cosh( z ) − ≥ λ ( z ) ≥ DN [cosh( z/N ) − , (53)which is obtained using the linear response current J = D A with the Einstein relation for the diffusion constant -1.5-1-0.500.5 -8 -7 -6 -5 -4 -3 -2 -1 0 1 λ ( z ) / J z A = 7 A = 5 A = 3 FIG. 6. Generating function λ ( z ) scaled by the steady statecurrent J in unicyclic networks with N = 10 for three familiesof distinct affinities A . For each family, transition rates weresampled according to the procedure described in appendix B.The black curves refer to the lower and upper bound fromEqs. (50) and (51). D and letting A →
0. Thus, the bound (34) for the ratefunction of the variable x = X/t can be refined to˜ h ( x ) ≤ N (cid:104) x arsinh (cid:16) x DN (cid:17) + 2 DN − (cid:112) x + (2 DN ) (cid:105) (54)for unicyclic networks. B. Multicyclic networks
A formulation of an affinity dependent bound for mul-ticyclic networks is more involved. In this case, the affini-ties of the fundamental cycles are fixed, which means thatthe transition rates are constrained by relations of theform (48) for each fundamental cycle. The hyperboliccosine bound for the generating function of the entropychange reads λ s ( z ) ≥ σ cosh[( z + 1 / A ∗ /n ∗ ] − cosh[ A ∗ / (2 n ∗ )]( A ∗ /n ∗ ) sinh [ A ∗ / (2 n ∗ )] , (55)where the affinity A ∗ and the number of states n ∗ cor-responds to the smallest ratio A /n among all the cyclesin in the network. We note that a local evaluation of theform (22) of the bound (55) leads to [18] F s ≥ A ∗ n ∗ coth (cid:18) A ∗ n ∗ (cid:19) . (56)This bound can be used to estimate the number of inter-mediate states in enzymatic schemes from measurementsof the Fano factor in single molecule experiments, as dis-cussed in [18].In order to explain how to identify A ∗ /n ∗ we considerthe network of states in Fig. 1c. We arbitrarily choose thecycles (1 , , ,
1) with affinity A , (2 , , ,
2) with affinity A , and (1 , , ,
1) with affinity A as the three funda-mental cycles. Any other cycle in the network is justa composition of these fundamental cycles; for example,the cycle (1 , , , , A + A is the sum ofthe first and second fundamental cycles. If the affinitiesare A = (1 , ,
3) then the cycle with the smallest affinityper number of states is the fundamental cycle (1 , , , A ∗ = 1 and n ∗ = 3. If the affinitiesare A = ( − , , A in Fig. 1c changes from anti-clockwise to clockwise,the cycle with minimal affinity per number of states is(1 , , , , A ∗ = 1 and n ∗ = 4.The basic idea behind the bound in Eq. (55) is as fol-lows. Given a network of states with fixed affinities, thetransition rates that lead to the smallest possible λ s ( z ) /J are those for which the cycle with smallest A /n domi-nates the network. This cycle dominates the network ifthe transition rates within the cycle are large, transitionrates to leave the cycle are small, and transition rates toreturn to the cycle are large. With this choice for thetransition rates the multicyclic network is effectively aunicyclic network with affinity A ∗ and number of states n ∗ , for which the bound (50) holds. Any other choiceof rates will just add cycles with a smaller affinity pernumber of states, which cannot decrease fluctuations.Our numerical evidence presented in Fig. 7d shows thatthis hyperbolic cosine bound is also valid for any individ-ual current X α in the form λ α ( z ) ≥ σ cosh[( zJ α /σ + 1 / A ∗ /n ∗ ] − cosh[ A ∗ / (2 n ∗ )]( A ∗ /n ∗ ) sinh [ A ∗ / (2 n ∗ )] . (57)Hence, the hyperbolic cosine bound can be written in themore general form λ ( z ) ≥ σ cosh[( z · J /σ + 1 / A ∗ /n ∗ ] − cosh[ A ∗ / (2 n ∗ )]( A ∗ /n ∗ ) sinh [ A ∗ / (2 n ∗ )] . (58)The full numerical evidence for this conjecture is dis-cussed in appendix B. If the cycle relevant for the bound(57) has a rather small affinity per number of states,which is often the case in a large network of states, thebound is only slightly stronger than the parabolic bound(43), as visible in Fig. 7a and 7c. An often tighter boundfor this situation is derived in the next section. Our nu-merics indicates that an affinity dependent upper boundon the generating function in the multicyclic case doesnot exist. For fixed affinities, the generating function canbecome arbitrarily close to the trivial bound λ s ( z ) < − < z <
0, visible in Fig. 7a-c. A generalization ofthe equilibrium bound (54) to multicyclic networks is notdirectly possible, since the identification of a cycle withminimal A /n becomes ambiguous in the limit A → -0.200.2 -1 0 λ s ( z ) / σ z (a) -0.200.2 -1 0 λ s ( z ) / σ z (b)-0.200.2 -1 0 λ s ( z ) / σ z (c) -0.200.2 -1 0 λ α ( z ) / σ zJ α /σ (d) FIG. 7. Generating functions for entropy change (a-c) andrandomly selected individual currents X α (d) for the net-work of Fig. 1c. The affinities are A = (1 , ,
3) in (a), A = (11 , ,
13) in (b) and (d), and A = ( − , ,
13) in (c).The black curves correspond to the parabolic bound and thered dashed curves correspond to the hyperbolic cosine bound.The generating functions were generated as explained in ap-pendix B. In panel (a) A ∗ /n ∗ = 1 /
3, in panels (b) and (d) A ∗ /n ∗ = 11 /
3, and in panel (c) A ∗ /n ∗ = 1 /
4. For smallvalues of A ∗ /n ∗ , as in panels (a) and (c), the parabolic andhyperbolic cosine bound are closer to each other. VI. EXPONENTIAL BOUND
A rigorous lower bound on the largest eigenvalue λ ( z )of the matrix L ( z ) can be obtained from the algebraicproperties of positive matrices. Specifically, a remark-able theorem about the largest eigenvalue µ of an arbi-trary matrix with non-negative entries B ij from Ellis [20,Theorem IX.4.4] isln µ = sup τ ij (cid:88) i,j τ ij ln B ij ν i τ ij , (59)where ν i ≡ (cid:80) (cid:96) τ i(cid:96) and 0 ln 0 ≡
0. The admissible matri-ces τ ij must satisfy certain normalization and symmetryproperties given in appendix A. For any specific matrix τ ij , Eq. (59) provides a lower bound on µ .In order to apply Eq. (59) to the modified Markovgenerator L ij ( z ), we construct a positive matrix B ij ( z ) ≡ δ ij + η L ij ( z ) (60)with a sufficiently small parameter η >
0. Its largesteigenvalue is given by 1+ ηλ ( z ). Making use of the knowneigenvector of L ij (0), which is the stationary distribution p s i , we can choose τ ij such that the supremum (59) issaturated for z = 0. As we show in appendix A, fixing this choice for all values of z , Eq. (59) yields the bound λ ( z ) ≥ (e η z · J − /η (61)on the generating function.The largest possible η in Eq. (60) provides thestrongest bound. The maximal value that still complieswith the requirement for a non-negative entries B ij isthe inverse of the maximal exit rate η = 1 / max i ( r i ).Extending the proof of Eq. (59) in appendix A, we canshow that Eq. (61) is valid for larger values of η up to η = 1 /R (62)where R ≡ (cid:88) i p s i r i ≤ max i r i , (63)is the steady state activity of the network, i.e., the aver-age number of transitions per time interval in the steadystate. We note that a term related to activity also ap-pears in a fluctuation dissipation relation for nonequilib-rium steady states [43].Due to the Gallavotti-Cohen symmetry (11) the expo-nential bound can also be written as λ ( z ) ≥ R (e ( −A − z ) · J /R − , (64)where we set η = R − . This bound is sharper than (61)for z · J < − A · J / σ/
2. Combining Eqs. (61) and(64) we obtain the exponential bound λ ( z ) ≥ R (cid:104) e ( | σ/ z · J |− σ/ /R − (cid:105) . (65)For an individual current X α , the exponential boundreads λ α ( z ) ≥ R (cid:104) e ( | σ/ zJ α |− σ/ R − (cid:105) . (66)The choice z = z A in Eq. (65) leads to λ s ( z ) ≥ R (cid:104) e ( | σ/ zσ |− σ/ /R − (cid:105) (67)for the entropy change.An illustration of the exponential bound (65) is pro-vided in Fig. 8. This bound is typically tighter thanthe parabolic bound for far from equilibrium conditions,i.e., for large affinity. For example, for a random walkon a unicyclic network with N sites, a uniform forwardstepping rate k and a vanishing backward stepping rate,which implies divergent affinity, the bound in Eq. (65) issaturated. Specifically, in this case the generating func-tion is λ ( z ) = k (cid:16) e z/N − (cid:17) , (68)the activity is R = k and the cycle current J = k/N . Forvanishing current at equilibrium, the exponential boundreduces to the trivial statement λ ( z ) ≥ ✲✶✵✵✶✵✷✵✲✷✵ ✲✶✺ ✲✶✵ ✲✺ ✵ ✺③ ✵✶✵✷✵✸✵✹✵✺✵ ✲✶ ✵ ✶ ✷ ✸✘✕✭③✮❜♦✉♥❞ ❤✭✘✮❜♦✉♥❞ FIG. 8. Generating function (left) and rate function(right) for a five-state unicyclic network with rates ln k + i =(3 , , , ,
2) and ln k − i = ( − , , − , , A = 15, cur-rent J (cid:39) .
25, and activity R (cid:39) .
9. The functions are shownas solid lines and the exponential bound (65) as dashed lines.Analytic continuations of the piecewise defined functions areshown as dotted curves.
Our numerics indicates that the hyperbolic cosinebound is always tighter than the exponential bound inunicyclic networks. For multicyclic networks the expo-nential bound can be tighter. Furthermore, contrary tothe hyperbolic cosine bound, the exponential bound doesnot require knowledge of the topology of the network ofstates, only the average entropy production and the av-erage activity are required.In terms of the rate function of an individual current X α , corresponding to the generating function Eq. (66),the exponential bound reads h α ( ξ ) ≤ η [1 + ξ − ξ ln | ξ | ] − σξ, ξ ≤ − e − ησ/ , η [1 − ξ + ξ ln | ξ | ] , ξ ≥ e − ησ/ , η [1 − e − ησ/ ] − σξ/ , otherwise . (69)This bound on the rate function is illustrated in Fig. 8for a unicyclic network.Using (22) in the exponential bound (65) for an indi-vidual current leads to F α ≥ J α /R. (70)This new relation provides a lower bound on the disper-sion of an individual current, characterized by the Fanofactor F α , in terms of its average J α and the activity R . VII. ASYMPTOTIC BOUNDSA. Unicyclic networks
The asymptotic bounds discussed in the followingare exact results that become tighter than all previousbounds for large values of | z | . First we consider a uni-cyclic network with N states and affinity A . In this case,we can prove the following bound on the generating func- z z -40-30-20-10010203040-40 -30 -20 -10 0 10 20 30 40 00.20.40.60.81 FIG. 9. Asymptotic bound for the house-shaped network withtwo fundamental cycles shown in Fig. 1b. The color coderepresents the ratio (77) between the generating function andthe bound. Black dashed lines indicate the borders betweensectors with constant relevant cycles ˆ C ( z ). For each sector,the relevant cycle ˆ C ( z ) is shown in white. The affinity ofthe three-cycle is A = 8 and the affinity of the four-cycle is A = 6. tion: λ ( z ) ≥ Jλ ARW ( z, A , N ) + r ARW − N N (cid:88) i =1 r i , (71)where λ ARW ( z, A , N ) is defined in Eq. (37), r ARW ≡ k + + k − , and k ± ≡ (cid:32) N (cid:89) i =1 k ± i (cid:33) /N . (72)This bound is proved in appendix D by comparing theweight of a trajectory in the ensemble with transitionrates k ± i with the weight of a trajectory in the ensemblewith transition rates k ± . Our numerics indicate that withincreasing | z | the difference between this bound and theactual generating function tends to zero. This fact isquite remarkable given the exponential growth of bothfunctions. Unlike all other bounds presented so far, thebound (47) is not saturated at z = 0. Only for the caseof uniform rates, i.e., k ± i = k ± , the generating function(36) saturates the bound (71) globally. B. Multicyclic networks
In order to obtain an asymptotic bound also valid formulticyclic networks we define an arbitrary closed path C , which is a sequence of jumps that finishes at the stateit started, as C ≡ [ i (1) → i (2) → · · · → i ( n C ) → i (1)] , (73)1where n C is the length of the closed path. With this pathwe associate a geometric mean of the transition rates γ C ≡ ( k i (1) ,i (2) k i (2) ,i (3) . . . k i ( n C ) ,i (1) ) /n C (74)and integer winding numbers m β C that count how oftenthe elementary cycle β is completed within the path C .Applying a theorem valid for arbitrary non-negative ma-trices [44, Lemma 3.5.3] to the matrix L ij ( z )+ δ ij max (cid:96) r (cid:96) we obtain λ ( z ) + max (cid:96) r (cid:96) ≥ f ( z , C ) ≡ γ C exp n C (cid:88) β m β C z β (75)for any closed path C . The best bound on λ ( z ) in Eq. (75)is obtained by choosing an optimal path ˆ C ( z ), whichin principle depends on z , that maximizes the r.h.s ofEq. (75) in the large z regime.First we consider this optimal path for the unicyclicnetwork. In this case, the optimal path is a single cyclein the forward direction with m C = 1 if z >
0. If weconsider a path C with two cycles, i.e., m C = 2, the boundremains the same as the number of states n C also doubles.If the closed path is not a direct cycle but contains, forexample, one backward jump, then γ C can become larger.However, such a backward jump also makes n C larger andhence, the exponent in Eq. (75) smaller. Since we areinterested in the large z regime, this second effect shouldbe dominant. Hence, for z > λ ( z ) ≥ k + exp( z/N ) − max (cid:96) r (cid:96) . (76)Even though this bound is different from (71), they bothpredict the same exponential growth, with the same pref-actor, for large z >
0. The same reasoning is valid for z <
0, with the optimal path being a single cycle in thenegative direction.For multicyclic networks we consider the house-shapednetwork with five states shown in Fig. 1b. This networkconsists of a cycle with three states and affinity A anda cycle with four states and affinity A . We choose thesecycles to be the fundamental cycles. This network alsohas a third cycle, which is the cycle with five states andaffinity A + A . Given a vector = ( z , z ), the optimalpath is the cycle that maximizes the r.h.s of Eq. (75).For large enough | z | , this optimal path depends only onthe direction of the vector. Clearly a path that includesother cycles will lead to a weaker bound.A contour plot of the ratio f ( z , ˆ C ( z )) / [ λ ( z )+max (cid:96) ] forthis house-shaped network is shown in Fig. 9. Remark-ably, the r.h.s. of (75) captures the leading order of theasymptotics for large | z | , i.e., for large | z | f ( z , ˆ C ( z )) λ ( z ) + max (cid:96) r (cid:96) → . (77)Only in the lines separating regions dominated by differ-ent cycles in Fig. 9 does this ratio tend to slightly lower (a)(b) FIG. 10. Schematic illustrations of the WASEP (a) and theKMP (b) models. For the WASEP, in the bulk the parti-cles jump with rates p ≡ / ν/ (2 L ) to the right and q ≡ / − ν/ (2 L ) to the left, where the SSEP correspondsto ν = 0. At the boundaries particles are exchanged with thereservoirs. The model also has the exclusion principle, i.e.,the maximum number of particles in a site is one. For theKMP model energy flows from a hot reservoir at tempera-ture T L to a cold reservoir at temperature T R . In the bulka randomly chosen pair of sites exchange energy, which is acontinuous variable, in such a way that the total energy isconserved. At the boundaries energy is exchanged with thereservoirs. The precise rules of these models can be found in[29] for the WASEP and [28] for the KMP model. values. Along this line the dominant cycle is degener-ate. As shown in appendix E, relation (77) is valid forany multicyclic network. Hence, we conclude that ourasymptotic bound predicts the exponential growth of thegenerating function, apart from exceptional regions in z where the optimal cycle is degenerate. VIII. PARABOLIC BOUND IN DRIVENDIFFUSIVE ONE DIMENSIONAL SYSTEMS
We now consider one dimensional driven diffusive sys-tems, which unlike the cases considered so far have a di-vergent number of states L in the thermodynamic limit.Calculating the generating function for these systems is amajor challenge that can be overcome in some cases withthe additivity principle [25]. In this section we comparethe parabolic bound to the cumulant generating functionobtained from this additivity principle for three examplesof driven diffusive systems, for which the validity of theadditivity principle has been verified numerically [28, 29].First we consider the WASEP and the SSEP, whichis a particular case of the WASEP. These models areillustrated in Fig. 10 and their precise definition can befound in [26]. In the WASEP particles flow from the leftreservoir with constant density (cid:37) L to the right reservoirwith density (cid:37) R < (cid:37) L . The current of particles in thesystem is proportional to the entropy production, andthe affinity that drives the process out of equilibrium isgiven by [26, 29] A WASEP = − ln 1 − (cid:37) L (cid:37) L + ln 1 − (cid:37) R (cid:37) R + ( L −
1) ln 1 − ν/L ν/L . (78)2 λ s ( z ) / σ z / A parabolic boundSSEPWASEPKMP FIG. 11. Comparison of the parabolic bound (42) with gen-erating functions for driven diffusive systems. For the SSEPthe densities of the left and right reservoirs were chosen as (cid:37) L = 0 .
99 and (cid:37) R = 0 .
01. For the WASEP the parametersare ν = 10, (cid:37) L = 4 /
7, and (cid:37) R = 5 /
18, as in Ref. [29]. Forthe KMP model the parameters are T L = 2 and T R = 1, as inRef. [28]. The weak asymmetry of the bulk rates, which scales with1 /L , guarantees that in the thermodynamic limit L →∞ the affinity is finite. In Fig. 11, we have calculatedthe generating function using the additivity principle forthe SSEP, as explained in [26], and for the WASEP, asexplained in [29]. In both cases the generating functionsare inside the parabolic bound.The KMP model is a driven diffusive system for thetransport of energy from a reservoir at temperature T L to a reservoir at temperature T R < T L , as illustrated inFig. 10. A key feature of the KMP model is that there isno dissipation in the bulk. The precise definition of themodel can be found in [28]. The heat transfer from theleft to the right reservoir is proportional to the entropyproduction with the affinity given by [28] A KMP = ( T − − T − ) . (79)The generating function for this model, which is obtainedfrom the additivity principle as explained in [28], alsosatisfies the parabolic bound in Fig. 11 within the finitesupport − T − < z < T − of λ ( z ). As a consequence, therate function satisfies the corresponding parabolic boundglobally.These results demonstrate that our parabolic boundis even more universal: it seems to be valid for thesedriven diffusive systems in the thermodynamic limit, forwhich the number of states diverges. We expect thatthe parabolic bound is the only relevant one in the limit L → ∞ . The hyperbolic cosine bound (57) approachesthe parabolic bound for vanishing affinity per number ofstates in a cycle. The exponential bound (61) degener-ates with increasing activity to a linear function, whichreflects simply the convexity of the generating function.Another interesting issue will be to explore whetherthe bound is still valid in the L → ∞ limit if the sys- tem undergoes a dynamical phase transition as the KMPmodel in a ring-like geometry [45]. IX. CONCLUSIONS
We have obtained four global bounds on current fluctu-ations for Markov processes in steady states summarizedin table I. The parabolic bound from Sec. IV is the mostuniversal result of this paper. The simple knowledge ofthe average entropy production is enough to bound thewhole range of fluctuations of any individual current. Inother words, for nonequilibrium steady states, the gen-erating function associated with any fluctuating currentmust lie inside the parabola shown in Fig. 12. The uni-versality of the parabolic bound was further confirmed bythe fact that it also applies to the three driven diffusivesystems we analyzed in Sec. VIII, for which the numberof states diverges.This parabolic bound can be saturated only close toequilibrium. A bound that is generally tighter than theparabolic bound, particularly if the system is far fromequilibrium, is the hyperbolic cosine bound from Sec. V.This necessarily less universal bound also requires knowl-edge of the thermodynamic forces, i.e., the affinities, thatdrive the process out of equilibrium and of the topologyof the network of states.The exponential bound depends on the average entropyproduction and on the average number of transitions pertime. In contrast to the parabolic and hyperbolic co-sine bounds that are conjectures based on extensive nu-merical evidence, we have proven the exponential bound.It is typically tighter than the parabolic bound for farfrom equilibrium situations. While for a unicyclic net-work we observed that the hyperbolic cosine bound is al-ways tighter than the exponential bound, for multicyclicnetworks the exponential bound can be tighter.The fourth bound is an exact asymptotic bound thatpredicts the growth of the generating function for large z , as illustrated in Fig. 12. This bound requires knowl-edge of the particular transitions rates. Therefore, itsimportance arises in a situation where a Markov processwith all its transition rates is given but calculating thefull generating function is not possible.Summarizing, typical and large fluctuations for any in-dividual current in stationary Markov processes, whichare used to describe a large amount of nonequilibriumsystems ranging from enzymatic reactions to nanoscaleelectronic systems, have been shown to be bounded bythe average entropy production or the average entropyproduction and the average activity. Rigorous proofs ofthe parabolic bound and of the hyperbolic cosine boundremain as main open technical challenges. Note added:
A proof of the parabolic bound has re-cently appeared [46].3 -50050100150 -20 -15 -10 -5 0 5 λ ( z ) z λ ( z )paraboliccoshexponentialasymptotic FIG. 12. Summary of the four bounds for a unicyclic networkwith four states. Transition rates are ln k + i = (3 , , ,
4) andln k + i = (0 , − , , A = 15 and thecurrent J (cid:39) . Parabolic z · J (1 + z · J /σ ) (31) Hyperboliccosine σ cosh (cid:104) ( z · J σ + ) A ∗ n ∗ (cid:105) − cosh [ A ∗ / (2 n ∗ )]( A ∗ /n ∗ ) sinh [ A ∗ / (2 n ∗ )] (58) Exponential R (cid:104) e ( | σ/ z · J |− σ/ /R − (cid:105) (65) Asymptotic − max (cid:96) r (cid:96) + γ C exp n C (cid:88) β m β C z β (75)TABLE I. Summary of lower bounds on the generating func-tion. Appendix A: Proof of the exponential bound
The theorem by Ellis [20, Theorem IX.4.4] can bestated as follows. For any non-negative matrix B ij , theassociated maximum eigenvalue can be calculated asln µ = sup τ ij (cid:88) i,j τ ij ln B ij ν i τ ij , (A1)where ν i ≡ (cid:80) k τ ik and 0 ln 0 ≡
0. The admissible matri-ces τ ij must satisfy the following properties:1. Normalization, (cid:88) i,j τ ij = (cid:88) i ν i = 1 . (A2)2. Equal row- and column-sums, ν i ≡ (cid:88) (cid:96) τ i(cid:96) = (cid:88) (cid:96) τ (cid:96)i . (A3)3. Non-negative with the same (or less complex) struc-ture as B ij , i.e., τ ij > ⇒ B ij > . (A4) In order to apply Eq. (A1) to the modified Markovgenerator L ij ( z ), we consider the matrix B ij ( z ) ≡ δ ij + η L ij ( z ) , (A5)with a sufficiently small parameter η >
0. Its largesteigenvalue is 1 + ηλ ( z ). Lower bounds on the generatingfunction λ ( z ) can be obtained from (A1) by choosing anappropriate matrix τ ij . The choice τ ij = B ij (0) p s j (A6)saturates the bound for z = 0. The bound for the eigen-value (A1) then reads (with ν i = p s i )ln[1 + ηλ ( z )] ≥ (cid:88) i,j [ δ ij + η L ij (0)] p s j ln [ δ ij + η L ij ( z )] p s i [ δ ij + η L ij (0)] p s j . (A7)Since L ii ( z ) = L ii (0) the logarithm vanishes for i = j and the r.h.s. simplifies toln[1 + ηλ ( z )] ≥ (cid:88) i (cid:54) = j η L ij (0) p s j ln L ij ( z ) p s i L ij (0) p s j (A8)= η (cid:88) i (cid:54) = j p s j k ji (cid:32) z · d ji + ln p s i p s j (cid:33) . (A9)The term proportional to ln( p s i /p s j ) vanishes because p s i is the stationary distribution. Identifying the stationarycurrent J = (cid:80) i (cid:54) = j p s i k ij d ij we obtain the bound λ ( z ) ≥ η (cid:0) e η z · J − (cid:1) , (A10)which is Eq. (61) in the main text.The bound improves for larger η . The maximal valuethat still complies with the requirement for positive en-tries B ij is the inverse of the maximal escape rate η =1 / max i ( r i ). In this case, the proof of (A1) in Ref. [20]uses the equation (cid:88) i,j τ ij ln B ij ν i τ ij = (cid:88) i,j τ ij ln B ij ν i τ ij − (cid:88) i,j τ ij ln τ ij ν i ν i τ ij , (A11)where ν i = (cid:80) j τ ij . The matrix τ is given by τ ij = ˜ q i B ij q j /µ, (A12)where ˜ q and q are left and right eigenvectors of B , respec-tively, with the normalization (cid:80) i q i = 1 and (cid:80) i ˜ q i q i = 1.From relations (A2), (A3), and (A12) it follows that thefirst sum in Eq. (A11) is simply ln µ . The second sumwith a minus sign can be shown to fulfill the inequality (cid:88) i,j y ij x ij ln x ij ≥ (cid:88) i,j y ij ( x ij −
1) = (cid:88) i,j ( τ ij − y ij ) = 0 , (A13)4where y ij ≡ ν i τ ij ν i , x ij ≡ τ ij ν i ν i τ ij . (A14)Equality in Eq. (A13) is achieved for τ ij = τ ij , whichwith Eq. (A13) provides a proof of (A1).Actually, we can show that the bound (A10) is valideven for larger values of η up to η = R − , where R is theaverage activity in Eq. (63). In this case, the diagonalelements B ii can be negative. This property can affectEq. (A13), which requires that x ij ≥ y ij ≥
0. For B ij ( z ) = δ ij + η L αij ( z ), if1 + ηλ α ( z ) > x ij and y ij in Eq. (A14) for i (cid:54) = j are non-negative. The inequality (A13) is then valid for i (cid:54) = j with condition (A15). For the diagonal terms we canwrite (cid:88) i y ii x ii ln x ii = (cid:88) i p s i (1 − ηr i ) ln[1 + ηλ α ( z )] ≥ (cid:88) i p s i (1 − ηr i ) ηλ α ( z )1 + ηλ α ( z ) , (A16)where the inequality is valid for η ≤ R − and if condition(A15) is fulfilled. Therefore, if condition (A15) holds thenthe inequality (A14) is valid for η ≤ /R .Since this condition (A15) is valid for z = 0, it mustalso be valid for some finite range in z . Using a simpleself-consistency check, we can even prove that this “finiterange” must in fact always be infinite. Assume the func-tion 1+ λ α ( z ) /R crosses zero at some value z = z ∗ . Thenthe condition (A15) is violated for z = z ∗ − δz . On theother side, for z = z ∗ + δz , the condition is still satisfiedand we find1 + λ α ( z ∗ + δz ) /R ≥ e ( z ∗ + δz ) J α /R > , (A17)which contradicts the continuity of λ α ( z ). Appendix B: Numerical verification
The numerical verification of the conjectured boundswas performed on large sets of networks with differenttransition rates. The corresponding generating functionsare given by the largest eigenvalues of the matrices L ( z ).We calculated these eigenvalues using standard numeri-cal algorithms. The stationary distributions p s i , which arecomputed as the eigenvector for z = 0, are used to evalu-ate the steady state currents that appear in the bounds.The precise procedures are described below.
1. Unicyclic networks
For unicyclic networks with N states it is convenientto parametrize the transition rates (47) as k ± i = exp( φ i ± θ i A / . (B1)The global time scale can be fixed by requiring (cid:80) i φ i = 0.Thus we avoid numerical instabilities due to extremelylarge or small matrix entries. Moreover, in order tosample cycles with predefined affinity A , we require (cid:80) i θ i = 1. We generate vectors φ (cid:48) i and θ (cid:48) i of N indepen-dent and identically distributed random numbers. Theabove constraints are satisfied by setting φ i = φ (cid:48) i − φ (cid:48) and θ i = θ (cid:48) i /θ (cid:48) , where the overbar denotes the average of thevectors elements within the realization. Samples whereat least one of the | θ i | is above a certain value were dis-carded in order to avoid numerical instabilities. The cut-off value 1 turns out to be suitable for this purpose. Sincethe transition rates associated with the discarded samplesare extremely non-uniform, the corresponding generatingfunction lies close to the (proven) upper bound anyway.For the plots shown in Fig. 6, φ (cid:48) i and θ (cid:48) i were drawnfrom uniform distributions with 0 < φ (cid:48) i < − . <θ (cid:48) i < .
5, respectively. The generating functions werecalculated for a total of 10 000 samples for each affinity,of which only the first 100 are shown in the figure. Forcycles with many states and high affinity, it is virtuallyimpossible to cover the whole area between the upperand the lower bound using a single, simple distributionfor φ (cid:48) i and θ (cid:48) i . In Fig. 5, we show several families of sam-ple generating functions where the rates are drawn fromdifferent statistical ensembles. Specifically, φ (cid:48) i and θ (cid:48) i areboth Gaussian random variables with mean 1 (which isirrelevant for φ i ) and standard deviations (SD) reachingfrom 0 .
01 to 2.In principle the lengths sub-steps d i,i + i can be dis-tributed arbitrarily among the edges of the network. Inmost cases, the choice d i,i + i ∝ ln( k + i /k − i ) avoids numer-ical instabilities. In order to pre-assess the range of z we can make use of the proven asymptotic bound (71):the validity of the hyperbolic cosine bound has to bechecked only in the finite range where it is weaker thanthe asymptotic bound. In all cases the hyperbolic co-sine bound (50) is satisfied. Since the hyperbolic cosinebound implies the parabolic bound, this numerical evi-dence also allows us to conjecture the parabolic boundfor unicyclic networks.
2. Multicyclic networks
The bounds relevant for a numerical test for multicyclicnetworks are the parabolic bound and the hyperbolic co-sine bound. They exist in the formulation for entropychange [Eqs. (44) and (55)] and for arbitrary individualcurrents [Eqs. (43) and (57)]. The former type can be5checked by setting the matrix of increments in Eq. (7) to d sij = ln( k ij /k ji ), for the latter we use anti-symmetrizedGaussian random matrices for d αij .We have performed two types of tests. The first typerelies on random rate matrices k ij of dimension N × N ,each of them corresponding to a fully connected networkwith N states. The rates were generated according to k ij = exp[ a ( φ ij + φ ji ) / bθ ij ] , (B2)where φ ij and θ ij are independent Gaussian randomnumbers with zero mean and variance 1. The param-eters a and b can be used to tune the properties of thenetwork. While small values of a simulate fully connectednetworks, larger values of a typically suppress some ofthe transitions, so that the generated matrices effectivelycorrespond to partially connected networks with randomtopology. The parameter b introduces an asymmetry inthe transitions, that drives the system out of equilibrium.For smaller values of b the generating functions lie closerto the parabolic bound. For Fig. 4 we have calculated300 generating functions for each N = 4 and N = 6 with a = 5 and b = 2. In a more extensive computation, wehave checked the parabolic bound for a total of 10 gener-ating functions with a ranging from 0 to 5, b ranging from0.01 to 5, and N ranging from 4 to 50. The hyperboliccosine bound could be checked only up to N = 8, forlarger networks the determination of the relevant cyclein (57) becomes numerically expensive.The second type of test applies to small networks withgiven topology, where the fundamental cycles can beidentified by hand. The affinities of these cycles can befixed, so that the bounds (55) and (57) depend only onthe steady state currents. For example, for the networkshown in Fig. 1c, random rates were assigned to most ofthe transition. The random numbers were generated suchthat ln k ij were Gaussian random numbers with standarddeviations ranging from 0.01 to 3 and, at first, with zeromean. Only the three forward transition rates in the cy-cle (1,2,3,1) were determined algebraically from the othertransition rates and the constraints from the fixed cycleaffinities A , A and A . Typically, the generating func-tions obtained via this procedure are quite far from thehyperbolic cosine bound. In order to test also more crit-ical cases, we have added a bias ±A ∗ / (2 n ∗ ) to the log-arithms of the forward (+) and backward (-) transitionrates constituting the relevant cycle for the hyperboliccosine bound. Moreover, the rates for transitions exitingthe relevant cycle were gradually lowered by several or-ders of magnitude. For the plots in Fig. 7, we have usedattenuations of these rates between e and e − . Simi-lar tests were performed for a large varieties of networks(as the networks shown in Fig. 2 of the supplementarymaterial of Ref. [17]). xP ( x ) y x FIG. 13. The polynomial P ( x ) for a generic unicyclic networkwith N = 6 states. The tangent at x = 0 and the values y = y and x = x are shown as dashed lines. Appendix C: Proof of the upper bound on thegenerating function for unicyclic networks
For unicyclic networks with N states the tilted Markovgenerator has the tridiagonal shape L ( z ) = − r ˜ k . . . ˜ k N ˜ k − r ˜ k . . .
00 ˜ k − r ˜ k . . . 00 0 ˜ k − r . . . ...... ... . . . . . . . . . ˜ k N,N − ˜ k N . . . ˜ k N − ,N − r N (C1)with ˜ k ij ≡ k ij e zd ij . The displacements d ij must satisfy d ij = − d ji . If the observable of interest is the numberof turnovers, the displacements must add up to the cycleaffinity d + d + · · · + d N = 1. It should be kept in mindthat any statistical quantity in the long time limit (inparticular the generating function and the rate function)do not depend on the specific choice of the individual d ij .The characteristic polynomial associated with the ma-trix (C1) reads χ ( z, x ) ≡ det( L ( z ) − x N )= (cid:88) π ( − π N (cid:89) i =1 ( L iπ ( i ) ( z ) − xδ iπ ( i ) ) , (C2)where the sum runs over all permutations π of the indices i = 1 , . . . , N and N is the N × N identity matrix. Weidentify 0 ≡ N and N + 1 ≡ k i +1 ,i : the contribution from the next row i + 1 can either be ˜ k i,i +1 or ˜ k i +2 ,i +1 . For the former typethe z -dependence cancels out due to d i +1 ,i = − d i,i +1 and we end up with the constant factor ˜ k i +1 ,i ˜ k i,i +1 = k i +1 ,i k i,i +1 . Terms of the latter type must also contain˜ k i,i − as the only possible contribution from the previous6column i −
1. Iteratively, we see that there can be onlyone term of this type, namely the one that contains allforward transitions˜ k ˜ k . . . ˜ k N = k k . . . k N e z ≡ Γ + e z . (C3)An analogous argument can be set up for the lower off-diagonal of the matrix with the z -dependent term˜ k ˜ k . . . ˜ k N = k k . . . k N e − z ≡ Γ − e − z = Γ + e − ( z + A ) . (C4)All other terms in the determinant (C2) do not dependon z and we can write χ ( z, x ) = ( − N +1 [ Γ + e z + Γ − e − z − ( Γ + + Γ − ) − P ( x )](C5)with some polynomial P ( x ) that is independent of z andthe specific choice of the d ij . The alternating prefactor isdue to the fact that the permutations associated with theterms (C3) and (C4) are either odd or even, dependingon the number of states N . The generating function isthus given by λ ( z ) = P − (cid:0) Γ + e z + Γ − e − z − Γ + − Γ − (cid:1) = P − (cid:16) √ Γ + Γ − [cosh( z + A / − cosh( A / (cid:17) , (C6)where the function P − ( y ) returns the root of the poly-nomial P ( x ) − y that has the largest real part. Due tothe Perron-Frobenius theorem, this root must be real forall arguments occurring in (C6), i.e., for all y ≥ y ≡ √ Γ + Γ − [1 − cosh( A / y is x ≡ P − ( y ) = min z λ ( z ) = λ ( −A / P ( x ) (see Fig. 13)has the properties P (0) = χ (0 ,
0) = 0 andlim x →∞ P ( x ) = ( − N lim x →∞ χ ( z, x )= lim x →∞ ( − N det( − x N ) = + ∞ . (C7)Since the matrix L ( −A /
2) can be brought to a symmetricform by choosing d ij = ln( k ij /k ji ) / A , the correspondingcharacteristic polynomial P ( x ) − y has only real roots x i with x denoting the largest one. The second derivativeof P ( x ) is P (cid:48)(cid:48) ( x ) = d d x (cid:34) y + N (cid:89) i =1 ( x − x i ) (cid:35) = N (cid:88) i =1 N (cid:88) j =1 j (cid:54) = i N (cid:89) (cid:96) =1 i (cid:54) = (cid:96) (cid:54) = j ( x − x (cid:96) ) . (C8)For x > x this expression is positive so that P ( x ) isconvex. As a consequence, the inverted function P − ( y )is concave for the relevant arguments y > y . Hence itsatisfies P − ( y ) ≤ ( P − ) (cid:48) (0) y (C9)with equality for y = 0. This relation leads to the upperbound λ ( z ) ≤ √ Γ + Γ − ( P − ) (cid:48) (0) [cosh( z + A / − cosh( A / z = 0. The prefactor in thisbound is equal to the one in Eq. (51), as can be seen bycalculating steady state current from Eq. (C6), J = λ (cid:48) ( z ) = 2 √ Γ + Γ − ( P − ) (cid:48) (0) sinh( A / . (C11) Appendix D: Proof of the asymptotic bound for unicyclic networks
First we restrict to a unicyclic network with N states, affinity A and transition rates k ij ≡ δ i,i +1 k + i + δ i +1 ,i k − i +1 . (D1)A stochastic path n ( τ ) is defined by the sequence of jumps n (cid:96) → n (cid:96) +1 between adjacent states that occur at times τ (cid:96) . The weight of this path is given by P [ n ( τ )] = (cid:89) (cid:96) k n (cid:96) n (cid:96) +1 exp[ − r n (cid:96) ( τ (cid:96) +1 − τ (cid:96) )] , (D2)where the sum runs over all jumps. The weight of a path with modified transition rates ˜ k ij reads˜ P [ n ( τ )] = (cid:89) (cid:96) ˜ k n (cid:96) n (cid:96) +1 exp[ − ˜ r n (cid:96) ( τ (cid:96) +1 − τ (cid:96) )] . (D3)For these modified transition rates we choose an asymmetric random walk, i.e.,˜ k ij ≡ δ i,i +1 k + + δ i +1 ,i k − (D4)7with k ± ≡ (cid:32) N (cid:89) i =1 k ± i (cid:33) /N , (D5)which leads to ˜ r ≡ k + + k − . Ensemble averages using the path weight ˜ P [ n ( τ )] are denoted as (cid:104) . . . (cid:105) ARW . The generatingfunction can be rewritten as λ ( z ) = 1 t ln (cid:68) e zX [ n ( τ )] (cid:69) = 1 t ln (cid:90) D n ( τ ) e zX [ n ( τ )] P [ n ( τ )]˜ P [ n ( τ )] ˜ p [ n ( τ )]= 1 t ln (cid:42) e zX [ n ( τ )] N (cid:89) i =1 (cid:18) k + i γ + (cid:19) m + i (cid:18) k − i γ − (cid:19) m − i e − ( r i − ˜ r i ) T i (cid:43) ARW , (D6)where the integration in the first line is over all stochastic trajectories. The path dependent variables m ± i count thejumps out of state i in forward or backward direction and T i is the total sojourn time in state i . These variables areidentically distributed in the ARW-ensemble.The probability ˜ P ( X ) is the probability that the fluctuating current is X in the ARW-ensemble. It is the sum ofthe weight of all trajectories for which the current is X . Using this ˜ P ( X ) Eq. (D6) can be written as λ ( z ) = 1 t ln (cid:88) X ˜ p ( X )e zX (cid:42) exp (cid:34) N (cid:88) i =1 m + i ln( k + i /γ + ) + N (cid:88) i =1 m − i ln( k − i /γ − ) − N (cid:88) i =1 ( r i − ˜ r ) T i (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X (cid:43) ARW ≥ t ln (cid:88) X ˜ p ( X )e zX exp (cid:34) N (cid:88) i =1 (cid:10) m + i | X (cid:11) ARW ln( k + i /γ + ) + N (cid:88) i =1 (cid:10) m − i | X (cid:11) ARW ln( k − i /γ − ) − N (cid:88) i =1 ( r i − ˜ r ) t/N (cid:35) , (D7)where the conditioned average in the first line represents a functional integration over all trajectories with fluctuatingcurrent equal to X and we used Jensen’s inequality from the first to the second line. Due to (D5) the terms with thelogarithms vanish, leading to the final result in Eq. (71). Appendix E: Proof of the asymptotic limit (77) for multicyclic networks
For general Markov generators L ( z ), as defined in Eq. (7), the determinant (C2) can be written as0 = χ ( z , λ ( z )) = N (cid:89) i =1 [ − r i − λ ( z )] + (cid:88) C ( − C γ n C C e m C · z (cid:89) j / ∈C [ − r j − λ ( z )] , (E1)where the sum runs over all combinations of disjoint cycles in the underlying network and ( − C denotes the sign ofthe corresponding permutations in the determinant. For each C , the quantities n C , γ C and m C are defined as for theindividual cycles in Sec. VII of the main text. Dividing Eq. (E1) by λ ( z ) N leads to0 = N (cid:89) i =1 [ − r i /λ ( z ) −
1] + (cid:88) C ( − C f ( z , C ) n C λ ( z ) − n C (cid:89) j / ∈C [ − r j /λ ( z ) − , (E2)where f ( z , C ) is defined in Eq. (75). We now analyze the limit | z | → ∞ with the direction z / | z | kept fixed. Makinguse of the (already proven) lower bound (75) with the optimal path ˆ C ≡ ˆ C ( z ), we see that r i /λ ( z ) and the terms with( m C · z ) /n C < m ˆ C · z /n ˆ C vanish in this limit. Provided that the optimal cycle is unique, we are left with0 = ( − N + lim | z |→∞ ( − n ˆ C f ( z , ˆ C ) n ˆ C λ ( z ) − n ˆ C ( − N − n ˆ C , (E3)which leads to lim | z |→∞ f ( z , ˆ C ) λ ( z ) = 1 . (E4)8In Eq. (77), the constant max (cid:96) r (cid:96) is added to the denominator without harm, in order to make the ratio positiveeverywhere. The essential ingredient in this proof is the uniqueness of the optimal cycle ˆ C . Only in peculiar regionsthe vector z leads to more than one cycle with the same value of m C · z /n C . For example, these regions show up inFig. 9 as the lines along which the ratio (77) differs from 1. [1] D. J. Evans, E. G. D. Cohen, and G. P. Morriss, “Proba-bility of second law violations in shearing steady states,”Phys. Rev. Lett. , 2401 (1993).[2] D. J. Evans and D. J. Searles, “Equilibrium microstateswhich generate second law violating steady states,” Phys.Rev. E , 1645 (1994).[3] G. Gallavotti and E. G. D. Cohen, “Dynamical ensem-bles in nonequilibrium statistical mechanics,” Phys. Rev.Lett. , 2694 (1995).[4] J. Kurchan, “Fluctuation theorem for stochastic dynam-ics,” J. Phys. A: Math. Gen. , 3719 (1998).[5] J. L. Lebowitz and H. Spohn, “A Gallavotti-Cohen-typesymmetry in the large deviation functional for stochasticdynamics,” J. Stat. Phys. , 333 (1999).[6] U. Seifert, “Stochastic thermodynamics, fluctuation the-orems, and molecular machines,” Rep. Prog. Phys. ,126001 (2012).[7] T. Speck, V. Blickle, C. Bechinger, and U. Seifert, “Dis-tribution of entropy production for a colloidal particle ina nonequilibrium steady state,” EPL , 30002 (2007).[8] J. Mehl, B. Lander, C. Bechinger, V. Blickle, andU. Seifert, “Role of hidden slow degrees of freedom inthe fluctuation theorem,” Phys. Rev. Lett. , 220601(2012).[9] J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto,“Heat fluctuations in a nonequilibrium bath,” Phys. Rev.Lett. , 200602 (2011).[10] S. Ciliberto, A. Imparato, A. Naert, and M. Tanase,“Heat flux and entropy produced by thermal fluctua-tions,” Phys. Rev. Lett. , 180601 (2013).[11] J. P. Pekola, “Towards quantum thermodynamics in elec-tronic circuits,” Nat. Phys. , 118–123 (2015).[12] K. Hayashi, H. Ueno, R. Iino, and H. Noji, “Fluctuationtheorem applied to F -ATPase,” Phys. Rev. Lett. ,218103 (2010).[13] K. Feitosa and N. Menon, “Fluidized granular mediumas an instance of the fluctuation theorem,” Phys. Rev.Lett. , 164301 (2004).[14] N. Kumar, S. Ramaswamy, and A. K. Sood, “Symmetryproperties of the large-deviation function of the velocityof a self-propelled polar particle,” Phys. Rev. Lett. ,118001 (2011).[15] S. Joubaud, D. Lohse, and D. van der Meer, “Fluctua-tion theorems for an asymmetric rotor in a granular gas,”Phys. Rev. Lett. , 210604 (2012).[16] D. Andrieux and P. Gaspard, “A fluctuation theorem forcurrents and non-linear response coefficients,” J. Stat.Mech. P02006 (2007).[17] A. C. Barato and U. Seifert, “Thermodynamic uncer-tainty relation for biomolecular processes,” Phys. Rev.Lett. , 158101 (2015).[18] A. C. Barato and U. Seifert, “Universal bound on thefano factor in enzyme kinetics,” J. Phys. Chem. B , 6555–6561 (2015).[19] A. C. Barato and U. Seifert, “Skewness and kurtosis instatistical kinetics,” Phys. Rev. Lett. , 188103 (2015).[20] R. S. Ellis, Entropy, Large Deviations, and StatisticalMechanics (Springer-Verlag, Berlin, 2006).[21] H. Touchette, “The large deviation approach to statisti-cal mechanics,” Phys. Rep. , 1–69 (2009).[22] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio,and C. Landim, “Fluctuations in stationary nonequilib-rium states of irreversible processes,” Phys. Rev. Lett. , 040601 (2001).[23] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, andC. Landim, “Macroscopic fluctuation theory for station-ary non-equilibrium states,” J. Stat. Phys. , 635–675(2002).[24] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, andC. Landim, “Macroscopic fluctuation theory,” Rev. Mod.Phys. , 593–636 (2015).[25] T. Bodineau and B. Derrida, “Current fluctuations innonequilibrium diffusive systems: An additivity princi-ple,” Phys. Rev. Lett. , 180601 (2004).[26] B. Derrida, “Non-equilibrium steady states: fluctuationsand large deviations of the density and of the current,”J. Stat. Mech. P07023 (2007).[27] P. I. Hurtado and P. L. Garrido, “Test of the additiv-ity principle for current fluctuations in a model of heatconduction,” Phys. Rev. Lett. , 250601 (2009).[28] P. I. Hurtado and P. L. Garrido, “Large fluctuations ofthe macroscopic current in diffusive systems: A numer-ical test of the additivity principle,” Phys. Rev. E ,041102 (2010).[29] M. Gorissen and C. Vanderzande, “Current fluctuationsin the weakly asymmetric exclusion process with openboundaries,” Phys. Rev. E , 051114 (2012).[30] J. Schnakenberg, “Network theory of microscopic andmacroscopic behavior of master equation systems,” Rev.Mod. Phys. , 571 (1976).[31] A. C. Barato and R. Ch´etrite, “On the symmetry ofcurrent probability distributions in jump processes,” J.Phys. A: Math. Theor. , 485002 (2012).[32] Z. Koza, “General technique of calculating the drift ve-locity and diffusion coefficient in arbitrary periodic sys-tems,” J. Phys. A: Math. Gen. , 7637 (1999).[33] R. Chetrite and H. Touchette, “Nonequilibrium micro-canonical and canonical ensembles and their equiva-lence,” Phys. Rev. Lett. , 120601 (2013).[34] A. C. Barato, R. Chetrite, H. Hinrichsen, andD. Mukamel, “Entropy production and fluctuation rela-tions for a KPZ interface,” J. Stat. Mech. P10008 (2010).[35] A. C. Barato, R. Chetrite, H. Hinrichsen, andD. Mukamel, “A Gallavotii-Cohen-Evans-Morriss likesymmetry for a class of Markov jump processes,” J. Stat.Phys. , 294–313 (2012). [36] See Supplementary Material of Ref. [17].[37] J. Mehl, T. Speck, and U. Seifert, “Large deviation func-tion for entropy production in driven one-dimensionalsystems,” Phys. Rev. E , 011123 (2008).[38] S. Dorosz and M. Pleimling, “Entropy production in thenonequilibrium steady states of interacting many-bodysystems,” Phys. Rev. E , 031107 (2011).[39] T. Speck, A. Engel, and U. Seifert, “The large devia-tion function for entropy production: the optimal trajec-tory and the role of fluctuations,” J. Stat. Mech. P12001(2012).[40] E. Roldan and J. M. R. Parrondo, “Estimating dissipa-tion from single stationary trajectories,” Phys. Rev. Lett. , 150607 (2010).[41] E. Roldan and J. M. R. Parrondo, “Entropy productionand Kullback-Leibler divergence between stationary tra-jectories of discrete systems,” Phys. Rev. E , 031129 (2012).[42] J. R. Moffitt and C. Bustamante, “Extracting signal fromnoise: kinetic mechanisms from a michaelismenten-likeexpression for enzymatic fluctuations,” FEBS J. ,498–517 (2014).[43] M. Baiesi, C. Maes, and B. Wynants, “Fluctuations andresponse of nonequilibrium states,” Phys. Rev. Lett. ,010602 (2009).[44] R. B. Bapat and T. E. S. Raghavan, Nonnegative Matri-ces and Applications (Cambridge University Press, Cam-bridge, England, 1997).[45] P. I. Hurtado and P. L. Garrido, “Spontaneous Symme-try Breaking at the Fluctuating Level,” Phys. Rev. Lett. , 180601 (2011).[46] T. R. Gingrich, J. M. Horowitz, N. Perunov, andJ. L. England, “Dissipation bounds all steady-state cur-rent fluctuations,” Phys. Rev. Lett.116