Stochastic transitions Paths over higher energy barriers can dominate in the early stages S. P. Fitzgerald1A. Bailey Hass1G. D ıaz Leines2and A. J. Archer3y 1Department of Applied Mathematics University of Leeds Leeds LS2 9JT United Kingdom

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Stochastic transitions: Paths over higher energy barriers can dominate in the early stages

S. P. Fitzgerald,1, ∗A. Bailey Hass,1G. D´

ıaz Leines,2and A. J. Archer3, †

1Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

2Yusuf Hamied Department of Chemistry, University of Cambridge,

Lensﬁeld Road, Cambridge CB2 1EW, United Kingdom

3Department of Mathematical Sciences and Interdisciplinary Centre for Mathematical Modelling,

Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom

The time evolution of many physical, chemical, and biological systems can be modelled by stochastic transitions

between the minima of the potential energy surface describing the system of interest. We show that in cases

where there are two (or more) possible pathways that the system can take, the time available for the transition

to occur is crucially important. The well-known results of reaction rate theory for determining the rates of the

transitions apply in the long-time limit. However, at short times, the system can instead choose to pass over

higher energy barriers with much higher probability, as long as the distance to travel in phase space is shorter.

We construct two simple models to illustrate this general phenomenon. We also present an extension of the

gMAM algorithm of Vanden-Eijnden and Heymann [J. Chem. Phys. 128, 061103 (2008)] to determine the most

likely path at both short and long times.

The transition dynamics of complex systems having many de-

grees of freedom can often be reduced to one or two reaction

coordinates. These are the system degrees of freedom that

evolve the slowest in time [1]. All the other (maybe very

many) degrees of freedom are slaved to these slowest pro-

cesses [2]. The slow evolution of these systems is usually

characterized by rare transitions between metastable states

separated by signiﬁcant energy barriers. The identiﬁcation of

the reaction coordinates in high-dimensional (complex) sys-

tems remains extremely challenging [3]. For example, for

large molecules, the centre of mass is often a ‘slow’ degree

of freedom, whilst the ﬂuctuations of the individual atoms

within the molecule are the slaved ‘fast’ degrees of freedom.

This is the case e.g. in biomolecular conformation changes

such as protein folding [4, 5], nucleation-driven phase trans-

formations [6], chemical reactions in general and surfactant

molecules in a liquid transitioning from being freely dispersed

in the liquid or joined together in a micelle or adsorbing to in-

terfaces [7–9]. An example of recent work to identify the rel-

evant reaction coordinates is Ref. [10], which uses machine-

learning. Of course, for high-dimensional systems, the energy

landscape is often complex, with multiple critical points, bar-

riers of various sizes and multiple transition paths connect-

ing the stable states. For such systems, algorithms based on

simplifying assumptions such as no barrier recrossings, single

transition states, a smooth landscape, ‘long enough’ (inﬁnite)

times often fail to provide an straightforward and accurate es-

timation of the rate [6] or to even identify the most likely path

[11, 12] under perturbations of the energy landscape.

We consider here a class of such stochastic dynamical sys-

tems where there is a simple choice of two transition pathways

away from the initial state: one is over a smaller energy bar-

rier (the activation energy barrier for chemical reactions), but

the system has to evolve a greater distance in phase space (i.e.

has a longer reaction pathway), while the other path is over a

much higher energy barrier, but has a much shorter distance

to travel in phase space. Examples of such systems include

where a surfactant molecule in liquid has a choice between ad-

sorbing to an interface or forming micelles, or where a chem-

ical reaction can proceed via a catalyzed or non-catalyzed

route. Standard reaction-rate theory (RRT) which includes

transition state theory and other related approaches [2] pre-

dicts that the path over the lowest barrier is the most likely

and therefore dominates the dynamics, at least for simple en-

ergy landscapes. However, even for these simple cases we

ﬁnd the standard RRT picture does not hold and the behaviour

crucially depends on the timescale over which the system is

sampled. In particular, for shorter times (but still much longer

than the timescale of the ﬂuctuating ‘fast’ degrees of freedom)

the ﬂux over the higher barrier can completely dominate the

dynamics of the system and even at intermediate times, the

transition probabilities are very different from the predictions

of RRT approaches which do not consider the time taken; i.e.

RRT only applies in the long-time limit. The key ﬁnding of

our work is that the length of time over which barrier crossing

problems are allowed to proceed is critically important. In any

system where the reaction is stopped after a certain time, the

reaction pathway predicted by RRT may not be the one actu-

ally taken. For example, this may be the case in ﬂow reactors

such as catalytic converters. However, in any system that can

explore the long-time limit, the predictions of RRT are fully

recovered. Conversely, if the potential landscape and reaction

coordinates are unknown, and are inferred via densities and

rates measured from experiments or simulations necessarily

performed on a ﬁnite timescale, then the dominant long-time

dynamics of the system may be missed entirely.

Additionally, we develop a method for calculating the most

likely path (MLP) through the potential energy landscape,

useful for analysing systems with two or more dimensions.

Various techniques to compute the minimum energy (and

hence most probable) path between two minima exist, in-

cluding the string method and the geometric minimum ac-

tion method (gMAM [13, 14]; see also [15, 16]). Such paths,

sometimes known as the instanton, are everywhere parallel to

the potential gradient, and correspond to the inﬁnite-time tran-

sition. In this work, we extend the gMAM approach to ﬁnite-

arXiv:2210.11280v1 [cond-mat.stat-mech] 20 Oct 2022

time transitions, and derive a modiﬁed algorithm to compute

ﬁnite-time, out-of-equilibrium paths. These are no longer par-

allel to the potential gradient, and correspond to the most

probable path conditioned on a ﬁnite duration. We ﬁnd that

these can be radically different from the instanton, and may

pass through completely different intermediate states. We ex-

plain how these paths are connected to the full transient dy-

namics of the system given by the Fokker-Planck equation.

We demonstrate our ﬁndings with two simple generic toy

models. The ﬁrst is one-dimensional (1D) and the potential

energy landscape has three minima. The system is initiated in

the middle one and then has a choice to evolve either to the left

or to the right. Our second model potential is two-dimensional

(2D). It has two minima and two saddles, meaning two differ-

ent classes of path linking one minimum to the other. One

path is shorter, but over a high barrier in the potential, while

the other is further, but over a much lower barrier. RRT would

suggest that the second is the dominant transition pathway, but

we ﬁnd that this is not the case if one only considers the sys-

tem for sufﬁciently short times. These systems are described

by the overdamped stochastic equation of motion

Γ−1dx

dt =−∇φ(x) + η,(1)

where xis the ‘slow’ relevant degree of freedom of the system

(1D or 2D in the cases considered here), φ(x)is the potential

energy of the system (strictly speaking in systems where irrel-

evant ‘fast’ degrees of freedom have been integrated out, φis a

constrained free energy), Γ−1is a friction constant that hence-

forth we set equal to one (i.e. absorb it into the timescale) and

ηis a random force originating from thermal ﬂuctuations in

the system. This is modelled as a white noise with zero mean

hηi(t)i= 0 and correlator hηi(t)ηj(t0)i= 2kBT δij δ(t−t0),

where kBis Boltzmann’s constant and Tis the temperature

(i.e. the amplitude of the random ﬂuctuations).

The Fokker-Plank equation for the probability density ρ(x, t)

corresponding to Eq. (1) is [17]

∂ρ

∂t = Γ∇ · [kBT∇ρ+ρ∇φ].(2)

When φ= 0 this becomes ∂ρ

∂t =D∇2ρ, the diffusion equa-

tion, with diffusion coefﬁcient D= ΓkBT. Note that Eq. (2)

can be written as a gradient dynamics

∂ρ

∂t =∇ · Γρ∇δF

δρ ,(3)

with the Helmholtz free energy functional

F[ρ] = Zρ[kBTln ρ+φ] dnx,(4)

which is a Lyapounov functional for the dynamics. Note

that these are the equations of dynamical density functional

theory [18–20]. For a given potential φ(x), the equilibrium

density is ρ(x) = ρ0e−βφ(x), where β= (kBT)−1and ρ0

is a constant determined by the normalisation of ρ(x); i.e.

ρ−1

0=Re−βφ(x)dnx.

When φ(x)has at least two minima, the quantity of interest

is the typical waiting time to observe transitions between the

minima. Standard RRT states that this transition rate kis given

by the Arrhenius (or Kramers) relation k=νexp(−β∆φ)

where ∆φ≡φ(xs)−φ(xA)is the height of the barrier, with

xAbeing the position of the minimum and xsthe maximum

(more generally saddle-point) on the barrier. The prefactor ν

depends on various factors [2], but it is the exponential that

crucially determines the rate and can be thought of as origi-

nating from the ratio ρ(xs)/ρ(xA), which is the probability of

ﬁnding the system on the barrier divided by the probability of

it being at the minimum. However, this ratio ∝exp(−β∆φ)

only in the long time t→ ∞ limit. Solving Eq. (2) with the

initial condition ρ(x, t = 0) = δ(x−xA), we ﬁnd that the

RRT result can be completely wrong in some cases, if consid-

ering transitions with only a short time to occur.

We consider ﬁrst the 1D potential φ(x)in Fig. 1(a); the equa-

tion for φ(x)is given in the supplementary information (SI).

This potential has 3 minima [labelled A, B and C in Fig. 1(a)],

at xB≈ −2,xA≈1and xC≈2.5and two maxima [labelled

D and E] at xD≈ −1and xE≈2. We initiate the system in

the minimum at A. It can then either move to the right, over

the much higher energy barrier at E, or it can go to the left

over the lower barrier at D. Going left, it has further to travel.

In Fig. 1(b) we plot the density proﬁle ρ(x, t)obtained from

solving Eq. (2) for a sequence of different times t. Rather than

initiating the system with the Dirac δ-distribution centred at

xA, we use a narrow Gaussian corresponding to a free diffu-

sion for the short initial time t= 0.01. By the time t= 0.5

we see a sizable peak in ρ(x, t)at C, the right hand minimum

in φ(x), but very little density has made it to the minimum at

B. This is because B is further away, so in the early stages the

system is more likely to cross the barrier at E, despite it being

higher than the barrier at D. It takes until t≈30 for ρ(x, t)

to cease evolving in time and the system to reach the equilib-

rium distribution. Note also that at t= 5 the density at C is

higher than its eventual equilibrium value. Once the system

has ‘found’ the lower-energy minimum at B, density moves

back over the high barrier at E to approach ρ0e−βφ(x).

In Fig. 1(c) we plot the densities at the points D and E over

time. These are the locations of the two potential maxima (the

barriers). We see that at early times t∼0.1the probability of

being at the highest maximum E is sizeable and well above the

RRT probability ∼exp[−βφ(xE)], whilst the probability of

being at the lower maximum D is still ≈0, in contrast to the

RRT prediction that the probability ∼exp[−βφ(xD)]. Even

at t∼1, the RRT predictions are still incorrect.

We also consider a system evolving in the 2D potential dis-

played in Fig. 2; the precise expression for this potential is

given in the SI. Fig. 2(a) shows a contour plot, whilst 2(b)

is a surface plot. This potential has a local minimum at

point A: (xA, yA) = (0.38,−0.47) and the global minimum

at B: (xB, yB) = (0.42,0.47). There is a local maximum

near the origin. We initiate the system at A. There are two

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Stochastictransitions:PathsoverhigherenergybarrierscandominateintheearlystagesS.P.Fitzgerald,1,A.BaileyHass,1G.D´azLeines,2andA.J.Archer3,y1DepartmentofAppliedMathematics,UniversityofLeeds,LeedsLS29JT,UnitedKingdom2YusufHamiedDepartmentofChemistry,UniversityofCambridge,LenseldRoad,CambridgeCB21EW...

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Stochastic transitions Paths over higher energy barriers can dominate in the early stages S. P. Fitzgerald1A. Bailey Hass1G. D ıaz Leines2and A. J. Archer3y 1Department of Applied Mathematics University of Leeds Leeds LS2 9JT United Kingdom.pdf

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Stochastic transitions Paths over higher energy barriers can dominate in the early stages S. P. Fitzgerald1A. Bailey Hass1G. D ıaz Leines2and A. J. Archer3y 1Department of Applied Mathematics University of Leeds Leeds LS2 9JT United Kingdom

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