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RESTART: THE SCIENCE OF STARTING ANEW

Resetting, i.e., interrupting a process to return it to its initial state, permeates our daily lives. It is the act of hitting the refresh button on a browser when a page stalls, or the decision to retrace our steps when the search for misplaced car keys takes too long. Crucially, the concept also extends beyond mundane activities and it has recently blossomed into a burgeoning field of scientific inquiry that explores the implications and applications of resetting [J. Phys. A: Math. Theor. 53 193001, 2020; J. Phys. A: Math. Theor. 57 060301, 2024New Vistas in Stochastic Resetting]. Work in the field delves into the intricate dynamics and potential benefits of strategic resetting in complex systems, marking an exciting era of discovery and understanding in the science of starting anew. 

 

One motivation to study resetting phenomena comes from connections with first-passage processes. Many basic questions, as well as a wide array of applications, have turned first-passage processes into a long-standing focal point of scientific interest. Indeed, for two molecules to react they first have to meet, and for an animal to eat it must first find food. In fact, any process that has a start and an end can be viewed as a first-passage process—which in a nutshell explains why this field attracted so much attention and why it is so widely explored.

 

Recently, we and others observed that there are numerous situations where first-passage processes are stopped and started anew, which initiated a rapidly moving and extremely vibrant research front. Intuitively, one expects that restart will act to hinder the completion of a process. Yet, as illustrated below, there are cases where starting fresh and making another try helps get to the end goal faster. 

 

For example: (a) A molecule that was previously prepared at an excited state decays to a low energy state. A pulse of laser can bring the molecule back to its excited state and restart the reaction. This time, the desired product may be formed; (b) Protein folding is often envisioned as a random walk on a rugged potential landscape. Misfolded proteins are formed when the walk gets “stuck” at a local energy minimum. Chaperones save misfolded proteins by restarting the folding process, which gives another shot at success; (c) In enzymatic catalysis the formation of an enzyme-substrate (ES) complex is a necessary step en-route to product formation (E+P). Unproductive substrate unbinding may, however, reset the reaction and send it back to first base (E+S). Such an event can delay product formation, but may also speed it up(!).

 

Previously, we conducted a series of studies in an attempt to better understand how stopping a process in its midst—only to start it all over again—will affect its completion [Proc. Natl. Acad. Sci. 111 (12), 4391, 2014; Phys. Rev. E 92, 060101(R), 2015; Phys. Rev. Lett. 116, 170601, 2016]. These studies formed the basis for an overarching theoretical and conceptual framework that we developed to treat first-passage processes under restart [Phys. Rev. Lett. 118, 030603, 2017]. Applications of the theory range from random search processes that are reset by home returns, to enzymatic reactions which are inherently stochastic at the single-molecule level and are naturally subject to restart by virtue of substrate unbinding. We continue to the develop the theory and its applications as described below. 

First-passage under restart with branching

We generalized the theory of first-passage under restart to further include branching [Phys. Rev. Lett. 122, 020602, 2019]. Our analysis revealed that two widely applied measures of statistical dispersion—the coefficient of variation and the Gini index—come together to determine how restart with branching affects the mean completion time of an arbitrary stochastic process. The universality of this result was demonstrated and its connection to extreme value theory was pointed out and explored. Applications of the theory range from search processes, to the spread of epidemics, and can also help in better design of randomized computer algorithms.

Péclet number governs transition to acceleratory restart in drift-diffusion

We showed that the Péclet number, i.e., the ratio between the rates of advective and diffusive transport, governs how restart affects the first-passage time of drift-diffusion to a target [J. Phys. A. 52, 255002, 2019]. This finding is of interest since drift-diffusion is used to describe a myriad of naturally occurring processes, e.g., biological evolution, charge transport, polymer translocation, the dynamics of neuron firing, and many more. A figure from this paper was featured on the cover of the Journal of Physics A. 

Local time of diffusion with stochastic resetting

The local time of a Brownian particle, e.g., a molecule immersed in fluid, is defined as the total time the particle spends in a small vicinity of its initial position. Since Brownian trajectories are stochastic—the local time is a random quantity which fluctuates round and about its mean value. We characterized how the statistics of this random variable is altered due to stochastic resetting [J. Phys. A. 52, 264002, 2019]. The opening figure of this paper was featured on the cover of the Journal of Physics A.

Time-dependent density of diffusion with stochastic resetting is invariant to return speed

The canonical Evans-Majumdar model for diffusion with stochastic resetting [Phys. Rev. Lett. 106, 160601, 2011] assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. We considered a more realistic situation where the particle returns to the origin at a finite (rather than infinite) speed [Phys. Rev. E 100, 040101(R), 2019]. This creates a non-trivial coupling between the particle's random position at the moment of resetting and its return time. However, whether returns were slow or fast, we always found that the time-dependent distribution of the particle's position is identical to that obtained in the case of instantaneous returns. This surprising discovery was generalized in a subsequent paper (see below). 

Invariants of motion with stochastic resetting and space-time coupled returns

Motion under stochastic resetting serves to model a myriad of processes in the sciences, but in many cases resetting is assumed to take zero time or a time decoupled from the spatial position at the resetting moment. Yet, in our world, getting from one place to another always takes time; and places that are further away take more time to be reached. We thus extended the theory of stochastic resetting to account for this inherent spatio-temporal coupling [New J. Phys. 21, 113024, 2019]. We considered a particle that starts at the origin and follows a certain law of stochastic motion until it is interrupted at some random time. The particle then returns to the origin

via a prescribed protocol to start its motion anew. We showed that the shape of the steady-state distribution which governs the stochastic motion phase in this wide class of models does not depend on the return protocol. We then utilized this shape invariance to build a simple, and generic, recipe for the computation of the full steady state distribution. Several case studies were analyzed and a class of processes whose steady state is completely invariant with respect to the speed of return was highlighted. For processes in this class, one recovers the same steady-state obtained for resetting with instantaneous returns—irrespective of whether the actual return speed is high or low. 

Diffusion with resetting in a logarithmic potential

We analyzed the effect of resetting on diffusion in the logarithmic potential, which naturally arises in various physical scenarios [J. Chem. Phys. 152, 234110, 2020]. We showed that this analytically tractable model system exhibits a series of phase transitions as a function of a single parameter: the ratio of the strength of the potential to the thermal energy, βU0. We also provided a closed-form expression for the first-passage time to the origin in such a system; and showed that resetting can expedite arrival to the origin when -1 < βU< 5, but that it does the exact opposite when  βU0 > 5.

Log-Potential.png

Search with home returns provides advantage under high uncertainty

Many search processes are conducted in the vicinity of a favored location, i.e., a home, which is visited repeatedly. Foraging animals return to their dens and nests to rest, scouts return to their bases to resupply, and drones return to their docking stations to recharge or refuel. Yet, despite its prevalence, very little was known about search with home returns because its analysis is much more challenging than that of unconstrained, free-range search. 

We developed a theoretical framework for search with home returns [Phys. Rev. Research 2, 043174, 2020]. This makes no assumptions on the underlying search process and is furthermore suited to treat generic return and home-stay strategies. We showed that the solution to the home-return problem can always be given in terms of the solution to the corresponding free-range problem---which not only reduces overall complexity but also gives rise to a simple and universal phase-diagram for search. The latter reveals that search with home returns outperforms free-range search in conditions of high uncertainty. Thus, when living gets rough, a home will not only provide warmth and shelter but also allow one to locate food and other resources quickly and more efficiently than in its absence. 

Sharp restart

Restart protocols can use either deterministic or stochastic timers. Restart protocols with deterministic timers—`sharp restart'—assume a principal role: if there exists a restart protocol that improves mean-performance, then there exists a sharp-restart protocol that performs as good or better. To deepen our understanding of sharp restart, we conducted a comprehensive analysis of first-passage with sharp restart. This analysis covered mean [J. Phys. A: Math. Theor. 53 405004, 2020] and tail [J. Phys. A: Math. Theor. 54 125001, 2021] behaviors, and also unraveled deep relations between measures of statistical inequality and the effect of sharp restart [J. Phys. A: Math. Theor. 54, 355001, 2021]. More recently, we have also explored how the entropy [J. Phys. A: Math. Theor. 56, 024002, 2023] and diversity [J. Phys. A: Math. Theor. 56, 024003, 2023] of first passage times are affected by sharp restart.

Experimental realization of diffusion with stochastic resetting

Despite a long catalogue of theoretical studies dedicated to diffusion with stochastic resetting, there was surprisingly no attempt to experimentally study this process in a controlled laboratory environment (prior to 2020).

Collaborating with the group of Prof. Yael Roichman, we developed the first experimental setup that allows the study of colloidal particle diffusion and resetting via holographic optical tweezers [J. Phys. Chem. Lett. 11, 7350, 2020]. We provided experimental corroboration of central theoretical results and went on to measure the energetic cost of resetting in steady-state and first-passage scenarios. While one naively expects that this cost would gradually vanish as resetting is performed less and less frequently, we showed that the energetic cost is bounded from below and that it cannot be made arbitrarily small because of fundamental constraints on realistic resetting protocols. The methods we developed open the door to future experimental study of resetting phenomena.

Listen to a short radio interview about this work (Hebrew):

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Thermodynamic uncertainty relation for systems with unidirectional transitions

The last two decades have seen fundamental advancements in our understanding of non-equilibrium processes. One of the most important results in the field is the Thermodynamic Uncertainty Relation (TUR), which gives a thermodynamic bound on the fluctuations of stochastic currents. Since its discovery, a large body of research has

been devoted to study the validity, properties, and possible applications of this important result. Yet, the applicability of the TUR has been limited as it does not hold for models and systems with unidirectional transitions which are commonly used to describe many physically interesting processes, e.g. spontaneous emission in atoms, enzymatic catalysis, stochastic resetting, and more. We collaborated with Prof. Saar Rahav to derive a generalized TUR that applies to systems of this kind [Phys. Rev. Research 3, 013273, 2020]. Our derivation does not assume a steady-state, and the result holds equally well to transient processes with arbitrary initial conditions. In addition, the generalized TUR we derived can also be used to bound fluctuations of non-current observables, e.g., residence times. We thus extend the applicability of the TUR well beyond its current reach. 

Thermodynamic uncertainty relation for first-passage times on Markov chains

Many interesting phenomena occurring at the molecular scale can be viewed as first-passage processes. From chemical reactions to enzyme catalysis and molecular search processes — one is interested in the exact moment at which a distinguished event occurs: the first passage time. Yet, the

thermodynamics of first-passage time processes remains poorly understood owing to two fundamental challenges: transient dynamics, and irreversible transitions which render entropy production ill defined. We again collaborated with Prof. Saar Rahav to show how these issues can be overcome to derive a thermodynamic uncertainty relation that poses a lower bound on first-passage time fluctuations [Phys. Rev. Research 3, L032034, 2021]. The bound reveals that fluctuations can only be reduced at the expense of higher fluxes and/or entropy production; which has immediate applications to chemical and enzymatic reactions, and to molecular search processes, where first-passage time fluctuations are of central importance. Our work thus opens new research avenues connecting first-passage with stochastic thermodynamics. 

Diffusion with local resetting and exclusion

Our current understanding of stochastic resetting stems from the renewal framework, which relates systems subject to global resetting to their nonresetting counterparts. Yet, in interacting many-body systems, even the simplest 

scenarios involving resetting give rise to the notion of local resetting, whose analysis falls outside the scope of the renewal approach. A prime example is that of diffusing particles with excluded volume interactions that independently attempt to reset their position to the origin of a one-dimensional lattice. With renewal rendered ineffective, we instead tackled this system via a mean-field approach whose validity was corroborated via extensive numerical simulations. The emerging picture shed first light on the nontrivial interplay between interactions and resetting in interacting many-body systems [Phys. Rev. Research 3, L012023, 2021]

Resetting transition is governed by an interplay between thermal and potential energy

A dynamical process that takes a random time to complete, e.g., a chemical reaction, may either be accelerated or hindered due to resetting. Tuning system parameters, such as temperature, viscosity, or concentration, can invert 

the effect of resetting on the mean completion time of the process, which leads to a resetting transition. Although the resetting transition has been recently studied for diffusion in a handful of model potentials, it remained to be understood whether the results follow any universality in terms of well-defined physical parameters. To bridge this gap, we developed a general framework which revealed that the resetting transition is governed by an interplay between the thermal and potential energy [J. Chem. Phys. 154, 171103, 2021]. This general result was illustrated for different classes of potentials that are used to model a wide variety of stochastic processes with numerous applications. 

The inspection paradox in stochastic resetting

The remaining travel time of a plane shortens with every minute that passes from its departure, and a flame diminishes a candle with every second it burns. Such everyday occurrences bias us to think that processes which have already begun will end before those which have just started. Yet, the inspection paradox teaches us that the converse can also happen when randomness is at play. The paradox comes from probability theory, where it is often illustrated by measuring how long passengers wait upon arriving at a bus stop at a random time. Interestingly, such passengers may on average wait longer than the mean time between bus arrivals — a counter-intuitive result, since one expects to wait less when coming some time after the previous bus departed.

We review the inspection paradox and its origins. The insight gained is then used to explain why, in some situations, stochastic resetting expedites the completion of random processes [J. Phys. A: Math. Theor. 55, 021001, 2022]. Importantly, this is done with elementary mathematical tools which help develop a probabilistic intuition for stochastic resetting and how it works. This paper can thus be used as an accessible introduction to the subject.

Mitigating long queues and waiting times with service resetting

What determines the average length of a queue, which stretches in front of a service station? The answer to this question clearly depends on the average rate at which jobs arrive at the queue and on the average rate of service. Somewhat less obvious is the fact that stochastic fluctuations in service and arrival times are also important, and that these are a major source of backlogs and delays. Strategies that could mitigate fluctuations-induced delays are

thus in high demand as queue structures appear in various natural and man-made systems.  We demonstrate that a simple service resetting mechanism can reverse the deleterious effects of large fluctuations in service times, thus turning a marked drawback into a favorable advantage [PNAS Nexus 1, 3, 2022]. This happens when stochastic fluctuations are intrinsic to the server, and we show that service resetting can then dramatically cut down average queue lengths and waiting times. Remarkably, this strategy is also useful in extreme situations where the variance, and possibly even mean, of the service time diverge—as resetting can then prevent queues from “blowing up.” We illustrate these results on the M/G/1 queue in which service times are general and arrivals are assumed to be Markovian. Yet, the main results and conclusions coming from our analysis are not specific to this particular model system and can thus be carried over to other queueing systems: in telecommunications, via computing, and all the way to molecular queues that emerge in enzymatic and metabolic cycles of living organisms.

Mitigating long queues and waiting times with service resetting

We collaborate with Yael Roichman and Ofir Tal-Friedman to generalize the model of diffusion with resetting to account for situations where a particle is returned only a fraction of its distance to the origin, e.g., half way [Phys. Rev. E 106, 054116, 2022]. We show that this model always attains a steady-state distribution which can be written as an infinite sum of independent, but not identical, Laplace random variables. As a result, we find that the steady-state transitions from the known Laplace form which is obtained in the limit of full resetting to a Gaussian form, which is obtained close to the limit of no resetting. A similar transition is shown to be displayed by drift-diffusion whose steady state can also be expressed as an infinite sum of independent random variables. Finally, we extend our analysis to capture the temporal evolution of drift-diffusion with partial resetting, providing a bottom-up probabilistic construction that yields a closed-form solution for the time-dependent distribution of this process in Fourier-Laplace space. 

Stochastic resetting for enhanced sampling

We collaborate with Barak Hirchberg and Ofir Blumer to present a method for enhanced sampling of molecular dynamics simulations using stochastic resetting [J. Phys. Chem. Lett. 13, 11230, 2022]. Various phenomena, ranging

from crystal nucleation to protein folding, occur on time scales that are unreachable in standard simulations. They are often characterized by broad transition time distributions, in which extremely slow events have a non-negligible probability. Stochastic resetting, i.e., restarting simulations at random times, can significantly expedite processes that follow such distributions. We take advantage of this to employ resetting for enhanced sampling of molecular simulations for the first time. We show that it accelerates long time scale processes by up to an order of magnitude in examples ranging from simple models to a molecular system. Most importantly, we recover the mean transition time without resetting, which is typically too long to be sampled directly, from accelerated simulations at a single restart rate. Stochastic resetting can be used as a standalone method or combined with other sampling algorithms to further accelerate simulations.

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