The Dissipation-Time Uncertainty Relation

The dissipation-time uncertainty relation Gianmaria Falasco∗ and Massimiliano Esposito† Complex Systems and Statistical Mechanics, Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg (Dated: March 16, 2020) We show that the dissipation rate bounds the rate at which physical processes can be performed in stochastic systems far from equilibrium. Namely, for rare processes we prove the fundamental tradeoff hS˙eiT ≥ kB between the entropy flow hS˙ei into the reservoirs and the mean time T to complete a process. This dissipation-time uncertainty relation is a novel form of speed limit: the smaller the dissipation, the larger the time to perform a process. Despite operating in noisy environments, complex sys- We show in this Letter that the dissipation alone suf- tems are capable of actuating processes at finite precision fices to bound the pace at which any stationary (or and speed. Living systems in particular perform pro- time-periodic) process can be performed. To do so, we cesses that are precise and fast enough to sustain, grow set up the most appropriate framework to describe non- and replicate themselves. To this end, nonequilibrium transient operations. Namely, we unambiguously define conditions are required. Indeed, no process that is based the process duration by the first-passage time for an ob- on a continuous supply of (matter, energy, etc.) currents servable to reach a given threshold [20–23]. We first de- can take place without dissipation. rive a bound for the rate of the process r, uniquely spec- Recently, an intrinsic limitation on precision set by dis- ified by the survival probability that the process is not sipation has been established by thermodynamic uncer- yet completed at time t [24]. Then, for rare nonequi- tainty relations [1–6]. Roughly speaking, these inequali- librium processes that posses a constant rate, we obtain ties state that the squared mean-to-variance ratio of cur- an uncertainty relation between the average duration of rents is upper bonded by (a function of) the entropy pro- the process = 1/r and the mean dissipation rate in T duction. Despite producing loose bounds for some spe- the reservoirs S˙e . This novel speed limit applies to far cific models [7, 8], their fundamental importance is unde- from equilibriumh i system affected by weak fluctuations. niable as they demonstrate that thermodynamics broadly The latter can arise in presence of weak noise [25], as constrains nonequilibrium dynamics [9]. found in transition state theory [26] or in macroscopic For speed instead, an equivalent limitation set by dis- fluctuation theory [27], and when the processes is set by sipation can only be speculated. For example, we know large thresholds [20]. from macroscopic thermodynamics that thermodynamic We start by considering stochastic trajectories ωt of machines will produce entropy to deliver finite power. ′ duration t—a list of states x ′ with t [0,t]—in a Yet, a constraint on a par with thermodynamic uncer- t space Ω with a stationary probability measure∈ P (ω )= tainty relations, only based on dissipation, is still lacking. t t P (ω x )ρ(x ), with ρ(x ) the stationary probability den- Efforts in this direction have appeared lately [10, 11], t 0 0 0 sity of| state x . We can think of ω as a diffusion or a inspired by research on quantum speed limits which are 0 t jump process describing a nonequilibrium system sub- bounds on the time needed to transform a system from jected to the action of nonconservative forces which may one state into another [12]. When extended to classical be mechanical or generated by reservoirs with different stochastic dynamics, these relations acquire a somewhat temperature or chemical potentials, for instance. Also formal appearance [13–16]. In their most explicit form non-Markovian dynamics [28, 29] or unravelled quan- they bound the distance between an initial state and tum trajectories of open systems [30] may fit into the a final one at time t—technically, the 1-norm between following framework. We introduce the stopping time the two probability distributions—in terms of the chosen τ := inf t 0 : O(ω ) D as the minimum time for time t, the dissipation, and other kinetic features of the t an observable{ ≥ O :Ω R∈ to} reach values belonging to a system [13]. However, many systems of interest, espe- t specific domain D R→, loosely referred to as ‘threshold’. arXiv:2002.03234v3 [cond-mat.stat-mech] 13 Mar 2020 cially biological ones, operate under stationary (or time- The observable O, the⊂ threshold D, and the time τ, define periodic) conditions [17]. They do not involve any (net) the physical process and its duration. For concreteness, transformation in the system’s state. The changes are τ may be the minimum time to displace a mass, or to confined to the reservoirs that fuel the nonequilibrium exchange a given amount of energy or particles with a dynamics via mass or energy exchanges, for instance. reservoir. In general, it represents the time needed for a Furthermore, the kinetic features of these systems are specific physical process to be carried out by the system. hardly known [18, 19]. We next identify the space of ‘survived’ trajectories at s s time t, Ωt, such that if ωt Ωt then O(ωt) D. They correspond to trajectories∈ in which the process6∈ is not ∗ [email protected] completed. The associated probability that the process † [email protected] is not yet completed at time t is expressed by the survival 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 intoh i 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.

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