Dissipation Bounds the Amplification of Transition Rates Far from Equilibrium

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BIOPHYSICS AND APPLIED PHYSICAL COMPUTATIONAL BIOLOGY SCIENCES them from equilibrium with both deterministic and autonomous this notion to systems driven away from equilibrium by an arbi- forces. trary time-dependent force λ(t) (11). For a nonequilibrium system, microscopic reversibility is manifested by Pλ[X(t)|x(0)] = Stochastic Thermodynamics of Rate Enhancement P˜λ[X˜(t)|x(t)] exp [β (Q[X(t)|x(0)] + Qrev([x(t), x(0)]) ]. The first To derive Eq. 1, we consider systems driven by a time-dependent term in the exponential, βQ, is what we refer to as dissipation, as force, λ(t), either externally controlled or coupled to an addi- it is the excess heat transferred from the system to the bath along tional nonthermal noise source that evolves independently of the a trajectory driven from equilibrium over the corresponding heat system state, precluding feedback. Extensions to systems evolv- transferred for a reversible process. For an equilibrium system, ing in boundary-driven nonequilibrium steady states, although the reversible contribution to the heat is equal to the Shannon likely possible, are not explored in this work. entropy, βQrev = ln ρ0[x(t)]/ρ0[x(0)]. It is generally derivable as In the presence of the time-dependent force, the rate, kλ, of the change in energy due to the conservative forces and thus transition between two long-lived states is the probability that a depends on only the trajectory’s boundaries. transition occurs per unit of time. For a system described by a Coupling the system to a dissipative process will generally configuration, x(t), at time t, we will consider initial and final change its dynamics. Using trajectory reweighting, we relate the states, A and B, that are collections of configurations defined by transition rate in the presence and absence of the nonequilib- the indicator functions, rium force λ(t). We consider two path probability distributions with support on the same X(t), so that the relative action ( 1 if x(t) ∈ i hi (t) = , [2] 0 Pλ[X(t)|x(0)] else β∆Uλ[X(t)|x(0)] = ln , [6] P0[X(t)|x(0)] where i ∈ {A, B}, and we assume A and B are not intersecting. For times longer than the characteristic local relaxation time and relating one to the other, is well-defined. Performing a change of measure, for a constant distribution of initial conditions, we much shorter than the inverse rate, kλ derives from a ratio of path partition functions, express ratios of path partition functions in either ensemble as

−1 ZAB (λ) D β∆U E D −β∆U E d ZAB (λ) = e λ = e λ , [7] kλ(A → B) = . [3] ZAB (0) 0 λ dt ZA(λ)

Here, where the brackets denote a conditional average in a transition- Z path ensemble connecting states A and B in time t, with path ZAB (λ) = D[X(t)]hA(0)hB(t)Pλ[X(t)] [4] probability P0[X(t)] in the first equality or Pλ[X(t)] in the second equality. is the number of transition paths, X(t) = {x(0), ... , x(t)}, start- When transitions in both path ensembles are rare, kλt and ing in A and ending in B at time t, weighted with probability k0t 1, the overwhelming majority of paths originating from A Pλ[X(t)], and will remain there on the timescales where the rate is time- independent, so that ZA(λ) = ZA(0). In SI Appendix, S1. Gener- Z alized Bound on Rate Enhancement, we consider generalizations ZA(λ) = D[X(t)]hA(0)Pλ[X(t)] [5] away from this limit, showing that the ratio ZA(0)/ZA(λ) cancels contributions in Eq. 7 due to different distributions of initial conditions. Thus, under mild assumptions, combining Eq. 3 with Eq. A is the corresponding number of paths starting in (8). 7, we find The ratio in Eq. 3 is simply the conditional probability of k D E D E−1 the system being in state B given it started in A. Provided λ = eβ∆Uλ = e−β∆Uλ , [8] the transition is rare, consistent with A and B representing k0 0 λ metastable states, there is a range of time over which ZAB (λ) which is an exact relation between transition rates in the presence increases linearly and kλ is constant. Specifically, the rate con- −1 or absence of the dissipative process. Lower and upper bounds stant is defined for observation times τA . t kλ , where the can be read off by applying Jensen’s inequality to each of these transition-path time is typically on the order of τA, the char- expressions, acteristic relaxation time within state A, and shorter than the

timescale required for global relaxation. The probability of a kλ path is the product of a distribution of initial conditions, ρλ[x(0)], β h∆Uλi0 ≤ ln ≤ β h∆Uλiλ , [9] k0 and the conditional transition probability Pλ[X(t)|x(0)], such P [X(t)] = P [X(t)|x(0)]ρ [x(0)] that λ λ λ . While in general away constituting a fundamental envelope for the rate enhancement. ρ [x(0)] P [X(t)|x(0)] from thermal equilibrium, λ is unknown, λ This result is general, provided any two trajectory ensembles can be inferred, provided an equation of motion. For the spe- have common support. In a suitably defined linear-response P [X(t)|x(0)] cific model calculations discussed below, λ will take regime, the ensembles are approximately equal, h∆U i ≈ an Onsager–Machlup form (13). λ λ h∆Uλi0, so the bounds are saturated. This corresponds to a Stochastic thermodynamics gives structure to path ensem- near-equilibrium regime where driving is small. While Eq. 9 is bles and relations to thermodynamic quantities. In an equilib- derived for constant initial conditions for simplicity of notation, rium system, the principle of microscopic reversibility implies the impact of different initial conditions on an upper bound is that the probability of a trajectory is equal to its time-reverse. ˜ ˜ to add a positive constant equal to a symmetrized Kullback– Specifically, let Pλ[X(t)] denote the probability of observing Leibler divergence between initial distributions in the driven and a time-reversed trajectory X˜(t) = {˜x(t), ... , ˜x(0)}, where ˜x(t) equilibrium ensembles (SI Appendix, S1. Generalized Bound on is a time-reversed configuration of the system at t, labeled Rate Enhancement). In the linear-response regime, and when in the forward time direction. In the absence of the dis- transition paths are long enough to lose memory, a common sipative protocol, λ = 0, the system is in equilibrium and occurrence for rare transitions that quickly relax in both A and P0[X(t)] = P˜0[X˜(t)]. The Crooks fluctuation theorem extends B, changes to initial conditions can be neglected.

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BIOPHYSICS AND APPLIED PHYSICAL COMPUTATIONAL BIOLOGY SCIENCES In order to understand the physical processes that determine with a recent study in which an effective potential approach was whether or not the bound is saturated, we focus on two cases used to derive kλ/k0 for an active Ornstein–Uhlenbeck process of the model presented above. First, we set p = 0, which corre- in a cubic well (22). As shown in Fig. 2B, our bound is closest sponds to an active Brownian particle in an external potential. to the true rate enhancement when driving is small compared Active Brownian particles provide a canonical realization of to the maximum force needed to surmount√ the barrier in equi- how autonomous athermal noise can drive novel steady states librium, f < Fm, where Fm = 8∆VA/3 3lA. In that regime, the without simply imparting an effective temperature (17). These heat and rate enhancement both scale with f 2, as predicted by self-propelled agents exhibit dynamical symmetry breaking and linear-response theory. When the protocol and gradient forces collective motion (18, 19), and previous studies have shown are comparable, the transition ceases being a rare event, and that the escape of active particles from a metastable potential further increasing f has little effect on the rate but increases exhibits interesting behavior arising from an interplay between the heat. the driving force, persistence time statistics, and the shape of As a second test case, we consider underdamped dynamics the potential (20, 21). For simplicity, we take a symmetric poten- with time-periodic driving, p = 1. This model, known as the Duff- tial, with lA = lB = 1 and β∆VA = β∆VB = 10, setting γ = 1 and ing oscillator (23), is the simplest model of a stochastic pump kBT = 1/2. and one whose nonequilibrium behavior is marked by significant Fig. 2A shows the dependence of the rate enhancement on the nonlinearity. As an underdamped process, its barrier-crossing rotational diffusivity, Dθ, at fixed f = 1. The small Dθ limit is a behavior is determined by both spatial as well as energy diffusion, quasistationary regime corresponding to an equilibrium system in which both position and velocity correlations play a role. The with an additional linear force added to the potential. Increasing √equation of motion is given by mx¨ = −γx˙ − ∂x V (x) + λ(t) + the rotational diffusivity decreases the persistence of the driving, 2kBT ηx , where the mass m reflects the change to under- and as a consequence, hQiλ and kλ/k0 fall off in this limit. In damped dynamics. Again, we take a symmetric potential, lA = the large Dθ limit, the system is effectively in equilibrium at an lB = 1, now with β∆VA = β∆VB = 7 and m = β = γ = Dx = 1. elevated temperature, as λ averages to 0. Lower rate enhance- Fig. 3A shows that for a moderate force, f /Fm ≈ 0.13, there is ment and little dissipation are observed across this range of Dθ, an optimal driving frequency, denoted here as ω∗, which greatly and our bound is uniformly close. These results are consistent enhances the transition rate. This phenomenon is known as stochastic resonance (23). For slow driving, ω ω∗, the particle typically makes a transition before the external force reaches its maximum. For ω ω∗, driving is inefficient and, on average, requires multiple forcing cycles before presenting a chance to cross the barrier with the help of a positive force within the time of a typical transition. The approximate shape of the rate- enhancement profile is Lorentzian, a trait inherited from the absorption lineshape of an underdamped harmonic oscillator. In this case, the resonant frequency coincides with the curvature of the equilibrium double-well potential driven with a quasistatic ∗ p force, ω ≈ 8(∆V − f ) − γ2/4 (24). We find near saturation of the bound throughout a wide range of frequencies and across even such nonlinear behavior as stochastic resonance. In Fig. 3B, we plot kneq/keq and βhQiλ/2 against the driving amplitude relative to Fm. As in the other examples, the bound on rate enhancement is tight so long as the metastability of state A is preserved. Thus far, we have characterized rates with transition paths, which lend themselves to a natural response theory for ln kλ and, therefore, the ratio of rates. However, the survival probability Pλ(t) = exp(−kλt) = 1 − ZAB (λ)/ZA(λ) contains similar information in the case of two metastable states, when hA + hB = 1. Falasco and Esposito recently (25) worked with this quantity to prove a speed limit on escape processes. Extending these results, we derive analogous bounds on absolute reaction rates (SI Appendix, S4. Bounds on Survival Probabilities). Assuming rare rates, we bound the ratio of survival probabilities Pλ(t)/P0(t), and thus the difference of rates kneq − keq, from above and below using the relative action and Jensen’s inequality. The final result reads D E D E β ∆U˙ λ ≤ kλ − k0 ≤ β ∆U˙ λ , [13] λ 0

where the rate of change of the relative action forms an envelope around the change in the transition rate. If k0 kλ, then Eq. 13 acts as a speed limit on the forced process. On the other hand, if kλ k0, Eq. 13 reports on the minimum dissipation required Fig. 2. Rate enhancement for an active Brownian particle. (A) Rate 2 to slow down a fast equilibrium process. The envelope in Eq. enhancement as a function of rotational diffusivity constant Dθ lA/Dx for f = 1. (B) Rate enhancement as a function of the magnitude of active driving 13 is similar to the bound derived by Falasco and Esposito in relative to the maximum force opposing the transition in equilibrium f/Fm that it relates differences of rates to conditioned path ensemble for Dθ = 1/2. In both, the rate enhancement (black diamonds) is bounded averages. However, in Eq. 13, two forward rates under differ- by the dissipated heat (red circles). ent dynamics are considered, while in ref. 25, a forward rate is

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compute We mixed the including studied, systems overdamped the for Simulations the for framework general a build investigations our Overall, one-dimensional simple on focused have examples our While [ D saerlxto time relaxation A-state ,1] 0, k 0 x A = ycutn h ubro rniin from transitions of number the counting by to log , ,w nfrl draw uniformly we 1, B n ty hr tlato h re f2.75∆t of order the on least at there stays and f ∈ λ(t [ ,log −1, ) = − ,for 0, 5 × f max t 10 τ = ] A 7 and , λ ∆t 1.5 = β na ie0 h yaisaefis evolved first are dynamics the 0, time at on ∆ rniin r one when counted are Transitions . l x A × V / = D A 10 √ θ https://doi.org/10.1073/pnas.2020863118 , ,adteprmtr ittn its dictating parameters the and 0, 8∆V β 3 ∈ ∆t ∆V [ hr ecntanthe constrain we where 10], 0.1, oeulbae ecalculate We equilibrate. to , A B o h nedme Duff- underdamped the For . ∈ hQi ω 3 7), (3, min λ neednl fthe of independently , = A ∆t l i 2 and 3.5 to ∈ = B 8∆V 10 n iiigby dividing and PNAS −2 i ω (ω 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BIOPHYSICS AND APPLIED PHYSICAL COMPUTATIONAL BIOLOGY SCIENCES parameters, corresponds to 103 to 104 transitions. At each point in time, we The rates are small quantities, prone to high statistical uncertainty, which record the instantaneous rate of heat dissipation as well as an indicator is amplified when the log of their ratio is taken. Because of this, we rerun function to coarse-grain x(t) into either state A, state B, or the transi- all points that have greater SE, in either enhancement or dissipation, than tion state, which, in this example, we define as x ∈ A ∪ B, the intersection the mean. The majority of these points correspond to lightly forced sys- between states. Immediately after each simulation, we collect and save a tems where the maximum protocol amplitude is usually much smaller than −1 list of times when transitions started, t1, and ended, t2. Once all simula- the thermal energy scale, β|λ| ∼ 10 . In this regime, ln kλ/k0 1 is well tions are complete, we histogram transition-path times, t2 − t1. In order within linear response. Since we are focused on the case where the equi- to account for all transitions when integrating over the heat flux to get librium rate changes appreciably, and know that by time-reversal symmetry Pn−1 Q(t = n∆t) ≈ i=0 λi(xi+1 − xi), we choose an initial observation time t arguments, the only contribution in this near-equilibrium case comes from equal to that required for 99% of transition paths to proceed start to dissipation, throughout the article, we only show points with ln kλ/k0 larger finish. For each transition, we position a window of length t so that it than a small number, which we choose to be 10−2. ends immediately after t2 and integrate over it. We slide this window by a small O(∆t) coarse-graining time, and then, given the transition still Data Availability Analysis and simulation codes have been deposited in occurs, repeat the integration. This procedure is repeated until the tran- GitHub (https://github.berkeley.edu/ben-kuznets-speck/Dissipation-bounds- sition no longer resides within the sliding window. The coarse-grained the-amplification-of-transition-rates-far-from-equilibrium). time ≈ ∆t controls how highly correlated reactive trajectories are. If, at any point, a window passes through a transition region without seeing ACKNOWLEDGMENTS. This material is based upon work supported by NSF a transition, due to coarse-graining, the observation time is increased by Grant CHE-1954580 and by a Scialog Program sponsored jointly by Research a small factor of t 7→ t(1 + 1/4) and the integration is restarted from the Corporation for Science Advancement and the Gordon and Betty Moore beginning. Foundation.

1. C. Bustamante, Y. R. Chemla, N. R. Forde, D. Izhaky, Mechanical processes in 19. T. GrandPre, D. T. Limmer, Current fluctuations of interacting active Brownian biochemistry. Annu. Rev. Biochem. 73, 705–748 (2004). particles. Phys. Rev. E. 98, 060601 (2018). 2. M. L. Mugnai, C. Hyeon, M. Hinczewski, D. Thirumalai, Theoretical perspectives on 20. E. Woillez, Y. Zhao, Y. Kafri, V. Lecomte, J. Tailleur, Activated escape of a biological machines. Rev. Mod. Phys. 92, 025001 (2020). self-propelled particle from a metastable state. Phys. Rev. Lett. 122, 258001 3. A. Mashaghi, G. Kramer, D. C. Lamb, M. P. Mayer, S. J. Tans, Chaperone action at the (2019). single-molecule level. Chem. Rev. 114, 660–676 (2014). 21. E. Woillez, Y. Kafri, V. Lecomte, Nonlocal stationary probability distributions and 4. Y. L. Wu, D. Derks, V. van. Blaaderen, A. Imhof, Melting and crystallization of colloidal escape rates for an active Ornstein–Uhlenbeck particle. J. Stat. Mech. Theor. 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2 Supplementary Information for

3 Dissipation bounds the ampliﬁcation of transition rates far from equilibrium

4 Benjamin Kuznets-Speck and David T. Limmer

5 David T. Limmer. 6 E-mail: [email protected]

7 This PDF ﬁle includes:

8 Supplementary text 9 Figs. S1 to S3 10 SI References

Benjamin Kuznets-Speck and David T. Limmer 1 of8 11 Supporting Information Text

12 S1. Generalized bound on rate enhancement

13 Our main results were obtained under the assumption that the probability of starting in A does not change much under 14 the inﬂuence of driving ZA(λ) ≈ ZA(0). In this section, we derive a more general bound that relaxes this assumption, and 15 discuss in what cases we expect initial conditions to play a signiﬁcant role. The factor ZA(0)/ZA(λ) in question is the ratio of 16 single-time probabilities and has played an important role in the development of Monte-Carlo sampling on the space of driving 17 protocols (S1). As with the transition path partition function, it is still possible to express such a ratio in terms of the moment 18 generating function of the relative action

−1 ZA(0) ρ0 −β∆Uλ ρλ β∆Uλ 19 = e = e , [S.1] ZA(λ) ρλ ρ0 A,λ A,0

20 which implies

ZA(0) 21 − βh∆UλiA,λ − DKL(ρλ||ρ0) ≤ ln ≤ −βh∆UλiA,0 + DKL(ρλ||ρ0) [S.2] ZA(λ)

22 where the paths of any length are conditioned to start in state A. After restoring initial conditions, Eq. 9 of the main text 23 then implies

kλ 24 β(h∆Uλi0 − h∆UλiA,λ) − J(ρλ||ρ0) ≤ ln ≤ β(h∆Uλiλ − h∆UλiA,0) + J(ρλ||ρ0), [S.3] k0

25 which is our most general result. Initial conditions ρλ(x0) and ρ0(x0) enter in through a symmetric variant of the Kullback–Leibler 26 divergence known as the Jeﬀreys divergence, Z ρλ 27 J(ρλ||ρ0) = DKL(ρλ||ρ0) + DKL(ρ0||ρλ) = dx0(ρλ − ρ0) ln ≥ 0, [S.4] ρ0

28 and we will discuss shortly under what conditions it can be neglected. Even when its most likely state changes, if A remains 29 metastable, the system will start in an effective stationary state with flux and force that both vanish on average, leaving the 2 30 negative, λ piece of the dynamical activity as the only portion of h∆UλiA,0 left. However, this piece does not depend on the 31 trajectory since under our assumptions, the external force is independent of the system configuration, so it cancels in the 32 relative action with an identical contribution from the driven ensemble. We can therefore deduce that only the trajectory 33 dependent portions of h∆Uλiλ, and J(ρλ||ρλ) are likely to contribute to our upper bound.

34 To understand the influence of initial conditions, we begin by noting that they cease to enter into the bound when the initial 35 configuration is deterministic or the system is prepared according to the same distribution, either ρλ or ρ0. Next, that ρλ and 36 ρ0 are equivalent in linear response means J(ρλ||ρλ) vanishes in this regime. Thus, if making the transition is correlated with 37 the initial distribution in state A, then whenever that correlation remains relatively invariant under driving, initial conditions 38 can be neglected. Perhaps most relevant to the present work and the study of rare events is the situation when initial conditions 39 becoming uncorrelated with the ’bulk’ of the transition path, and are forgotten. The reason why loosing memory of initial 40 conditions is far from exotic for such reactive trajectories stems directly from the fact that state A is assumed, by definition of 41 the rate constant, to be metastable, harboring a local minima of the free energy landscape. The initial portion of a typical 42 rare event transition path from A to B will therefore relax exponentially quickly in this local basin and loose memory over a 43 short timescale τA determined by the curvature of the basin. Since the rate constant is defined for times that are order τA, as 44 detailed in the paragraph following Eq. 5 in the main text, the brunt of reactive paths are marked by boundary portions of 45 rapid relaxation on either side of the transition region. When state A is suitably defined so that this memory loss occurs in 46 both the driven and reference ensembles, all that matters is that the particle starts somewhere in state A, which is guaranteed 47 by the transition portion of the path probability. In other words, knowledge about the character of state A is contained directly 48 in h∆Uλiλ when both rate constants are defined with a common observation time, which is the topic of Sec. S4. When the 49 driving is so extreme that a common observation time ensuring relaxation before escaping state A cannot be obtained, the 50 rate constant effectively looses meaning. Such atypical cases can be addressed taking into account initial conformational 51 distributions, or preferably, redefining A so that k0 and kλ are comparable by memory-less transition paths of the same length.

52 S2. Conditions for neglecting the dynamical activity

53 In order to understand when the contribution of the dynamical activity to the rate enhancement bound can be neglected, 54 we consider both simplified limiting cases that are analytically tractable, as well as additional numerical experiments on the 55 systems considered in the main text. In general, the dissipative bound is valid in cases where the activity can be neglected 56 due to its size or if it can be shown to be strictly negative. The former case can be argued to occur near-equilibrium or when 57 specific spatial symmetries exist that result in its average being small. The latter case occurs when a particle traverses both 58 large, narrow barriers as well as broad diffusive barriers.

2 of8 Benjamin Kuznets-Speck and David T. Limmer 59 A. Limiting cases of high symmetric and broad diffusive barriers. For simplicity and concreteness, we consider an overdamped 60 particle in a 1d symmetric double well potential. The equation of motion is analogous to that considered in the ﬁrst two 61 examples in the main text,

p 0 0 62 γx˙(t) = Fλ(x) + 2kBT γη, hη(t)i = 0 hη(t)η(t )i = δ(t − t ), [S.5]

0 63 where Fλ(x) = −V (x) + λ(t) is the total force, including the gradient and non-gradient contributions. From the equation of 64 motion, it is straight-forward to show that the conditioned transition probability takes the form ln Pλ[x(t)|x(0)] = βUλ[x(t)]+C, 65 where C is a constant and the Onsager-Machlup path-action is Z t 1 0 0 0 2 66 Uλ = − dt γx˙(t ) − Fλ[x(t )] [S.6] 4γ 0

67 for an ensemble with finite nonequilibrium driving and path of length t. 68 In the limit that the barrier separating two metastable states is large, such that transitions between them are rare, the 69 rate can be computed by extremizing the path action, δUλ/δx = 0. Preforming the functional differentiation, the instantonic 70 trajectory satisfies the second order differential equation

2 0 71 γ x¨ − F˙λ[x(t )] − Fλ[x(t)]∂xFλ[x(t)] = 0, [S.7]

72 which can, in principle, be solved subject to the boundary conditions of starting in state A and ending in state B. Here we will 73 consider both states being defined at specific points x = xA and x = xB for states A and B, respectfully. However, solving this 74 equation is difficult for nonconservative potentials and time dependent driving forces. We consider the case of a large, sharply 75 peaked barrier, such that the maximum force due to the potential, Fm, is much larger than the magnitude of the applied 76 driving force, f = max |λ(t)| Fm. In this limit, the instanton equation simplifies to

2 0 00 77 γ x¨ ≈ V (x)V (x) [S.8]

78 which results in the trajectory traced out by the particle in equilibrium. Its ﬁrst integral is a constant of motion, yielding

2 2 0 2 79 γ x˙ = [V (x)] , [S.9]

80 that provides both branches of the instanton, or extremal path (S2). The positive root yields the trajectory beginning at 81 xA and ending at the maximum of the potential separating the two metastable states, x = xm, and results in a positive 82 contribution to the action. The negative root yields the trajectory beginning at xm and ending at xB , with zero action. The 83 resultant total action is approximated in this limit as

Z t/2 Z t/2 Z x‡ 0 0 0 0 0 0 0 84 Uλ ≈ dt x˙(t )Fλ[x(t )] = dt x˙(t )λ(t ) − dx V (x) [S.10] 0 0 xA

2 85 where consistent with the assumption of an equilibrium trajectory minimizing action, we have neglected terms of order O(λ ) 86 that are strictly negative and invoked time reversal symmetry to set the domain of the integrals. The ﬁrst term is the 87 equilibrium change in energy, while the second is the heat. Using this result, we can compute the averaged relative action 88 between equilibrium and nonequilibrium path ensembles

89 h∆Uλiλ ≈ hQiλ/2 [S.11]

90 which has no contributions from the activity and coincides with our main result. This calculation clariﬁes a relevant linear 91 response limit in which the dissipative bound is tight. It is one in which the typical reactive trajectory in and out of equilibrium 92 are similar, following a gradient path, which occurs in cases where the conservative forces experienced during a transition are 93 large relative to the nonequilibrium driving forces, f Fm. For a second example, we consider a barrier that is locally parabolic in the vicinity of its maximum,

2 V (x) ≈ −kx /2 + V0

94 where k denotes its local curvature and V0 is the oﬀset from the minimum in the A state. In the limit that βV0 1, we can 95 assume that the majority of the action required to overcome the barrier is localized to the region around the maximum. In 96 such a case, the stochastic action in the presence of the external force is extremized by the solution of

2 2 97 γ x¨(t) = k x(t) + kλ(t) + γλ˙ (t) , [S.12]

98 for the instantonic trajectory subject to boundary conditions which we take to be symmetric about the maximum, x(0) = −x0 ∗ 99 and x(t ) = x0. It is convienent to introduce the charactoristic relaxation time, τ = γ/k. This linear ordinary diﬀerential 100 equation can be solved by the method of Laplace transforms, yielding

∗ ∗ x0 + x0 cosh(t /τ) f(t )/γ 101 x(t) = −x cosh(t/τ) + sinh(t/τ) + f(t)/γ − sinh(t/τ) [S.13] 0 sinh(t∗/τ) sinh(t∗/τ)

Benjamin Kuznets-Speck and David T. Limmer 3 of8 102 where f(t) is the convolution of the external force with the Green’s function, Z t 0 0 (t−t0)/τ 103 f(t) = dt λ(t )e [S.14] 0

104 and acts as an inhomogenious source. From the deﬁnitions in the main text, the heat and activity are given by

∗ ∗ Z t 1 Z t 105 Q = dt x˙(t)λ(t) Γ = − dt λ(t) [2kx(t) + λ(t)] [S.15] 0 2γ 0

106 whose averages are computed within the instantonic trajectory. The heat has a familiar form. The activity includes a 107 contributions due to the product of the gradient force and the external force, and a contribution due to the external force 108 squared. While the latter is strictly negative, path independent, and can be dropped while still satisfying the bound, the former 109 may be positive or negative. 110 In the limit of small λ(t), it can be verified that the first contribution to the activity vanishes, as the instanton trajectory 111 spends equal time on the left and right side of the barrier with corresponding equal and opposite forces from the external 2 112 potential. The second term enters proportional to −λ (t) which can also be neglected if λ(t) is small. This is identical to 113 the near equilibrium case considered above. Analogously, in the limit that λ˙ ≈ 0, such that over the transition the external 114 force is well approximated by a constant, by symmetry the activity will be strictly negative. Consider for concreteness a 115 periodic force λ(t) = f cos(ωt) with characteristic amplitude f and frequency ω. In the limit that ωτ 1, the heat will be 2 ∗ 116 hQiλ = 2fx0 and the activity hΓiλ = −f t /2γ < 0. In the opposite limit that ωτ 1, many cycles of the driving force will 117 elapse during the instantonic trajectory and its influence on breaking the symmetry around the barrier will vanish. In the case 2 ∗ 2 ∗ 118 of a monochromatic driving force, the heat is hQiλ = f t /2γ and the activity hΓiλ = −f t /4γ < 0 can be neglected in the 119 bound. Away from these cases, the trajectory need not be symmetrically distributed around the barrier, and the activity maybe 120 finite. However, this occurs when the driving is large, in which case the second term in the activity will dominate leading to it 121 being negative. Finally, in the case where the barrier is broad and flat, we can approximate the motion at the top as free diffusion conditioned on a set of starting and ending points, xA and xB . Specifically, when V (x) = 0, for an arbitrary external force, the average heat and activity simplify to Z t∗ Z t∗ Z t∗ (xB − xA) −1 ¯ ∗ 2 1 2 hQiλ = ∗ dt λ(t) + γ dt λ(t) − λ(t ) hΓiλ = − dt λ (t) t 0 0 2γ 0

∗ ∗ 122 where λ¯(t ) denotes the external force averaged over the transition time t . The activity is manifestly negative and can be 123 dropped from the dissipative bound. As above, it is negligible when the scale of λ is small as it scales quadratically. Taking ∗ 124 |xB − xA| and t large, the heat becomes the displacement times the average force, hQiλ ≈ (xB − xA) λ¯ and the activity ∗ ¯2 ¯2 125 hΓiλ ≈ −t λ /2γ where λ is the average squared size of the external driving.

126 B. Relative magnitudes of the heat and activity in the transition path ensemble. Here, we discuss the results of additional 127 simulations on systems featured in the main text Figs. 2 and 3. Shown in Figs. S1 and S2 are the heat (top row), and 128 the activity (bottom row) throughout the course of the transition. Each column represents a driving speed: low, medium 129 and, high, from left to right, and we fix all other parameters as in the main text. The active Brownian particle in Fig. S1 130 is driven at a speed Dθ equal to its inverse auto-correlation time, and ω is the analogous variable in the Duffing Oscillator 131 depicted in Fig. S2. The quantities plotted in these figures as a function of time, in units of the reactant (A) relaxation R τ−t/2 R τ−t/2 132 time, are averages of Q˙ and Γ˙ , which are not part of the transition path ensemble until crossing occurs at −t/2 −t/2 −1 133 time zero. For the overdamped active Brownian particle, Γ˙ = −γ λ(t)Fλ[x(t)] and in the underdamped Duffing oscillator −1 134 Γ˙ = γ λ(t)(γx¨(t) − Fλ[x(t)]), which can be derived by constructing the symmetric part of the corresponding path action. 135 As the reaction proceeds from beginning to end, heat and activity are computed in the path ensemble defined by the ‡ 136 additional constraint that the particle position coincides with the transition state, x at time zero. That is, we average over 137 all trajectories starting at time −t/2 in A before crossing at time 0 and ending up in B a time t/2 after that. Since both 138 the underlying system and the driving are symmetric in time in the long-time limit, it is reasonable to believe the instanton 139 connecting A to B is also, so this scheme should sample the AB ensemble as the number of trajectories becomes large. We 8−9 140 collect long trajectories of Q˙ and Γ˙ , integrating from −t/2 to τ −t/2 for τ ∈ (0, t). Specifically, we run simulations of ∼ 10 ∆t 3−4 141 until ∼ 10 reactions are observed. The majority of paths lie above hQiλ but there are a number of outlying negative heat 142 trajectories, wherein the particle crosses the barrier while also opposing the driving force, which is a consequence of the integral 143 fluctuation theorem. 144 In Heat (top, red) rises past the rate enhancement (black) in the immediate vicinity of this time, an observation consistent 145 with the non-equilibrium transition-state theory we develop below. On the other hand, the activity need not serve as an upper 146 bound, as is the case for particles driven at the half-maximum speed and resonance frequency in in the center column of 147 Figs. S1 and S2, respectively. Like we discussed in the section prior, the activity accumulated over a symmetric barrier should 148 vanish by symmetry in the adiabatic limit, a signature which is approximately realized in Fig. S1 (bottom left). Finally, when 149 driving varies so quickly that it couples effectively as a second bath, Fig. S1, the activity is negative, as would be expected for 150 free diffusion.

4 of8 Benjamin Kuznets-Speck and David T. Limmer Fig. S1. Additional examples of active barrier crossing. Rate ampliﬁcation ln kλ/k0 (black, from counting transitions), dissipation (red, top row), and dynamical activity (red, bottom row) along the typical reaction trajectory, in units of the reactant (state A) relaxation time τA. All parameters except for active diffusivity Dθ relative to that ∗ ∗ −3 at which the enhancement is half its maximum Dθ are set according to Fig. 3 a). Left: slow driving with a long persistence time Dθ /Dθ ≈ 2 × 10 . Center: close ∗ ∗ half-maximum persistence Dθ /Dθ ≈ 1. Right: low-persistence forcing Dθ /Dθ ≈ 7.

Fig. S2. Revisiting stochastic resonance. Rate ampliﬁcation ln kλ/k0 (black, from counting transitions), dissipation (red, top row), and dynamical activity (red, bottom row) along the typical reaction trajectory, in units of the reactant (state A) relaxation time τA. Parameters apart from driving frequency ω are set according to FIG. 3 a) in the main text. Left: slow, quasi-adiabatic driving ω/ω∗ ≈ 0.42. Center: close to resonance ω/ω∗ ≈ 1.04. Right: very fast forcing ω/ω∗ ≈ 1.55.

Benjamin Kuznets-Speck and David T. Limmer 5 of8 151 S3. Separation of timescales in the driven and equilibrium ensembles

152 The existence of a time-independent transition rate between two metastable states requires a separation of timescales between 153 local relaxation within a metastable state and the characteristic time to transition between the two states (S3). The rate 154 enhancement relation in the main text additionally requires that separation exists for both the driven and equilibrium ensembles, 155 and further that these two intervals defined by the local relaxation and typical transition waiting times have some amount of 156 overlap, starting at tmin. 157 Crucially, the observation time t past which our bounds are defined and valid in the conditioned path ensemble must be 158 chosen at least as large as the minimum overlap time t > tmin. Before tmin, heat will increase approximately linearly. An 159 observed transition to different behavior, typically occurring around the reactant relaxation time τA, can be employed as a 160 signature of when our bound is tightest in situations wherein the true rate enhancement is unknown. 161 Heat does not accumulate in a quasi-equilibrium state, so if both A and B remain deeply metastable, one can expect there 162 to exist an interval of time starting around tmin when the particle commits to state B and hQiλ varies relatively slowly. To test 163 these predictions, we consider the mixed driving system studied in Fig. 1. Specifically, Fig. S3 shows the log ratio of the rates 164 in and out of equilibrium as a function of time for that system, together with the accumulated heat, where τ is an intermediate 165 time between 0 and the observation time t. In all cases where there is a large barrier separating states A and B, and the 166 applied force f is smaller than the maximum force Fm due to the potential, the ratio of rates plateaus within a time of O(1). 167 Further, over the time window when the rate ratio is time independent, heat gains support at a much smaller rate than prior 168 to this window, plateauing in the case of dual metastability in Fig. S3a. Therefore, under these conditions, we find the path 169 ensemble averages in question to be somewhat insensitive to the precise time at which they are taken.

Fig. S3. Rate ampliﬁcation kλ/k0 (black) and dissipation (red) for three examples of heterogeneously driven two-state systems. The shape of the equilibrium potential for each system is inset (blue) in the upper-left hand corner, and time is taken in units of the relaxation time for state A (the state on the left side of each inset). a, transitions in an approximately symmetric double-well driven by equal parts deterministic and active forces, amplitudes summing to around half the equilibrium well-depth. b Escape from a basin of attraction to a less-stable state driven by mostly active forces with total max-amplitude around 0.35∆VA. c Excursions to a state with greater stability, forces mainly by a time-periodic protocol with maximum amplitude about 0.75∆VA.

170 S4. Bounds on survival probabilities

171 The recently proposed dissipation-time uncertainty relation (S4) showed that the rate of steady state entropy production 172 bounds the diﬀerence of forward to backward, denoted here with a tilde, transition rates, out of and into a metastable state A, 173 from above 174 kλ − k˜λ ≤ βhQ˙ iλ. [S.16]

175 In this section, we follow the same line of thinking, but use the definition of ∆U in place of the traditional fluctuation theorem. 176 For a two-state system, the indicator functions (Main: Eq. 2) defining states A and B are simply related by hA(t) + hB (t) = 1. 177 In this case, the cumulative probability p(t) that no transitions occur up to time t, the survival probability, is related to the 178 probability that at least one occurs in the same way:

ZAA(λ, t) ZAB (λ, t) 179 pλ(t) = = 1 − . [S.17] ZA ZA

180 From Eq. 1. in the main text, this relation implies that the transition rate is given by kλ = −d ln pλ(t)/dt. Bounding pλ(t)/p0(t) 181 from above and below in the AA path ensemble, and subsequently using the fact that hA(t) = 1 − hB (t) to convert to the AB 182 ensemble yields the envelope presented in Eq. 13. For example,

p0 ZAA(0, t) −β∆U −βh∆UiAA,λ −β(1−h∆UiAB,λ) 183 = = he iAA,λ ≥ e = e =⇒ kλ − k0 ≥ βh∆U˙ λiAB,λ [S.18] pλ ZAA(λ, t)

184 yields an upper bound, where, to be explicit, we label averages by which path ensemble they belong to (AA and AB). Again, 185 bounds in Eq. 13 assume the meta-stability of the A state is suﬃciently preserved, though this can be relaxed by following the 186 procedure laid out in SM section 3. We leave it to future work to delve into the utility of bounding the change of the rate 187 kλ − k0 in this fashion.

6 of8 Benjamin Kuznets-Speck and David T. Limmer 188 S5. Connection to transition-state theory

189 Our main focus is on rates calculated within the transition path ensemble, obtained by approximating the sum over time- 190 dependent trajectories. The benefit of this approach is that all that one needs is a definition of two states, A and B. Detailed 191 mechanistic knowledge of the kinetic bottlenecks through which the transitions pass is unnecessary, alleviating the potentially ‡ 192 hard problem of finding a good reaction coordinate, q, and relevant transition state, q , dividing metastable states A and B. 193 However, if a pertinent reaction coordinate is known, the rate can be estimated by much simpler means using transition-state 194 theory, which trades in the path partition function ZAB (λ) for the average flux through a transition state times the probability ‡ 195 of being at q (S5). Generalizing the traditional result, for a nonequilibrium system,

‡ h|q˙|iλ,q‡ ρλ(q ) 196 kλ ≤ . [S.19] 2 ρλ(A)

‡ 197 where h|q˙|iλ,q‡ is the average flux through the transition state, and ρλ(q )/ρλ(A) is the probability of finding the system at the 198 transition state relative to the total probability of finding the system in state A. In equilibrium, the equipartition theorem 199 dictates that the average flux is independent of q, but that is not necessarily true away from equilibrium. The transition-state 200 theory approximation is an underestimate of the true rate, as it neglects those trajectories that switch direction after already ‡ 201 passing q , and can be made exact by computing a correction termed the transmission coefficient. ‡ 202 For rare, instantonic transitions, the relative probability to reach the transition state, ρλ(q )/ρλ(A), is by far the dominant 203 contribution to the transition-state theory rate. In equilibrium, this term gives rise to the Arrhenius law

‡ ρ0(q ) −β∆F 204 k0 ∝ = e [S.20] ρ0(qA)

‡ ‡ 205 where ∆F = F (q ) − F (qA) is the height of the free energy barrier separating q from the most probable state qA ∈ A. Out of 206 equilibrium, rate estimation via transition state theory becomes a much more diﬃcult problem, because while a time averaged 207 stationary distribution ρλ(q) still exists, one cannot in general express it in a simple closed form. 208 However, using the Kawasaki distribution equation, we can approximate the nonequilibrium steady-state distribution in 209 terms of a product of the equilibrium distribution and a correction dependent on the mean dissipation. As formulated by 210 Crooks (S6) for stochastic dynamics and similar to that derived by Evans and Searles (S7), the Kawasaki distribution provides 211 a relation for the probability of being in state q having started in an equilibrium distribution, and cumulant generating function 212 of the accumulated dissipated heat. Speciﬁcally,

−βQ 213 ρ0(q) = ρλ(q) e [S.21] λ,q

214 where Q is the heat dissipated to the environment, ρ0(q) is the initial equilibrium probability of state q, ρλ(q) is the 215 nonequilibrium probability of state q, and the brackets h... iλ,q denote an average under the driving force λ conditioned on 216 ending at state q. Taking the saddle point, and applying Jensen’s inequality,

ρ (q) λ βhQiλ,q 217 ≤ e [S.22] ρ0(q)

218 we arrive at a bound for the ratio of the nonequilibrium to equilibrium distributions.

219 The bound of the steady state distribution cannot be applied directly to the estimation of rates, but under mild assumptions 220 it can provide an estimate similar to the result in the main text. To proceed, we first assume that the probability of being in 221 state A is unchanged between the equilibrium and nonequilibrium ensemble, ρλ(A) = ρ0(A). From the Jarzynski equality (S8) 222 and integral fluctuation theorems (S9), this is true if the protocol is cyclic, because then hexp(βQ)iλ = 1. This can be obtained 223 by taking the observation time long enough for the driven system to relax back to the original equilibrium distribution by the 224 final time, since in that case, the conservative heat and the change in free energy both cancel, leaving the excess heat behind. 225 Assuming like initial distributions is approximately valid in the limit that A is deeply metastable. Next, we assume that we 226 can ignore the change to the flux over the transition state due to coupling to λ. Moreover, the fluctuation-response inequality p 227 (S10) implies the first order change to the flux is expected to scale as ∼ βhQiλ, which is subdominant to the exponential 228 dependence from the change in the probability distribution. Finally, we assume that the transmission coefficient in and out of 229 equilibrium are the same. This is likely a good assumption in the limit that the original transition state theory estimate in 230 equilibrium is tight, and the transition is instantonic such that the protocol is slowly varying relative to the typical transition 231 path time. Under those assumptions, we find

kλ βhQi ‡ 232 . e λ,q [S.23] k0

233 where we have applied Eq. S.22 to the probability of reaching the transition state. Note here, as previously, the activity does 234 not enter the approximate bound, and we have a purely mechanical relationship between the enhancement speed of a process ‡ 235 and the energy required. Further, the dissipated heat is that accumulated in going from state A to the transition state, q . In 236 an instantonic limit, and near equilibrium, we expect the heat accumulated in reaching the top of the barrier to be half that to 237 reach state B, which would result in an analogous expression as the main text.

Benjamin Kuznets-Speck and David T. Limmer 7 of8 238 References

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