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[ for different dynamics potentials, with non-Markovian chemical reservoirs by or may generated temperature which or forces sub- mechanical nonconservative system of be action nonequilibrium the a to jected describing process jump h atrcnaiei rsneo eknie[ noise weak of presence in fluctuations. arise weak can by latter affected The system equilibrium from ieabudfrtert fteprocess the of rate the for bound a rive olwn rmwr.W nrdc h tpigtime stopping the introduce We [ τ systems framework. open following of trajectories tum [ theory state transition in found reservoirs the of duration average obtain process the we the between rate, relation constant uncertainty a an posses that processes librium utainter [ theory fluctuation fidb h uvvlpoaiiyta h rcs snot is process time the at that completed probability survival yet the by ified i ( notersror n h entime mean the and reservoirs the into =inf := a etemnmmtm odslc as rto or mass, a displace to time minimum the be may enx dniytesaeo srie’taetre at trajectories ‘survived’ of space the identify next We esatb osdrn tcatctrajectories stochastic considering by start We eso nti etrta h ispto ln suf- alone dissipation the that Letter this in show We ω t | t x to sanvlfr fsedlmt the limit: speed of form novel a is ation Ω , 0 t ) cess. { rcse epoetefundamental the prove we processes e ρ t t s ihasainr rbblt measure probability stationary a with t ( uhta if that such , als fstates of list —a x ≥ 0 T x ,with ), yisadMtrasScience, Materials and hysics ia rcse a eperformed be can processes sical : 0 h 0 O S 1 = ecntikof think can We . O ˙ D e Ω : h threshold the , 20 O i hsnvlsedlmtapist far to applies limit speed novel This . ⊂ ( /r 27 ]. t ω ρ R → ( t ,adwe h rcse sstby set is processes the when and ], † x ) n h endsiainrt in rate dissipation mean the and osl eerdt s‘threshold’. as to referred loosely , 0 t R ∈ ω h ttoaypoaiiyden- probability stationary the ) t [ orahvle eogn oa to belonging values reach to 24 D ∈ 28 t } .Te,frrr nonequi- rare for Then, ]. sepesdb h survival the by expressed is Ω x , stemnmmtm for time minimum the as t t s 29 ′ D then with n h time the and , rurvle quan- unravelled or ] 26 ω 30 t 20 ri macroscopic in or ] sadffso ra or diffusion a as a tit the into fit may ] O – T t r 23 ( ′ nqeyspec- uniquely , ω to .W rtde- first We ]. t ∈ ) 6∈ [0 D t , τ P —na ]—in They . define , 25 ( ω ω ,as ], t t = ) of 2 probability ps(t) := Prob(τ >t), which satisfies ps(0) = 1 and analogously forr ˜(t). Because (5) holds for any pos- by definition. We can formally write it as itive time t′, we find that the entropy production rate bounds the difference of the process rates s p (t)= P (ωt)= χ(O(ωt))P (ωt) (1) s 1 ωt∈Ω ωt∈Ωt ˙ ′ ′ ′ X t X Σ ˜s(t ) r(t ) r˜(t ). (7) kB h i ≥ − where χ(O(ωt)) equals 1 if O(ωt) D and zero otherwise. 6∈ When the mean time of the process exists, i.e. when We then consider the (involutive) transformation ωt 7→ τ = ∞ dt ps(t) =: is finite [24], (7) can be turned ω˜t that time-reverses the order of the states xt′ (possibly 0 hintoi a bound on .T First, we assume that both pro- changing sign, according to their parity). This allows to R T define the log ratio cesses are rare so that their survival probabilities read ps(t) = e−rt andp ˜s(t) = e−rt˜ , which implies = 1/r. t Second, we choose the threshold D such thatr ˜ T r. This P (ωt) ′ ˙ kB log =: dt Σ(ωt). (2) implies thatp ˜s(t′) = 1 for all times t′ 1/r˜ and≪ the en- P (˜ωt) Z0 tropy production rate coincides with the≪ (constant) mean We note that P (˜ωt) = P (˜ωt x˜0)ρ(˜x0) is the probability entropy flux of the stationary dynamics, Σ˙ ˜s = S˙e . measure of time-reversed trajectories| evolving with the Under these assumptions, (7) simplifies toh ouri mainh re-i original dynamics and starting from the stationary prob- sult: the dissipation-time uncertainty principle ability distribution ρ(˜x0). If the dynamics obeys local ˙ ˙ ′ S˙e kB. (8) detailed balance [31, 32], Σ= Se + dS/dt is the entropy h iT ≥ production rate at time t′, which splits into the entropy The fundamental implication of this result is that to re- flux in the reservoirs, S˙ , plus the time derivative of the e alize a rare nonequilibrium process in a given (average) system entropy S = log ρ(x ′ ). t time at least k / must be dissipated in the reser- Applying the time-reversal− to (A1) and using (2) we B voirs.T Rare processesT are effectively Poissonian processes find on long timescales compared to the fast microscopic dy- t ′ ˙ s − R dt Σ(ωt)/kB namics. They arise in two broad classes of problems. p (t)= χ(O(˜ωt))e 0 P (ωt) (3) First in the presence of weak noise, which is typically the ωt∈Ωt X case for problems described by transition state theory, or after the relabelingω ˜ ω . Notice that the sum in by macroscopic fluctuation theory where fluctuations are t → t (A10) is restricted by χ(O(˜ωt)) to a subset of trajecto- exponentially suppressed in the system size. Second, for ˜ s s ˜ ries Ωt which differs from Ωt if O(ωt) := O(˜ωt) = O(ωt). large thresholds, i.e. when the domain D can be reached This defines a different process, named reversed6 pro- only by atypical fluctuations. We will illustrate these two cess, whose associated survival probability isp ˜s(t) := cases on paradigmatic models. ˜ The first example represents overdamped particle ωt∈Ωt χ(O(ωt))P (ωt). Hence, we arrive at the modi- fied integral fluctuation relation transport in one spatial dimension. The dynamics fol- P lows the Langevin equation s s − R t dt′Σ˙ /k p (t)=˜p (t) e 0 B . (4) ˜s x˙ = U ′(x)+ 2/βξ (9) D E − s Hereafter, F := ˜ s F (ωt)P (ωt)/p˜ (t) denotes the p h i˜s ωt∈Ωt with periodic boundary conditions in x [0, 2π]. Here, −1 ∈ normalized average of the generic observable F on the (kBβ) denotes temperature, ξ is a zero-mean Gaussian P ˜ s set of survived trajectories Ωt. One should note that (4) white noise of unit variance, and U(x) = a cos(x) fx appears in implicit form in Ref. [33–35] as a general- is a periodic potential superimposed to a constant− non- ized fluctuation theorem holding when a subset of for- conservative tilt f > 0. This model describes a wealth of ward trajectories have no time-reversal equivalent [36]. transport processes ranging from loaded molecular mo- Our crucial new ingredient is to define Ωs and Ω˜ s via the tors [38] to electrons across Josephson junctions [39]. For choice of an observable and a threshold, and to assign weak noises, (8) applies in the stationary regime for the stopping times to trajectories in that subset. process of transporting the coordinate x over N > 0 pe- As a consequence of Jensen’s inequality, (4) yields riods, i.e.
t ′ s s − R dt hΣ˙ i˜s/kB t p (t) p˜ (t)e 0 , (5) ′ ≥ O = x˙ t′ dt ,D = [2πN, ), (10) ∞ which gives a bound on the pace at which the two pro- Z0 cesses proceed. Since survival probabilities are positive Indeed, when β−1 is small with respect to the smaller and monotonically decreasing, one can define the instan- energy barrier ∆Umin, escaping from the tilted potential taneous rate r(t) of the process as [37] well is a weak noise problem [40]. Namely, the process s (10) unfolds with a single rate, which can be roughly es- 1 dp timated as the product of N Arrhenius factors, i.e. r r(t) := s (t), (6) −p (t) dt e−Nβ∆Umin (see Fig. 1). Similarly, the reversed process∼ 3
10−2
103 −3 −1 10 10 0 10−6 10 −1 −15 10 10−4 10 10−2 −2 t −5 1 ′ ˙ ′ 10 10 t R dt hSei˜s(t ) t 0 0 0.2 0.4 0.6 0.8 1 1 ′ ˙ ′ 2 4 6 8 10 1 t ′ ′ t R0 dt hSei˜s(t ) t R0 dt r(t ) f 1 t ′ ′ ∆β t R0 dt r(t ) 10−6 10−3 1,000 2,000 3,000 4,000 0 100 200 300 400 500 t t
FIG. 1. Speed limit for the dynamics (9) with process (10), FIG. 2. Speed limit for the two-state system of the main obtained by numerical averages over 2 · 103 trajectories with text with process (12), obtained by numerical averages over −1 5 a = 1, β = 0.07, f = 0.6, N = 2, kB = 1. Dashed and 10 Gillespie trajectories with 1/βh = 1.5, 1/βc = 0.5, ǫ2 = dotted lines correspond to the analytical estimation of time- 1, ǫ1 = 0,E = 5, δ = 6,kB = 1. The dashed line corresponds ˙ integrated hS˙ei and r, respectively, while the dots denote their to hSei. At short times t . 50 the numerical estimation of numerical values. At short times t . 103 the numerical esti- r(t) is impeded by the finite statistics. Inset: time-averaged mation of r(t) is impeded by the finite statistics. Inset: The value of hS˙ei˜s and process rate r as function of ∆β := βc − βh analytical approximation of hS˙ei (dashed) and r (dotted) as at fixed average temperature (1/βc + 1/βh)/2 = 1. function of the tilt f.
bath ν = c: defined by the trajectories realizing a negative current δ c c dN2→1 dN1→2 ˜ −Nβ∆Umax ′ O = 2πN takes place with rater ˜ e r O = (ǫ2 ǫ1) dt ′ ′ ,D = [E, ), − ∼ ≪ − 0 dt − dt ∞ (where Umax is the larger energy barrier), which is negli- Z (12) gible for sufficiently large f. Thus, the uncertainty prin- ciple (8) applies with the stationary entropy flow given One may think of E as an activation energy (e.g., of re- by action [42]) and of δ as the timescale over which it may be dissipated. For large δ−1 and/or E > 0 (with respect ν S˙ = k fβ f +a dxρ(x) sin(x) . (11) to the rates wi→j and the energy gap ∆ǫ := ǫ1 ǫ2 , h ei B | − | Z respectively) the process is realized by large fluctuations and, therefore, is rare. The rate of the reverse process, Here ρ is the stationary probability distribution asso- defined by extracting an energy larger than E in the time ciated to (9), which is well approximated by the local- δ from the cold reservoir, is in comparison negligible. −βU(x) equilibrium distribution ρ(x) e (see Fig. 1)[41]. Since the rate r and the entropy flow S˙e ˜s S˙e = ∼ ∆ǫβ ∆ǫβ ∆ǫβ h∆ǫβi ≃ h i Note however that these conditions on r and S˙ break h c h + c −1 h ei kB(1/βc 1/βh)(e 2 e 2 )(e 2 2 + 1) are down in two limits, when f 1 and when f 0. When constant,− as shown in Fig.− 2, equation (8) holds true. f 1, i.e. the value for which→ ∆U 0,→ the weak → min → These two examples suggest that currents are natural noise assumption becomes invalid, so the process is no observables to which our theory applies. For integrated longer rare. When f 0, i.e. close to detailed balance ˜ → currents, i.e. O(ωt)= O(ωt), our results can also be di- dynamics, the reverse process is no longer negligible, so rectly derived from the− steady state fluctuation theorem that (7) cannot be simplified to (8). Nonetheless, within [36]. Note that in a similar setting, i.e. a large threshold its range of validity, the analytical estimation shows that for O, a complementary result based on large deviations (8) becomes tighter as the distance from equilibrium in- theory is known, which holds when O is an integrated creases (see inset in Fig. 1). current [20] (resp. counting observable [21]) scaling lin- The second example represents energy transfer be- early with t. It lower bounds 2 by the variance of τ T tween two heat baths (at inverse temperatures kBβh and times the entropy production (resp. the dynamical ac- kBβc, respectively) mediated by a two-level system. The tivity). Our theory, instead, provides an upper bound latter performs Markovian jumps (corresponding to Pois- only based on entropy production and allows to consider son processes dN ν ) between the two states i = 1, 2 i→j { } more general observables O, e. g., that are not extensive ν −βν (ǫj −ǫi)/2 of energy ǫi with rates wi→j = e associated in the trajectory duration t. However, our theory does to the baths ν = h, c . We define the process as the not impose any constraint on processes defined by time { } s transfer of an energy E in a fixed time δ into the cold symmetric observables, i.e. O˜(ωt)= O(ωt). For them, p 4 andp ˜s coincide so that (4) and (5) only give (for survived tightness of the bound [44]. Concerning tightness, it is trajectories) the usual integral fluctuation theorem and known from the theory of deterministic dynamical sys- the positivity of entropy production, respectively. tems that escape rate is proportional to the dissipation We then come back to the general theory, seeking ex- rate in open Hamiltonian systems where particle leak- tensions of the inequalities (7) and (8) beyond station- age on large scales is compatible with a drifted diffusive arity. Systems subject to time-dependent driving can be process [45, 46]. This result can be obtained within our treated with a slight modification of the above deriva- framework considering the dynamics (9) with a = 0 and tion which includes time-reversal of the driving protocols the process of transporting the coordinate x to either 0 λ(t′) λ(t t′) both in the dynamics and in the ini- or L. Since Ωs = Ω˜ s for such process, (8) does not di- tial probability7→ − of time-reversed trajectories [43], which rectly apply but it can be easily generalized by using an we take as the periodic steady state ρλ(t)(xt) at the final appropriate auxiliary dynamics in (A10), which is differ- value of the protocol λ(t). Then, (7) generalizes to ent from time reversal [36]. Actually, this general strat- egy can be applied to derive system-specific bounds for 1 ˙ B ′ F ′ B ′ processes that break symmetries other than time-reversal Σ ˜s (t ) r (t ) r˜ (t ), (13) kB h i ≥ − (such as reflection). which bounds the difference of the rates of the for- In summary, resorting to the concept of first passage ward and backward process with the dissipation in the time we have defined the physical process that a stochas- backward dynamics. Importantly, (7) is recovered for tic system can perform. This allowed us to describe sta- time-symmetric driving protocols for which forward and tionary (resp., periodic) processes, that do not involve backward dynamics coincide. Moreover, rare processes (resp., any net) transformation of states, and to define with sufficiently fast driving of small amplitude are still unambiguously the pace at which the process advances. characterized (for t′ much larger than the driving pe- Irrespective of the stochastic dynamics, we have pro- riod t ) only by the constant time-averaged rater ¯ := d vided an upper bound on the rate at which a process 1 td dt′r(t′) −1, so that ps(t) e−tr¯, and by the td 0 ≃ T ≃ is performed. This result becomes particularly useful for ˙ 1 td ′ ˙ ′ entropyR flow rate Se = dt Se (t ). For such pro- rare processes—e.g. realized by weak fluctuations or de- h i td 0 h i cesses, (8) holds in the integrated form fined by large thresholds—far from equilibrium, where R the bound reduces to the entropy flow in the reservoirs. S˙ k , (14) This result suggests that the connection between our h eiT ≥ B framework and the escape-rate theory of deterministic if the threshold D is chosen such that the rate of the systems [47] should be deepened, in particular for hyper- reversed process is negligible. Our approach still holds bolic ones where the role of weak noise is taken up by in the most general case of driven transient dynamics chaos to produce a constant escape rate r. Furthermore, with arbitrary threshold. Eqs. (5) and (13) remain valid it calls for an extension of stochastic thermodynamics to but their use can be impractical. On one hand, for pro- systems with escape, given the recent work on the second cesses with a survival probability decaying as a power- law at stopping times [23], and the renewed fundamental law ps(t) 1/tγ, the definition (A15) is rather ad hoc interest in unstable dynamics [48, 49]. since no characteristic∼ timescale exists (the average time even diverges for γ 1). On the other hand, the Finally, the dissipation-time uncertainty relation, to- entropyT production rate≤ is hard to estimate, being de- gether with the recent thermodynamic uncertainty rela- pendent on the survival probability itself. This clarifies tions show manifestly how thermodynamics constrains how our results are complementary to the other classical nonequilibrium dynamics. It also hints at a general speed limits, since our formalism is specifically tailored to emerging tradeoff between speed, precision, accuracy and different processes, which are marked by transformations dissipation [50–53], in which the role of information [54– in the reservoirs instead of in the system. 57] only awaits to be explicitly uncovered. Our main result, (8), asserts that a large dissipation allows for a fast process. But this does not imply that in- This research was funded by the European Research creasing dissipation will necessarily speed up the process. Council project NanoThermo (ERC-2015-CoG Agree- As for thermodynamic uncertainty relations [7, 8], kinet- ment No. 681456). We thank Ptaszy´nski Krzysztof for ics aspects of the dynamics are essential to determine the valuable comments on an earlier draft of this work.
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Appendix A: Derivation of the bound
′ We consider stochastic trajectories ω = x ′ : 0 t t of duration t corresponding to an ordered list of states t { t ≥ ≥ } xt′ . On the trajectory space Ωt ωt we consider the probability measure P (ωt) = P (ωt x0)ρ(x0), with ρ(x0) the stationary probability density of state∋ x . The stopping time τ := inf t 0 : O(ω ) D is| the minimum time for an 0 { ≥ t ∈ } observable O :Ωt R to reach values belonging to a specific domain D R, loosely referred to as ‘threshold’. The observable O, the→ threshold D, and the time τ, define the physical process⊂ and its duration. s s We identify the space of ‘survived’ trajectories at time t,Ωt, such that if ωt Ωt then O(ωt) D. They correspond to trajectories in which the process is not completed. The associated probability∈ that the process6∈ is not yet completed at time t is expressed by the survival probability ps(t) := Prob(τ >t), which by definition satisfies ps(0) = 1. We can formally write it as
s p (t)= P (ωt)= χ(O(ωt)), P (ωt). (A1) s ωXt∈Ωt ωXt∈Ωt where χ(O(ωt)) is an indicator function giving 1 if the process defined by O is not complete and 0 otherwise:
1 if O D χ(O)= 6∈ (A2) 0 if O D ( ∈ ′ We then reweight the average in (A1) considering the time reversed trajectoriesω ˜t = θxt−t′ : 0 t t , where θ accounts for the parity of x: { ≥ ≥ }
s P (ωt) p (t)= χ(O(ωt)) P (˜ωt). (A3) P (˜ωt) ωXt∈Ωt Here P (˜ω ) = P (˜ω x˜ )ρ(˜x ) is the probability of time reversed trajectories starting from the stationary probability t t| 0 0 ρ(˜x0). When the dynamics satisfies local detailed balance, we can introduce the entropy production rate Σ(˜˙ ωt) of the stationary time-reversed dynamics as
P (˜ωt|x˜0) P (ωt) − log +log ρ(˜xt)−log ρ(˜x0) = e P (ωt|x0) (A4) P (˜ωt) = e−(Se(˜ωt)+∆S(˜ωt))/kB (A5) 1 t ˙ dS − R (Se+ ′ )(˜ωt) = e kB 0 dt (A6) 1 t − R Σ(˜˙ ωt) = e kB 0 . (A7)
Here we have used the decomposition of Σ˙ at time t′ into the entropy flow in the reservoirs, S˙ (˜ω ) := k log P (˜ωt|x˜0) , e t B P (ωt|x0) and the derivative of the system entropy S := log ρ(xt′ ). Using (A7) and rewriting all terms appearing− in (A3) in terms of reversed trajectories we find
1 t ˙ s − k R Σ(˜ωt) p (t)= χ(O(ωt))e B 0 P (˜ωt) (A8) ωXt∈Ωt 1 t ˙ − k R Σ(˜ωt) = χ(O(ωt))e B 0 P (˜ωt) (A9) ω˜Xt∈Ωt 1 t ˙ − k R Σ(ωt) = χ(O˜(ωt))e B 0 P (ωt). (A10) ωXt∈Ωt
For the second equality we used that ωt = ω˜t and for the third we renamedω ˜t as ωt and defined the time reversed observable O˜(ω ) := O(˜ω ). For example, if O is an integrated current in [0,t], then O˜(ω ) = O(ω ) and t t P P t t ˜ s s − the set of time-reversed survived trajectories Ωt is different from Ωt. If, instead, the observable is time-symmetric, ˜ s s O(˜ωt)= O(ωt), we have Ωt =Ωt. In order to use Jensen’s inequality, we define the normalization of the average in (A10),
s p˜ (t)= χ(O˜(ωt))P (ωt)= P (ωt), (A11)
ωt∈Ωt ˜ s X ωXt∈Ωt 7 which is the survival probability of the process defined by O˜. Then we can normalize the average in (A10) multiplying and dividing by (A11),
1 t ˙ − k R Σ(ωt) χ(O˜(ωt))e B 0 P (ωt) ps(t)=˜ps(t) ωt∈Ωt . (A12) χ(O˜(ω ))P (ω ) P ωt∈Ωt t t P Defining the normalized average of a generic observable F (ωt) on ‘survived’ trajectories of the reversed process as
1 F := χ(O˜(ω )) F (ω ) P (ω ), (A13) h i˜s p˜s(t) t t t ωXt∈Ωt equation (A12) reads
t ′ 1 t ′ ˙ s s − R dt Σ˙ /k s − R dt hΣi˜s p (t)=˜p (t) e 0 B p˜ (t)e kB 0 , (A14) ˜s ≥ D E where we used Jensen’s inequality in the last passage. Since survival probabilities are positive and monotonically decreasing, one can define the instantaneous rates of the processes as
1 dps 1 dp˜s r(t) := (t), r˜(t) := (t). (A15) −ps(t) dt −p˜s(t) dt
Since (A14) holds for all times t, it gives the bound
1 ′ ′ ′ Σ˙ ˜s(t ) r(t ) r˜(t ). (A16) kB h i ≥ −
It is worth noticing the relation with previous results involving absolutely irreversible dynamics, in which some trajectories lack their time-reversed ones. In Ref. [33] the authors exponentiate also the indicator function χ which select absolutely continuous trajectories, namely,
1 t ˙ s − k R Σ(ωt)+log χ(O(˜ωt)) p (t)= e B 0 P (ωt). (A17) ωXt∈Ωt s log χ(O(˜ω ))P (ω )= Application of Jensen’s inequality in (A17) would give a trivial bound on p (t) because ωt t t due to the ‘absorbed’ trajectories. This is explicitly stated in their last example where a Langevin dynamics with−∞ absorbing condition is considered. In Refs. [34] and [35] the authors do not exponentiatePχ and so keep the sum over a subset of trajectories, similarly to our approach. Application of Jensen’s inequality, requires to normalize the average to 1, and so to introduce the survival probability of the reversed process,p ˜s(t). It is unclear whether in Refs. [34] and [35] Jensen’s inequality is applied to the unnormalized average, which would be erroneous, or if the implicitly definition of average therein contains the normalization factorp ˜s(t). Hence, to the best of our knowledge, the modified integral fluctuation theorem and the bound in (A12) never appeared before in a clear explicit form. We conclude our proof of the dissipation-time uncertainty relation introducing 3 hypotheses:
i) Both processes (defined by O and O˜, respectively) are rare. This means that their survival probabilities read ps(t)= e−rt andp ˜s(t)= e−rt˜ .
ii) The choice of D is such that the rates satisfyr ˜ r. Then, for times t 1/r˜,p ˜s(t) 1 and F coincides with ≪ ≪ ≃ h is˜ the unconstrained mean F := F (ωt)P (ωt). h i ωt∈Ωt iii) The mean entropy production containsP an entropy flow contribution which is extensive in time on the above mentioned timescale, namely,
t dt′ Σ˙ = S˙ t. (A18) h i h ei Z0
Hence, (A16) turns into our final result S˙ k . h eiT ≥ B 8
Appendix B: Alternate derivation for integrated currents
We consider a system in a steady state for which the fluctuation relation holds for the vector of time-averaged currents J in the (long) time span t,
P (J)= P ( J)etA·J , (B1) − where P (J) is the probability density of the current J and A is the vector of thermodynamic affinities (in units of ∗ kB). We define the process of realizing a scalar current larger than Ji , namely, we set the observable equal to the ith ∗ component of the vector J, O = Ji, and take D = [Ji , + ). The survival probability, i.e. the probability that the ∗ ∞ current Ji was not yet realized in the time t, is
ps(t)= Θ(J ∗ J (ω ))P (ω ) (B2) i − i t t ωXt∈Ωt = Θ(J ∗ J (ω )) dJδ(J J(ω )) P (ω ) (B3) i − i t − t t ωt∈Ωt Z X =1 = dJΘ(J ∗ J ) δ|(J J({zω ))P (ω }) (B4) i − i − t t ω Z Xt =:P (J)
= dJP (J). | {z } (B5) J ps(t)= dJP ( J)etA·J (B6) J Appendix C: Entropy flow and escape rate in drifted diffusion We consider within our theory the results of [45, 58] on the proportionality between escape rate and entropy production rate in large diffusive systems with constant drift. First, we reproduce the direct analytical calculation of the two quantities. Then, we show that their proportionality does not descends from the Eq. (8) of the main text, but instead it corresponds to a saturated bound obtained by comparison with an auxiliary dynamics other then time-reversal. We examine the Langevin dynamics x˙ = f + 2/βξ, (C1) p 9 where f is a constant drift and ξ a Gaussian white noise with zero mean and unit variance. We impose absorbing conditions in x = 0 and x = L> 0 which amounts to consider the process O = x ′ D = ( , 0] [L, ) (C2) t −∞ ∪ ∞ ∞ −rnt The associated state probability at time t is given by the series ρt(x)= n=1 cne exp(fxkBβ/2) sin(nπx/L) with 2 2 2 2 decay rates rn = f βkB/4+ n π /(βkBL ) and coefficients cn depending on the initial conditions [58]. In the limit of weak noise, i.e. β , and/or large system size L , all modes n P= 1 are irrelevant and the probability decays → ∞ 2 → ∞ 6 with a single rate r1 = f βkB/4=1/ . This is 4 times smaller than the mean entropy flow associated to (C1) in T 2 absence of absorbing conditions, i.e. S˙e = f kBβ. Note that Ref. [45] incorrectly states the equality of escape rate and dissipation rate, which is due (inh ouri notation) to the erroneous additional factors 2 and 1/2 in the escape rate and entropy flow, respectively. Even if the bound S˙e =4kB kB is satisfied, our derivation does not apply since the process (C2) is invariant under time-reversal, i.e.h iTps =p ˜s. However,≥ the derivation can be modified considering the auxiliary dynamics given † by (C1) with f = 0. The probability P (ωt) of the associated trajectories can be used to write the survival probability of the process s P (ωt) † p (t)= χ(O(ωt)) † P (ωt) (C3) P (ωt) ωXt∈Ωt − 1 f 2βt+ 1 fβ R t dt′x˙ † = χ(O(ωt))e 4 2 0 P (ωt) (C4) ωXt∈Ωt s − 1 f 2βk t+ 1 fβk R t dt′hx˙ i p˜ (t)e 4 B 2 B 0 ˜s (C5) ≥ s χ(O(ω ))P †(ω ) is the survival probability of the (auxiliary) process ( wherep ˜ (t) := ωt∈Ωt t t C2) on the dynamics † s without drift, and F := χ(O(ωt))F (ωt)P (ωt)/p˜ (t) is the normalized average over the same dynamics. In P h i˜s ωt∈Ωt the limit β and/or L the processes are rare. In particular, the process without drift has rate r2 r1 so thatp ˜s(t) =→ 1 and∞ x˙ = xP˙→= ∞ 0. Hence, equation (C5) yields ≪ h i˜s h i s −r t − 1 f 2βk t p (t)= e 1 e 4 B . (C6) ≥ This is the system-specific bound f 2β = S˙ /k 4r =4/ that we found above to be realized in equality. h ei B ≥ 1 T