(2021) Reaction Rates and the Noisy Saddle-Node Bifurcation

PHYSICAL REVIEW RESEARCH 3, 013156 (2021)

Reaction rates and the noisy saddle-node bifurcation: Renormalization group for barrier crossing

David Hathcock and James P. Sethna Department of Physics, Cornell University, Ithaca, New York 14853, USA

(Received 22 October 2020; accepted 29 January 2021; published 17 February 2021)

Barrier crossing calculations in chemical reaction-rate theory typically assume that the barrier is large compared to the temperature. When the barrier vanishes, however, there is a qualitative change in behavior. Instead of crossing a barrier, particles slide down a sloping potential. We formulate a renormalization group description of this noisy saddle-node transition. We derive the universal scaling behavior and corrections to scaling for the mean escape time in overdamped systems with arbitrary barrier height. We also develop an accurate approximation to the full distribution of barrier escape times by approximating the eigenvalues of the Fokker-Plank operator as equally spaced. This lets us derive a family of distributions that captures the barrier crossing times for arbitrary barrier height. Our critical theory draws links between barrier crossing in chemistry, the renormalization group, and bifurcation theory.

DOI: 10.1103/PhysRevResearch.3.013156

I. INTRODUCTION quasiequilibrium state and the escape from that state [18]. In the limit of vanishing barrier, however, there is a qualitative In this paper we investigate deep connections between change in behavior. Particles instead slide down a monotonic barrier crossing, the renormalization group, and the noisy potential, spending the most time near its inflection point. To saddle-node bifurcation. In particular, we show that Kramers’ capture the low barrier escape rate, extensions to Kramers’ reaction rates can be understood as an asymptotic limit of theory have been developed (e.g., incorporating anharmonic the universal scaling near the continuous transition between corrections), but these have significant errors when the barrier high barrier and barrierless regimes. Applying methods from and thermal energy are comparable (E ≈ k T )[15]. stochastic processes theory we derive an analytical expression b B Finite barrier escape problems have garnered increasing for the universal scaling function for the mean barrier escape theoretical interest over the past decade, with several stud- time near the critical point, giving the crossover between high ies contributing further low barrier refinements of existing and low barrier limits. The renormalization group provides a theories [19–24] or focusing directly on the saddle-node bi- framework within which this result can be understood and furcation where the barrier vanishes [25,26]. Such escape systematically improved by perturbative calculations of cor- processes are relevant to certain high precision measurements. rections to scaling, some of which we give explicitly. For instance, force spectroscopy experiments apply a force Barrier crossing arises in applications across physics, on a single bond in a biomolecule until it breaks [20,27]. For chemistry, and biology. In 1940, Kramers computed the typical molecules, the critical force, at which the energy bar- barrier crossing rate for particles in both overdamped and un- rier for breaking vanishes and Kramers’ theory breaks down, derdamped regimes [1]. This result and others [2–4] provided is now well within the reach of atomic force microscopy the theoretical explanation for the Arrhenius equation describ- and optical tweezers [27]. Another exciting application is in ing chemical rate coefficients k ∼ exp(−E /k T ), where E b B b micro- and nanoelectromechanical devices, which sensitively is the energy barrier for activation [5]. More recent efforts switch oscillation amplitude in response to an input signal have established the escape rate at arbitrary damping, giv- by operating near the barrierless critical point [25,28]. Here ing the crossover between the low- and high-damping limits an analytical theory of low barrier crossing would help to [6,7], and have accounted for the effects of state-dependent distinguish between noise and signal activated switching. [8,9], non-Gaussian [10–12], and colored [12–14] noise, an- We develop a critical theory for barrier crossing with a harmonic corrections [15,16], and fluctuating barriers [14,17]. renormalization group approach that gives a complete scal- Most transition-state calculations assume a large barrier ing description of the noisy saddle-node bifurcation. We are limit. This means the barrier escape is a rare event, with inspired by previous work on the “intermittency” [41] route to a separation of timescales between the relaxation into a chaos [30–32], where the renormalization group coarse grains in time, then rescales the system to fix a certain term in the potential. In chaos theory, this procedure involves iterating and rescaling a discrete map [31,32], leading to a different Published by the American Physical Society under the terms of the fixed point for the same renormalization group equations used Creative Commons Attribution 4.0 International license. Further by Feigenbaum to study period doubling [33]. We take the distribution of this work must maintain attribution to the author(s) continuous time limit, reducing the renormalization group to and the published article’s title, journal citation, and DOI. a series of elementary rescalings and yielding a simplified

2643-1564/2021/3(1)/013156(9) 013156-1 Published by the American Physical Society DAVID HATHCOCK AND JAMES P. SETHNA PHYSICAL REVIEW RESEARCH 3, 013156 (2021) description applicable to barrier escape problems. Our pro- ( ) ∗( ) cedure organizes what amounts to dimensional analysis, V x V x providing an elegant renormalization-group framework that ∞ unifies Kramers’ theory for Arrhenius barrier crossing with + − ˜ ( − )n the dynamical systems theory of a noisy saddle-node bifurca- V0 Eb Vn x xmax tion. n=2 ∞ Why do we frame our analysis in terms of the renor- n+1 − nx malization group, if the scaling form can be justified using n+1 =0 dimensional analysis and the analytical methods we use are n Eb drawn from more traditional stochastic analysis? On the one g(x) hand, this forms a wonderful case study, unifying and illu- minating bifurcation theory, the renormalization group, and chemical reaction theory. Second, barrier crossing forms the ∞ n solvable limiting case of much more complex phenomena: V0 + Vn(x − xmin) coupling to colored-noise heat baths, nucleation of abrupt n=2 phase transitions, and depinning transitions in disordered systems (see Sec. V). The scale invariance of random walks does not demand a renormalization group proof of the central limit theorem, and the one-dimensional Ising model can be solved without the machinery of flows in Hamiltonian space. FIG. 1. Typical potentials in the high barrier Arrhenius limit But framing the problems in terms of the renormalization (solid curve) and at the renormalization group fixed point (dashed group provide excellent pedagogical exercises, and the natural curve). Kramers’ theory utilizes a two point series expansion at xmin framework for extensions to self-avoiding random walks and in the potential well and at xmax, the top of the barrier. For our Ising models in higher dimensions. renormalization group approach the natural description is in terms Using our scaling theory as an organizing framework, we of a single expansion at the origin parametrizing perturbations away ∗ derive an analytical expression for the scaling form of the from the fixed point potential V (x) ∝−x3. Also shown is the noise mean escape time near the saddle-node bifurcation. We also amplitude g(x), which generically has spatial dependence (dotted compute corrections to scaling due to anharmonicity in the curve). potential and finite initial or final position. Going beyond the mean, we develop an accurate approximation to the scaling Eq. (1). The effective potential along the reaction coordinate form for the full distribution of escape times by assuming the has a barrier separating species A and B. eigenvalues of the Fokker-Plank operator are equally spaced. As a starting point we consider the equation of motion for an overdamped particle in a general potential V (x) and driven II. RENORMALIZATION GROUP AND SCALING THEORY by spatially dependent white noise, To begin we parametrize Eq. (1) by the Taylor coefficients of g(x) and f (x), x˙ = f (x) + g(x) ξ (t ). (1) ∞ ∞ dx n n − = x + ξ (t ) g x . (2) Here f (x) =−η 1 dV/dx is the force exerted on the particle n n dt = = (divided by the damping coefficient η) and g(x) is the spatially n 0 n 0 varying noise amplitude (with the damping absorbed). The The renormalization group defines a flow in this space of noise ξ (t ) has zero mean ξ (t )=0 and is uncorrelated in systems described by a single reaction coordinate x. Near the time ξ (t )ξ (t )=δ(t − t ). With barrier crossing phenomena renormalization group fixed point, the behavior is most effec- in mind, we consider potentials with boundary conditions tively described by a single Taylor expansion at the origin. In V (x) →∞as x →−∞and V (x) →−∞as x →∞.The contrast, for large barriers in Kramers’ theory, the escape time potential either has a single barrier or is monotonically de- is characterized by two expansions, capturing the harmonic creasing (e.g., Fig. 1). The quantity of interest is the mean oscillations in the potential well and at the top of the barrier. barrier crossing time τ, defined as the time particles take to These two equivalent schemes are shown in Fig. 1.Giventhe reach +∞ from an initial position at −∞. later expansion at the two extrema, the expansion at the origin Besides the experimental systems discussed above, this can be reconstructed via a two-point Padé approximation [34]. model also serves as the natural description for a gen- As discussed above, the discrete renormalization group eral chemical reaction, involving the transition between coarse grains by iterating a map, evolving the equations for- metastable species A and B. These species are points in a 3N ward in time. Ignoring the noise for the moment, we consider dimensional configuration space defined by the locations of N a discrete approximation to Eq. (1), xn+1 = xn + δtf(xn ) ≡ reaction constituents. As derived by Hänggi et al. this system h(xn ). References [30–32] study this discrete equation using can be reduced to the one-dimensional model we study [18]. the Feigenbaum renormalization group transformation. This They coarse grain to a one-dimensional reaction coordinate, transformation iterates the map and rescales space, induc- which parametrizes the minimal gradient path between the ing a flow in function space, T [h](x) = ah[h(x/a)], where states A and B, neglecting effects of memory friction and a is the rescaling factor. The renormalization fixed point noise correlations, and taking the overdamped limit produces is the function h(x) that obeys T [h] = h for a particular

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∗ rescaling a. We simplify this calculation by taking a contin- until g0( ) = 1, we find that the escape time has the form uum limit. Expanding the renormalization group to first order τ = −2/3T / 2(2−n)/3 , / 1−2n/3 , in δt, T [h](x) = x + 2δt[af(x/a)]. Thus, in the continuous g0 n g0 gn g0 (5) time limit, the RG iteration becomes a simple rescaling of time where T is a universal scaling function, with n 1forthe and space. We will use this below to derive our scaling theory second term in brackets. for barrier crossing near a saddle-node bifurcation. While the scaling form Eq. (5) could have been written Within the context of the renormalization group for singu- down using dimensional analysis, the renormalization group lar perturbations developed by Goldenfeld, Oono, and others approach provides the natural structure and motivation for [35,36], our problem can be understood as having zero anoma- our approach. The parameter space flows indicate that, with lous dimension. For our calculations in the following sections a fixed quadratic force, the constant and linear force and we do not need to use the traditional renormalization-group noise terms {0,1, g0, g1} are relevant, growing under coarse machinery; instead we assemble a variety of tools from prob- graining and dominant on long timescales. Other variables ability theory and Markov processes to express the mean and are irrelevant and can be incorporated perturbatively. Of the distributions of escape times as universal scaling functions relevant variables, the linear force coefficient 1 can be set near the transition where the barrier vanishes. The scaling the- to zero by placing the origin at the inflection point of the ory provides a powerful and elegant structure which organizes potential. The spatial dependence of the noise (including the our understanding of the barrier crossing process. relevant linear term g1) can also be removed by a change of Following the above expansion of the discrete renormaliza- coordinates x → x˜ withx ˜ defined by [37] tion group, we “coarse grain” the system in time by scaling x˜ tˆ = t/b. As the timescale shrinks, the noise is amplified, g0 / x = dy, (6) ξˆ (tˆ) = b1 2ξ (t ) (the exponent 1/2 follows from the units of g(y) the correlation function). Our goal is to understand the scaling producing a system with constant noiseg ˜(˜x) = g and force properties near the critical point, where a qualitative change in 0 f˜(˜x) = f (˜x)/g(˜x) (hence g was relevant before we removed behavior occurs. For a generic analytic potential this happens 1 it, because it contributes to the linear term in the expansion when the barrier vanishes and V (x) =−x3 is locally a perfect of f˜). cubic. Therefore we rescale our system to fix the coefficient Systems near enough to the critical point therefore can be , corresponding to the cubic term in the potential. The cor- 2 modeled as a cubic potential with a linear perturbation V (x) = rect rescaling defines a new spatial coordinatex ˆ = bx.After −x3/3 − x and constant noise g . This is the “normal form” both coarse graining and rescaling we arrive at 0 0 used in bifurcation theory for the saddle-node transition, and ∞ ∞ might have been anticipated from Taylor’s theorem. The es- dxˆ 2−n n 3/2−n n = nb xˆ + ξˆ (tˆ) b gnxˆ . (3) cape time scaling form becomes dtˆ n=0 n=0 −2/3 4/3 τ = g T 0/g . (7) We can then read off how the parameters flow under the 0 0 2−n 3/2−n renormalization group ˆn = b n andg ˆn = b gn. These Thus, the problem asymptotically reduces to finding the uni- T α α = / 4/3 flows and exponents exactly match those found under the versal function of a single variable ( ), where 0 g0 . discrete-time renormalization group [31,32], indicating that The limiting form of the scaling function T (α) must give the the scaling of the “intermittency route to chaos” [29]isalso known solutions. In the limit α →−∞the barrier is large nonanomalous [36]. Taking the coarse-graining factor to be compared to the noise, so the Kramers approximation [1] close to 1, b = (1 + d), we obtain continuous flow equations, applies, so π d dg 8 |α|3/2 n n T (α) ∼ e 3 ,α→−∞. (8) = (2 − n)n, = (3/2 − n)gn. (4) |α|1/2 d d The eigenfunctions of the renormalization group in our For our choice of parameters, the energy barrier is given by / = / |α|3/2 α →∞ continuum theory are the monomials xn and noisy monomials Eb kBT 8 3 . In the opposite limit , the poten- ξ (t )xn. If the right-hand side of Eq. (2) is an eigenfunction, tial is downward sloping with gradient much larger than the it is scaled by a constant factor under the action of the noise level. The passage of particles over the inflection point renormalization group. These eigenfunctions are the much occurs even in the absence of noise (in contrast to the Kramers simpler continuous time limit of those for the discrete-time limit, which requires noise for barrier escape). Therefore, the renormalization group [32]. In particular, the cubic potential crossing time approaches that for a deterministic particle in V (x) ∝−x3 (without noise) is the fixed point. At the fixed the cubic potential. One can easily show that the limiting point, particle trajectories x(t ) ∼ 1/t exhibit scale invariance scaling form is in time as they approach the cubic inflection point at x = 0. π T (α) ∼ ,α→∞. (9) Perturbations away from the fixed point lead to dynamics α1/2 with nonpower-law decay to a locally stable state or over the inflection point. III. MEAN ESCAPE TIME The mean barrier crossing time is a function of the potential shape and the noise correlation, encoded through the A. Analytical escape time for relevant variables expansion coefficients n and gn. Thus, the escape time can We now turn our focus to obtaining an exact analytical be expressed as τ ({n}, {gn}), where n 0. If we coarse grain expression for T (α) that is valid for all α. To this end we study

013156-3 DAVID HATHCOCK AND JAMES P. SETHNA PHYSICAL REVIEW RESEARCH 3, 013156 (2021) the trajectories of particles injected at position xi and time ti 103 into a general potential V (x) with noise g0 and compute the π 8 | |3/2 e 3 α T (α) mean ﬁrst passage time to x f , following the standard approach |α|1/2 [18,30,38]. Let P(x, t ) be the distribution of particles over po- Deterministic P , = δ − sitions x at time t, with (x ti ) (x xi ). The probability Arrhenius that a particle has not reached x f at time t is Simulations x f 102 P(t ) = P(x, t ) dx. (10) −∞ τ 3

Note that P(0) = 1 and P(t ) → 0ast →∞as long as there / 2 0 is noise driving the system, which guarantees particles reach g x f . The distribution of first passage times is p(t ) =−dP/dt so that the mean first passage time is 101 ∞ ∞ τ (xi|x f ) = tp(t ) dt = P(t ) dt, (11) π 0 0 1 2 α / 2 2 where we integrate by parts for the second equality. To de- 21/3π2 Ai (−22/3α)+Bi (−22/3α) rive a differential equation for τ (xi|x f ), we start from the Kolmogorov backward equation for distribution P(x, t ) with −1 0 1 4/3 initial condition xi [39], α = 0/g0 2 dP(x, t ) dP(x, t ) 1 d P(x, t ) − =− + 2 . FIG. 2. Comparison of the universal scaling function T (α)(solid V (xi ) g0 2 (12) dti dxi 2 dxi curve) to the Arrhenius (dotted curve) and deterministic (dashed To write this equation in terms of the mean first passage time curve) limits. Also shown are the mean escape times for 500 sim- τ, we multiply both sides by t and integrate over x and t. ulations of the barrier escape process. For the simulations we fixed = =− = Using the relations in Eqs. (10) and (11) and the identity g0 1 while varying 0 and used boundary conditions x f xi 25. Agreement with our analytic expression for T (α) is excellent. dP(x, t )/dti =−dP(x, t )/dt,wearriveat The insets show snapshots of the barrier crossing simulations for 1 2 τ | − τ | =− . =± 2 g0 (xi x f ) V (xi ) (xi x f ) 1 (13) 0 1. This gives an ordinary differential equation for the first passage time from xi to x f of particles in potential V (x) two known limits when the barrier is large or the potential and constant noise with amplitude g0. The boundary con- is strongly downward sloping and agrees excellently with the ditions are τ (x f |x f ) = 0 and τ (−∞|x f ) = 0, which encode absorbing and reflecting boundaries, respectively. Writing the numerical results. solution to Eq. (13) in integral form, we arrive at the result Kramers’ escape rate for the cubic potential follows from obtained in Refs. [18,30,38], Eq. (15) and the asymptotic form of the second Airy function. α → x y As 0, however, contributions from the first Airy function f − 2 [V (z)−V (y)] 2 g2 τ | = 0 , become important so that Kramers’ theory and extensions (xi x f ) 2 dy dze (14) g0 xi −∞ involving anharmonic corrections break down. The difference which satisfies the boundary conditions as long as V (x) →∞ between Eqs. (8) and (15) is also related to the narrowing of as x →−∞. For large barriers it is known that Eq. (14) the spectral gap of the barrier crossing Fokker-Plank operator reproduces Kramers escape rate formula via a saddle point ap- (which has been measured numerically [40] and is discussed proximation that expands the potential around the maximum below in Sec. IV). and the minimum (as shown in Fig. 1) to second order [18]. Our renormalization group analysis allows us to restrict our focus to the relevant variables. For the cubic potential (sys- B. Corrections to scaling tems on the unstable manifold of the renormalization group 1. Finite launching and absorbing positions fixed point), the escape time can be computed analytically One limitation of our result Eq. (15) is that we assume =− →∞ using Eq. (14) in the limit x f xi . We find that initial and final states at infinity. In a real chemical or me- τ = −2/3T α g0 ( ) with the universal scaling function given by chanical system, the transition of interest generally occurs between states with finite coordinates. For systems with large T (α) = 21/3π 2[Ai2(−22/3α) + Bi2(−22/3α)], (15) barriers, the escape time is exponentially large compared to the time it takes to settle into a metastable state in the well. where Ai(x) and Bi(x) are the first and second Airy functions Thus, the scaling function Eq. (15) is universal: independent α = / 4/3 and 0 g0 as above. This solution is shown in Fig. 2, of initial and final conditions. When the potential is downward along with the Arrhenius and deterministic limits given in sloping, the barrier crossing time is still dominated by the Eqs. (8) and (9), respectively, and the mean barrier cross- time spent near the inflection point and we can systematically ing times from direct simulations of the Langevin process compute corrections to scaling due to the finite initial and final [Eq. (1)]. The universal scaling function T (α) reproduces the positions.

013156-4 REACTION RATES AND THE NOISY SADDLE-NODE … PHYSICAL REVIEW RESEARCH 3, 013156 (2021)

α = 4/3 χ = δV x =− x4/ − x5/ > Working in terms of scaling variables /g0 , − ( ) 3 4 4 5 with 4 0 and sufficiently large 2/3 2/3 so that the potential remains monotonically decreasing. To xi/g , and χ+ = x f /g , we can write the barrier crossing 0 0 β γ = 4/3 time for the cubic potential with arbitrary initial and final con- quadratic order in and 4g0 , the universal scaling τ = −2/3T α,χ ,χ function is [41] ditions as g0 ( − +) with the scaling function T α,χ ,χ = T α − T α,χ − T α,χ . T (α,β,γ ) ( − +) ( ) −( −) +( +) (16) α 0 π 15 √ 3π Here T±(α,χ±) are the universal corrections for finite final −−→ √ − β2 π α − √ and initial conditions, respectively. These have integral repre- α 8 4 γ sentations, √ 3 √ 5 35 −π γ + π αγ − παγ3/2 + πα3/2γ 2. χ− y (21) −2(y3/3+αy−z3/3−αz) 2 2 8 T−(α,χ−) = 2 dy dze (17) √ −∞ −∞ The term π/ α is just the deterministic limit of the√ scaling β2T α =− π αβ2/ and form for the cubic potential and 3( ) 15 8 ∞ y comes from the quartic perturbation to the inflection point. −2(y3/3+αy−z3/3−αz) T+(α,χ+) = 2 dy dze . (18) Other terms arise from quintic corrections or global changes χ+ −∞ γ in the potential. Here is a dangerous irrelevant variable√ [42, πβ2/ γ Assuming χ− < 0 and χ+ > 0 these can be expanded in pow- Secs. 3.6, 5.4, and 5.6], which has a pole 3 4 in the ersof1/χ±, expansion about 0, because it is needed to keep the potential monotonic (for β = 0). −1/2 −1 1/2 T±(α,χ±) =|α| tan (|α| /|χ±|) α α2 + 1 1 − + 3 IV. APPROXIMATING THE DISTRIBUTION 4 2 4 χ± 4 2χ± 4χ± OF ESCAPE TIMES 1 5 13α To completely characterize the escape times, we require ± − + χ −10 . 7 2 O( ± ) (19) their distribution, which captures the full range of outcomes χ± 14 9χ± we might expect from the stochastic dynamics of the Langevin The first term, which we can compute exactly, is simply equation. For high barriers (the Arrhenius limit) the escape is the correction due to the deterministic trajectory in the cu- dominated by the decay of a single (quasiequilibrium) mode bic potential between χ± and ±∞. The higher order terms in the bottom of the potential well, so the distribution is expo- describe the influence of noise on the corrections due to fi- nential with rate parameter given by 1/τ. On the other hand, nite initial conditions. These terms appear to have the form for small barriers or sloping potentials many modes contribute −1−3 j 2 χ± f j (α/χ±) for integers j and some functions f j.The and the mean may not be representative of the escape times in expansions of f1 and f2 are given in parentheses in the ex- general. In this section we develop an approximation to the pression above. distribution of escape times which is accurate for all α (i.e., Our corrections to scaling for finite launching and ab- any cubic potential). The approximate distribution is given sorbing positions are universal if the initial conditions are as an analytical scaling form parametrized by two variables sufficiently close to the cubic inflection point so that anhar- which are computed numerically for a given α. monic corrections are small. Thus, for an arbitrary potential, the universal escape time near a saddle-node bifurcation is A. First-passage distributions in Markov processes given by Eq. (15) corrected using Eq. (19). In the following section we will separately treat the anharmonic corrections To study the distribution we borrow a result from the theory to scaling. The interplay between corrections to scaling due of Markov processes. Consider a birth-death process, i.e., a to anharmonicity and initial conditions will be an interesting Markov chain with a one-dimensional state space and hopping subject for future studies. only between nearest neighbor states, that has one reflecting boundary and one absorbing boundary. This is precisely 2. Anharmonic corrections what we would obtain by discretizing space in the Langevin equation [Eq. (1)], which has a one-dimensional state space The scaling function Eq. (15) also serves as a starting with a reflecting boundary for large negative x values and an point from which the theory can be systematically improved absorbing boundary for large positive x values. For Markov by computing anharmonic corrections to scaling. The higher systems initialized near the reflecting boundary, the distribu- order terms in the potential are irrelevant variables under tion of times to reach the absorbing state can be characterized the renormalization group flows and hence can be treated completely in terms of the eigenvalues of the transition matrix. perturbatively. For instance, consider a quartic perturbation In particular the distribution p(t ) =−dP(t )/dt (using the δ =− 4/ β = 2/3 V (x) 3x 4 and let 3g0 . In the Kramers regime notation from the preceding section) can be shown to be a α →−∞ T α,β ≈ T α + β2T α we have that ( ) ( ) 3( ) to lead- convolution of exponentials [43], ing order, where p(t ) = E(λ ) ∗ E(λ ) ∗···∗E(λN ), (22) α / 0 1 0 8 |α|3 2 3/2 T3(α) −−→ π |α|e 3 (8|α| + 11)/8. (20) where λn are the negative eigenvalues of the Markov transition In the deterministic regime α →∞ we also add a quintic matrix, E(λn ) are exponential distributions, and ∗ denotes a term as a regulator on the boundary conditions of the potential convolution. As we will see below, it is useful to consider

013156-5 DAVID HATHCOCK AND JAMES P. SETHNA PHYSICAL REVIEW RESEARCH 3, 013156 (2021) the Fourier transformp ˜(ω) of the distribution p(t ) and the Λn(α) cumulant generating function 20 ∞ m logp ˜(ω) = κm(iω) /m! (23) 15 m=1 since the cumulants κm are easily expressed in terms of the eigenvalues of the Markov matrix, 10

N 1 κ = (m − 1)! . (24) 5 m λm Λ n=0 n α κ Note that the ﬁrst cumulant 1 is just the mean barrier crossing −2 −1 0 1 2 α time τ, which was the focus of the previous section.

FIG. 3. The scaling forms n(α) for the first seven eigenvalues. B. Spectra of the Fokker-Planck operator For large positive α the eigenvalues are approximately evenly spaced. For large negative α the leading eigenvalue approaches 0 and the When the state space becomes continuous, the process is gap to the second eigenvalue grows. The scaling form for the leading generated by the Fokker-Planck operator rather than a finite eigenvalue 0(α) ≡ α used in our approximation to the distribution transition matrix. Thus we want to understand the spectra of of escape times is shown in red. the right-hand side of the Fokker-Planck equation, 2 ∂P(x, t ) ∂ 1 ∂ P(x, t ) (α) is the decay rate of the slowest decaying eigenmode. = [V (x)P(x, t )] + g2 . (25) 0 ∂t ∂x 2 0 ∂x This approximation becomes exact in the limit α →∞,for For numerical evaluation of the eigenvalues it is conve- which the effective potential in Eq. (26) becomes harmonic. nient to change variables so that the differential opera- Though we can see in Fig. 3 that this approximation is clearly P , not correct for α →−∞, the behavior in this limit is domi- tor is Hermitian. To do this we rescale (x t )bythe square root of the Boltzmann factor, defining σ (x, t ) = nated by the vanishing leading eigenvalue α. For instance, P , / − / 2 σ , the cumulants in Eq. (24) are insensitive to α when α (x t ) exp[ V (x) g0]. Then (x t ) satisfies α and α → 0. Thus, this approximation will produce a ∂σ(x, t ) g2 ∂2 1 1 family of distributions which correctly captures the behavior = 0 + V (x) − [V (x)]2 σ (x, t ), ∂ ∂ 2 2 for both large negative and positive α. As we will see below, t 2 x 2 2g0 (26) the approximation is also quite accurate over the full range which has the form of a Schrödinger equation with imaginary of α. time. Importantly, even in the limit of continuous space, the One caveat of the equal spacing approximation is that T = K = ∞ −1 spectrum of the Fokker-Plank operator is discrete. This is easy the mean escape time 0 n=1 n always diverges. to see from Eq. (26) because the “effective quantum potential” Therefore, we must fix the mean to the value derived in 2/ 2 − / Sec. III A. On the other hand, the higher order cumulants, [V (x)] 2g0 V (x) 2 is bounded from below for the cubic potential as well as any polynomial potential that diverges which determine the shape of the distribution, always con- superlinearly as x →±∞. verge. Within the scaling theory developed in Sec. II, if we spe- To proceed, we sum Eq. (24) using our eigenvalue ansatz. cialize to the cubic potential (i.e., relevant variables of the For m 2 the result is renormalization group) the eigenvalues have scaling forms ψ (m−1) / K α = − m ( α α ), 2/3 m( ) ( 1) m (28) λ = α , α n g0 n( ) (27) / ψ (m) written in terms of the RG-invariant quantity α = /g4 3. where (x) is the polygamma function of order m.Ne- 0 glecting the mean, we can sum the series Eq. (23), then This also implies a scaling form for the cumulants κm = −2m/3 exponentiate and inverse Fourier transform to obtain the g K (α). The scaling forms (α) are plotted in Fig. 3 0 m n distribution p(t ). Writing the distribution in terms of its uni- for the first several eigenvalues. These curves describing the ,, = 2/3ρ ,α = − α dependence of the Fokker-Planck spectra are universal for versal scaling function p(t g0 ) g0 (s ) with s (t 2/3 4/3 systems near a saddle-node bifurcation (where α is the only τ )/g and α = /g , the final result is 0 0 relevant variable). For large positive α, the eigenvalues are α (0) ψ (α /α )−α s approximately evenly spaced, while for large negative α,a ρ(s,α) = exp e − αs (α/α ) single slowly decaying mode with eigenvalue very close to 0 dominates the behavior. α (0) + ψ (α/α ) . (29) α C. Evenly spaced eigenvalue approximation To use this family of distributions, we compute the leading To develop an approximation to the distribution of barrier eigenvalue α using a shooting method on the right-hand side escape times for the cubic potential, we assume the eigen- of Eq. (26). The effective α is chosen so that the second values are equally spaced, n(α) = α + nα, where α ≡ cumulant K2 (i.e., the variance) is exact. To do this, we follow

013156-6 REACTION RATES AND THE NOISY SADDLE-NODE … PHYSICAL REVIEW RESEARCH 3, 013156 (2021)

0.20 (a) α = −2 (b) α =0 2.0 0.15 ] 4 1.5 ) 10

× 0.10 )[ s, α

1.0 ( ρ s, α (

ρ 0.05 0.5

0.0 0.00 −5000 0 5000 10000 15000 −5 0 5 10 s s

0.8 3 ) ) 0.6 s, α s, α 2 ( ( ρ ρ 0.4

1 0.2

0.0 0 −1 0 1 2 −0.50 −0.25 0.00 0.25 0.50 s s

FIG. 4. The barrier crossing time distributions obtained using our evenly spaced eigenvalue approximation Eq. (29) (lines) and from direct simulation of the Fokker-Planck equation (symbols) for (a) α =−2, (b) α = 0, (c) α = 2, and (d) α = 7. In all cases agreement between the theory and simulations is excellent. In the large barrier limit (a) the distribution is approximately exponential and in the strongly sloped potential (d) it is nearly Gaussian.

3(1−m)/2 the same approach used in Sec. III A to write an integral like Km(α) ∼ α for α →∞. In particular, the ratio / expression for the variance. The result is K = g4 3(τ − τ 2 ) K /Km/2 → > 2 0 2 m 2 0form 2, i.e., the higher order cumulants with are small compared to the variance and the distribution ap- x y f − 2 [V (z)−V (y)] proaches a Gaussian for large α. This prediction is born out in 2 g2 τ | = τ | 0 , 2(xi x f ) 2 dy dz (z x f ) e (30) direct simulation of the Fokker-Plank equation [see Fig. 4(d)]. g −∞ 0 xi The distribution Eq. (29) combined with our analytical where τ (z|x f ) is the mean given in Eq. (14). After evaluating understanding of the mean escape time provides a complete these integrals numerically to obtain the variance, we solve description of the barrier crossing process. Our procedure K = ψ (1) /2 2 ( α ) α to fix the eigenvalue spacing α. allows for accurate approximations to the distribution by In Fig. 4 we compare the approximate distribution Eq. (29) evaluating just the two universal scaling forms α and α. to those obtained by direct numerical simulation of the Knowledge of these quantities allows for evaluation of the Fokker-Plank equation. Our approximation captures the shape distribution of barrier crossing times for any system near a of the distributions remarkably well for the full range of α. saddle-node bifurcation. We can also show analytically that this distribution reproduces the correct large α limits. For large negative α, taking V. DISCUSSION the α →∞limit of Eq. (29), gives the expected exponen- −sα −1 tial distribution ρ(s,α) → αe (shifted to have zero We expect our results will be directly applicable to mean). For large positive α we can estimate α and α barrier crossing processes in which thermal fluctuations using the harmonic approximation√ to the potential in Eq. (26). are comparable to the energy barrier including the afore- 2 We find α ∼ α and α ∼ α so that the cumulants scale mentioned experimental systems, narrow escape problems

013156-7 DAVID HATHCOCK AND JAMES P. SETHNA PHYSICAL REVIEW RESEARCH 3, 013156 (2021) in cellular biology [44], and downhill protein folding the colored noise under coarse graining (for example, barrier scenarios [45,46]. Our approximation to the distribution of crossing between two symmetric wells—a noisy pitchfork barrier escape times, combined with our analytical results for bifurcation—coupled to an Ohmic heat bath leads in the quan- the mean, provides an accurate and complete characterization tum limit to a critical point in the same universality class as of the barrier crossing process for systems near a saddle-node the Kondo problem [48]). We expect short-range correlations bifurcation. will be irrelevant under coarse graining, while those with A more thorough analysis of incorporating perturbative power-law decay will give rise to new anomalous scaling. For corrections from irrelevant variables into Eq. (14) would be some reactions, an underdamped model or multidimensional both theoretically interesting and useful in applications. The reaction coordinate may be required for an accurate descrip- interplay between anharmonic corrections and finite initial tion. Renormalization group scaling will provide a natural conditions is also important. Computing these corrections will organizing framework for these studies. extend the applicability of our theory to systems in which Nucleation of abrupt phase transitions (e.g., raindrop the boundary conditions correspond to positions with non- formation) is also described by Arrhenius rates. Here the negligible anharmonicity. It will be interesting to test the noiseless bifurcation underlying critical droplet theory is the accuracy of our evenly spaced eigenvalue approximation to spinodal line. This line—a discredited mean-field boundary the barrier escape time distribution when irrelevant variables between nucleation and spontaneous phase separation—could are incorporated. We conjecture that, at least perturbatively, play the role of our renormalization-group fixed point in a fu- Eq. (29) will still accurately parametrize the distributions ture generalization of this work to higher spatial dimensions. if α and α are corrected to account for the irrelevant Finally, the saddle-node bifurcation is the simplest exam- variables. ple of a depinning transition. One anticipates studying how Our analysis directly translates to higher order cuspoid adding noise would affect depinning of earthquakes, vor- catastrophes [47, Sec. 36.2], which form their own universal- tices in superconductors, plastic flow in crystals, raindrops on ity classes with different exponents (in fact, these have already windshields, coffee soaking into napkins, and other depinning been analyzed for the discrete iterated map [31,32]). For these phenomena. Each of these has anomalous exponents and RG bifurcations, the fixed point potential will be a higher order treatments [49] even without added noise. Our work provides monomial and our analysis can be used to identify the relevant a stepping stone toward understanding the universal scaling variables and develop a scaling theory for quantities like the near noisy depinning transitions in these more sophisticated barrier crossing time mean and distribution. systems as well. More generally, it would be useful to study the applicability of our renormalization group and scaling analysis to ACKNOWLEDGMENTS systems with colored noise, multiple dimensions, or in other damping regimes. The effects of colored noise are encoded in We thank Alen Senanian for helpful discussions. This work the correlation function ξ (t )ξ (t )=G(x, t − t ). The renor- was supported by an NSF Graduate Research fellowship Grant malization group transformation can be adapted to act on the No. DGE-1650441 to D.H. and by NSF Grant No. DMR- Fourier transform of this quantity G˜ (x,ω), giving flows of 1719490 to J.P.S.

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