Stochastic Resetting for Enhanced Sampling Oﬁr BlumerShlomi Reuveniand Barak Hirshberg School of Chemistry Tel Aviv University Tel Aviv 6997801 Israel.

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Stochastic Resetting for Enhanced Sampling

Oﬁr Blumer,†Shlomi Reuveni,†,‡,¶and Barak Hirshberg∗,†,‡

†School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel.

‡The Center for Computational Molecular and Materials Science, Tel Aviv University, Tel

Aviv 6997801, Israel.

¶The Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv

6997801, Israel.

E-mail: hirshb@tauex.tau.ac.il

Abstract

We present a method for enhanced sampling of molecular dynamics simulations us-

ing stochastic resetting. Various phenomena, ranging from crystal nucleation to protein

folding, occur on timescales that are unreachable in standard simulations. This is often

caused by broad transition time distributions in which extremely slow events have a

non-negligible probability. Stochastic resetting, i.e., restarting simulations at random

times, was recently shown to signiﬁcantly expedite processes that follow such distribu-

tions. Here, we employ resetting for enhanced sampling of molecular simulations for

the ﬁrst time. We show that it accelerates long-timescale processes by up to an order

of magnitude in examples ranging from simple models to molecular systems. Most

importantly, we recover the mean transition time without resetting – typically too long

to be sampled directly – from accelerated simulations at a single restart rate. Stochas-

tic resetting can be used as a standalone method or combined with other sampling

algorithms to further accelerate simulations.

arXiv:2210.00558v1 [physics.chem-ph] 2 Oct 2022

Introduction

Molecular dynamics (MD) simulations are very powerful, providing microscopic insights into

the mechanisms underlying physical and chemical condensed phase processes. However,

due to their atomic spatial and temporal resolution, standard MD simulations are limited

to events that occur on timescales shorter than ∼1µs.1,2 In many cases, the complex

dynamics of the system lead to longer timescales, through a very broad distribution of

transition times between metastable states, also known as ﬁrst-passage times3(FPT). To

demonstrate this, Fig. 1 presents the probability density, denoted by f(τ), of the FPT,

τ1, τ2, ..., τN, obtained from Nsimulations of transitions between the two conformers of an

alanine dipeptide molecule – a common model system.3,4 It shows that many transitions occur

on a timescale much shorter than 1 µs – more than 25% of them under 100 ns. However, the

tail of the distribution decays so slowly that the mean FPT is almost an order of magnitude

larger, 759 ns, and some trajectories fail to complete even after 4 µs. There is thus an ongoing

eﬀort to develop procedures for expediting such processes.5,6

Figure 1: (a) The two conformers of an alanine dipeptide molecule. (b) The FPT distri-

butions for transitions between them, starting from C7eq, without resetting (blue circles)

and with Poisson resetting at a rate of r= 0.1ns−1(green squares). The y axis is given

on a logarithmic scale. The full details of the simulation protocol and how the FPT was

determined are given in the SI.

Stochastic resetting (SR) is the procedure of occasionally stopping and restarting random

processes using independent and identically distributed initial conditions. The resetting

times are typically taken at constant intervals (“sharp resetting”) or from an exponential

distribution with a ﬁxed rate (“Poisson resetting”). The interest in SR has grown signiﬁcantly

since the pioneering work of Evans and Majumdar.7They showed that while a particle

undergoing Brownian motion between two ﬁxed points in space has an inﬁnite mean FPT, its

mean FPT with SR becomes ﬁnite. Therefore, the particle reaches the target point inﬁnitely

faster on average. This result has eﬀectively established an emerging ﬁeld of research in

statistical physics, to which a recent special issue was dedicated.8,9

The power of resetting in accelerating random processes has been widely demonstrated:

in randomized computer algorithms,10–12 in various search processes,13–20 experimentally

in systems of colloidal particles,21,22 in queuing systems,23,24 and in the Michaelis–Menten

model of enzymatic catalysis, where resetting occurs naturally by virtue of enzyme-substrate

unbinding.25,26 The latter ﬁnding was then leveraged to develop a general treatment of ﬁrst-

passage processes under restart.27 There, it was shown that the FPT distribution in the

absence of SR can be used to determine the FPT distribution with resetting. Moreover,

the mean and standard deviation of the FPT distribution without resetting are enough to

determine a suﬃcient condition for SR to expedite a random process.28 Speciﬁcally, if the

ratio of the standard deviation to the mean FPT (the coeﬃcient of variation, COV) is greater

than one, a small reset rate ris guaranteed to lower the mean FPT. The slowly-decaying

distributions that occur in molecular simulations of long-timescale processes can also have a

COV that is greater than one. For example, the distribution in Fig. 1 has a COV of ∼1.3.

This indicates that resetting can expedite MD simulations.

In this work, we use SR for the ﬁrst time for enhanced sampling of molecular simulations.

MD simulations are an exciting playground for the application of resetting, while raising

new fundamental questions that are of interest to both communities. In SR, the unbiased

kinetics (without resetting) are known, and the goal is to understand how much speedup can

be gained by restarting the random process. On the other hand, in the MD community, the

long-timescale processes cannot be accessed directly and enhanced sampling methods are

required to expedite them. Introducing SR for this purpose raises the question of inference –

can we obtain the free energy surfaces and the kinetics of reset-free processes from simulations

with SR? This question has not been explored in the SR community but is the natural goal

of enhanced sampling methods.

Various methods have been developed in the ﬁeld of molecular simulations to overcome

the long-timescale problem, such as umbrella sampling,29,30 Metadynamics,1,31–33 on-the-ﬂy

probability enhanced sampling (OPES),34–36 and adiabatic free energy dynamics.37–39 Many

of them rely on identifying suitable collective variables – eﬀective reaction coordinates that

ideally describe the slowest modes of the process.40 Below, we show that SR can be used

for enhanced sampling without ﬁnding suitable collective variables, which is highly non-

trivial for condensed phase processes.41,42 Most importantly, we demonstrate that the mean

transition times without resetting, that are often too long to be sampled directly, can be

recovered from accelerated simulations performed at a single restart rate. In this letter

we give a proof of concept for these desirable features using examples ranging from simple

models to a molecular system. We obtain a speedup by an order of magnitude in some cases.

Our method opens new avenues in both the MD and SR communities, hopefully promoting

a fruitful collaboration between the two.

Results and discussion

We begin by demonstrating that SR can indeed enhance the sampling of MD simulations.

Mathematically, we know that if the COV is greater than one, it is guaranteed that resetting

can expedite the process. But for what potential energy surfaces do we expect this to

occur? We answer this question using three illustrative model systems representing possible

scenarios in MD simulations. Resetting was successful in accelerating transitions in all of

them, and, for two of them, we obtained an order of magnitude speedup in the mean FPT.

To benchmark our approach, we chose the parameters of the model potentials such that

the mean FPT without resetting is accessible (∼1ns) to allow extensive sampling of the

unbiased process. Below, we brieﬂy describe the models while the full parameters are given

in the SI. The results for each model are given in a separate row in Fig. 2. In all cases, the left

panel shows the potential and the middle panel presents the FPT probability density f(τ)

without resetting. The right panel shows the speedup obtained by both Poisson and sharp

resetting, at diﬀerent restart rates r. All simulations are of a single particle initialized at

ﬁxed positions, denoted by stars in the left panels of Fig. 2, with an initial velocity sampled

from the Maxwell-Boltzmann distribution at 300 K. The dashed line in Fig. 2 deﬁnes the

spatial threshold for the ﬁrst passage. The simulations were performed in the Large-scale

Atomic/Molecular Massively Parallel Simulator (LAMMPS),43 with SR easily implemented

in the input ﬁles. Full details and input examples are given in the SI and the corresponding

GitHub repository.44

The ﬁrst model is presented in the top row of Fig. 2. It is a one dimensional double-well

potential that is composed of a trapping harmonic term and a Gaussian centered at x= 0 ˚

The model has two symmetric minima that are separated by a moderate barrier (1 kBT).

The harmonic spring constant was taken to be soft, such that the particle can explore areas

very far away from the center (∼100 ˚

A). This model, with a diﬀerent choice of parameters,

was previously used to describe the umbrella inversion in ammonia.45 The simulations were

initiated at the right minimum (x= 3 ˚

A) and the FPT was deﬁned as reaching the second

basin (x≤ −3˚

A). The distribution without resetting is broad, spanning about four orders

of magnitude (note the logarithmic timescale), and has a COV of ∼2.9. In the absence of

resetting, some transitions occur as fast as a few picoseconds while others take as long as tens

of nanoseconds. The median FPT is 125 ps and the mean FPT is 1325 ps. By introducing

SR, we were able to reduce the mean FPT by more than an order of magnitude, with a

speedup of 10.5 and 12.1 for Poisson and sharp resetting, respectively. The results agree

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StochasticResettingforEnhancedSamplingOrBlumer,yShlomiReuveni,y,z,{andBarakHirshberg,y,zySchoolofChemistry,TelAvivUniversity,TelAviv6997801,Israel.zTheCenterforComputationalMolecularandMaterialsScience,TelAvivUniversity,TelAviv6997801,Israel.{TheCenterforPhysicsandChemistryofLivingSystems,TelAvivU...

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Stochastic Resetting for Enhanced Sampling Oﬁr BlumerShlomi Reuveniand Barak Hirshberg School of Chemistry Tel Aviv University Tel Aviv 6997801 Israel.

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