First order transition for the optimal search time of Lévy flights with resetting

Abstract

We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at $x_0\ge 0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(\eta )$. In addition, with a probability $0\le r\le 1$ the position of the searcher is reset to its initial position $x_0$. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution $f(\eta )$, initial position $x_0$ and resetting probability $r$, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index $0<\mu <2$, we show that, for any given $x_0$, the MFPT has a global minimum in the $(\mu ,r)$ plane at $({\mu }^{\ast }(x_0),r^{\ast }(x_0))$. We find a remarkable first-order phase transition as $x_0$ crosses a critical value $x_0^{\ast }$ at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.

arXiv comment5 pages, 6 figures

References in corpus (12)

• Diffusion with Stochastic Resetting
• Intermittent search strategies
• Diffusion with Optimal Resetting
• Optimal search strategies for hidden targets
• Diffusion with resetting in arbitrary spatial dimension
• Random walks with preferential relocations to places visited in the past and their application to biology
• Fluctuating interfaces subject to stochastic resetting
• Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics
• Boosting search by rare events
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Statistical mechanics of the coagulation-diffusion process with a stochastic reset
• Effect of Partial Absorption on Diffusion with Resetting

Cited by in corpus (217)

• Stochastic Resetting and Applications
• First Passage Under Restart
• Optimal stochastic restart renders fluctuations in first passage times universal
• Experimental realization of diffusion with stochastic resetting
• Diffusion under time-dependent resetting
• Run and tumble particle under resetting: a renewal approach
• Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results
• Extreme value statistics of correlated random variables: a pedagogical review
• Dynamical transition in the temporal relaxation of stochastic processes under resetting
• Diffusion with stochastic resetting at power-law times
• Non-equilibrium steady states of stochastic processes with intermittent resetting
• The Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem
• First passage under stochastic resetting in an interval
• Optimal first arrival times in Lévy flights with resetting
• Stochastic resetting: A (very) brief review
• Stochastic thermodynamics of resetting
• Search with home returns provides advantage under high uncertainty
• Restart could optimize the probability of success in a Bernoulli trial
• Large deviations for Markov processes with resetting
• Resetting processes with non-instanteneous return
• Effects of refractory period on stochastic resetting
• The inspection paradox in stochastic resetting
• Scaled Brownian motion with renewal resetting
• First passage of a particle in a potential under stochastic resetting: a vanishing transition of optimal resetting rate
• Work fluctuations and Jarzynski equality in stochastic resetting
• Quantum dynamics with stochastic reset
• Péclet number governs transition to acceleratory restart in drift-diffusion
• Diffusion with resetting in a logarithmic potential
• Landau theory of restart transitions
• Random walks on networks with stochastic resetting
• Random walk with random resetting to the maximum
• Localization transition induced by learning in random searches
• Integral Fluctuation Theorems for Stochastic Resetting Systems
• First passage under restart with branching
• Symmetric Exclusion Process under Stochastic Resetting
• Non-renewal resetting of scaled Brownian motion
• Phase transitions in optimal search times: how random walkers should combine resetting and flight scales
• Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement
• Stochastic resetting with stochastic returns using external trap
• First-passage and first-hitting times of Levy flights and Levy walks
• Diffusion with stochastic resetting is invariant to return speed
• Ising model with stochastic resetting
• Transport properties and first arrival statistics of random searches with stochastic reset times
• Run-and-Tumble particles in Two-dimensions under Stochastic Resetting
• Spectral properties of simple classical and quantum reset processes
• First passage under restart for discrete space and time: application to one dimensional confined lattice random walks
• Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process
• On subdiffusive continuous time random walks with stochastic resetting
• Invariants of motion with stochastic resetting and space-time coupled returns
• Sisyphus random walk
• Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics
• Stochastic resetting in underdamped Brownian motion
• Properties of additive functionals of Brownian motion with resetting
• Spectral content of fractional Brownian motion with stochastic reset
• Continuous-time random walks with reset events: Historical background and new perspectives
• Intermittent resetting potentials
• Extreme statistics and spacing distribution in a Brownian gas correlated by resetting
• Stochastic resetting in interacting particle systems: A review
• Brownian motion under intermittent harmonic potentials
• Thermodynamic uncertainty relation for systems with unidirectional transitions
• Stochastic resetting by a random amplitude
• Extremal statistics for stochastic resetting systems
• Interacting Brownian Motion with Resetting
• Local time of diffusion with stochastic resetting
• Phase transitions in large deviations of reset processes
• Entropy production in systems with unidirectional transitions
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Space-dependent diffusion with stochastic resetting: A first-passage study
• Random acceleration process under stochastic resetting
• Record statistics for random walks and Lévy flights with resetting
• Diffusion with Resetting Inside a Circle
• Resetting with stochastic return through linear confining potential
• Continuous time random walks under power-law resetting
• Diffusive transport on networks with stochastic resetting to multiple nodes
• A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation
• The cost of stochastic resetting
• First-passage Brownian functionals with stochastic resetting
• Continuous time random walks under Markovian resetting
• Number of distinct sites visited by a resetting random walker
• Diffusion with Partial Resetting
• Diffusion-mediated surface reactions and stochastic resetting
• Resetting of fluctuating interfaces at power-law times
• First passage in the presence of stochastic resetting and a potential barrier
• Lotka-Volterra systems with stochastic resetting
• Anderson-like localization transition of random walks with resetting
• Optimization and universality of Brownian search in quenched heterogeneous media
• Comparison of two models of tethered motion
• Persistence-Speed Coupling Enhances the Search Efficiency of Migrating Immune Cells
• Mean-performance of sharp restart I: Statistical roadmap
• Mitigating long queues and waiting times with service resetting
• Slow Lévy flights
• Diffusion processes with Gamma-distributed resetting and non-instantaneous returns
• Synchronization in the Kuramoto model in presence of stochastic resetting
• Resetting transition is governed by an interplay between thermal and potential energy
• Diffusion with Local Resetting and Exclusion
• Asymmetric Stochastic Resetting: Modeling Catastrophic Events
• Mitigating long transient time in deterministic systems by resetting
• Universal framework for record ages under restart
• Time to reach the maximum for a stationary stochastic process
• Discrete space-time resetting model: Application to first-passage and transmission statistics
• Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting
• Emergent quantum correlations and collective behavior in non-interacting quantum systems subject to stochastic resetting
• Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes
• Continuous time random walks and Lévy walks with stochastic resetting
• Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting
• Exact extreme, order and sum statistics in a class of strongly correlated system
• Brownian motion under non-instantaneous resetting in higher dimensions
• Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain
• Non-equilibrium thermodynamics of diffusion in fluctuating potentials
• First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry
• First detection probability in quantum resetting via random projective measurements
• Robust random search with scale-free stochastic resetting
• Analysis and applications of the residual varentropy of random lifetimes
• Resetting mediated navigation of active Brownian searcher in a homogeneous topography
• Dynamical phase transition in the first-passage probability of a Brownian motion
• Brownian motion surviving in the unstable cubic potential and the role of Maxwell's demon
• Optimality of spatially inhomogeneous search strategies
• Unusual stationary state in Brownian systems with Lorentz force
• Drift-diffusion on a Cayley tree with stochastic resetting: the localization-delocalization transition
• Diffusion with two resetting points
• Accumulation time of stochastic processes with resetting
• Critical number of walkers for diffusive search processes with resetting
• Random walks with stochastic resetting in complex networks: a discrete time approach
• Median and Mode in First Passage under Restart
• Encounter-based approach to diffusion with resetting
• Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials
• Kuramoto model subject to subsystem resetting: How resetting a part of the system may synchronize the whole of it
• Mean-performance of Sharp Restart II: Inequality Roadmap
• Stochastic resetting in the Kramers problem: A Monte Carlo approach
• Non-linear diffusion with stochastic resetting
• Effect of tax dynamics on linearly growing processes under stochastic resetting: a possible economic model
• Role of dimensions in first passage of a diffusing particle under stochastic resetting and attractive bias
• Striking universalities in stochastic resetting processes
• Tail-behavior roadmap for sharp restart
• Expediting Feller process with stochastic resetting
• Discrete-time random walks and Lévy flights on arbitrary networks: when resetting becomes advantageous?
• Joint statistics of space and time exploration of $1d$ random walks
• Twitching Motility of Bacteria with Type IV Pili: Fractal Walks, First passage time and their Consequences on Microcolonies
• Learning how to find targets in the micro-world: The case of intermittent active Brownian particles
• Effect of stochastic resetting on Brownian motion with stochastic diffusion coefficient
• Optimal potentials for diffusive search strategies
• Optimal search in E.coli chemotaxis
• Emergence of Lévy walks from second order stochastic optimization
• Lévy flights and Lévy walks under stochastic resetting
• Autonomous Ratcheting by Stochastic Resetting
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime
• Effects of mortality on stochastic search processes with resetting
• Diversity of Sharp Restart
• Optimal Measurement Protocols in Quantum Zeno Effect
• Optimizing Leapover Lengths of Lévy Flights with Resetting
• Freezing transitions of Brownian particles in confining potentials
• Biased random walk on random networks in presence of stochastic resetting: Exact results
• Entropy rate of random walks on complex networks under stochastic resetting
• Stochastic resetting in discrete-time quantum dynamics: steady states and correlations in few-qubit systems
• Numerical analysis of homogeneous and inhomogeneous intermittent search strategies
• Search with stochastic home-returns can expedite classical first passage under resetting
• Efficiency of a microscopic heat engine subjected to stochastic resetting
• First passage in discrete-time absorbing Markov chains under stochastic resetting
• Target Search Optimization by Threshold Resetting
• Optimal conditions for first passage of jump processes with resetting
• Queues with resetting: a perspective
• Comparing the roles of time overhead and spatial dimensions on optimal resetting rate vanishing transitions, in Brownian processes with potential bias and stochastic resetting
• Stochastic resetting of a population of random walks with resetting-rate-dependent diffusivity
• Diffusion in a partially absorbing medium with position and occupation time resetting
• Manipulating phases in many-body interacting systems with subsystem resetting
• Optimizing the random search of a finite-lived target by a Lévy flight
• Survival probability of random walks and Lévy flights with stochastic resetting
• Universal performance bounds of restart
• Optimization of escape kinetics by reflecting and resetting
• Random resetting in search problems
• Partial versus total resetting for Lévy flights in d dimensions: similarities and discrepancies
• Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols
• The cost of resetting discrete-time random walks
• Non-homogeneous random walks with stochastic resetting: an application to the Gillis model
• Control of spatiotemporal chaos by stochastic resetting
• Entropy of Sharp Restart
• Emerging cost-time Pareto front for diffusion with stochastic return
• Adaptive active Brownian particles searching for targets of unknown positions
• Survival probability and position distribution of a run and tumble particle in $U(x)=\alpha |x|$ potential with an absorbing boundary
• Mean first-encounter times of simultaneous random walkers with resetting on networks
• Stochastic resetting in a nonequilibrium environment
• Drift-diffusive resetting search process with stochastic returns: speed-up beyond optimal instantaneous return
• Crossover from Anomalous to Normal Diffusion: Ising Model with Stochastic Resetting
• Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: Experiments, theory and numerical tests
• Stochastic resetting in a networked multiparticle system with correlated transitions
• Construction and heat kernel estimates of general stable-like Markov processes
• Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous and discrete time stochastic processes
• Unified approach to reset processes and application to coupling between process and reset
• Persistent search in confined domains: a velocity-jump process model
• Autocorrelation functions and ergodicity in diffusion with stochastic resetting
• Resetting induced multimodality
• Conditioned backward and forward times of diffusion with stochastic resetting: a renewal theory approach
• Continuous-Time Multi-Armed Bandits with Controlled Restarts
• Diffusion with stochastic resetting on a lattice
• The Pólya and Sisyphus lattice random walks with resetting -- a first passage under restart approach
• Analytic and Monte Carlo Approximations to the Distribution of the First Passage Time of the Drifted Diffusion with Stochastic Resetting and Mixed Boundary Conditions
• Local resetting with geometric confinement
• Returns to the origin of the Pólya walk with stochastic resetting
• The distribution of the maximum of independent resetting Brownian motions
• Exact fluctuation and long-range correlations in a single-file model under resetting
• Arcsine laws for Brownian motion with Poissonian resetting
• An exactly solvable predator prey model with resetting
• Abrupt transitions in the optimization of diffusion with distributed resetting
• Proxitaxis: an adaptive search strategy based on proximity and stochastic resetting
• Localization of information driven by stochastic resetting
• Ising Model with Power Law Resetting
• Concentration of Empirical First-Passage Times
• The one-dimensional telegraphic process with noninstantaneous stochastic resetting
• Search of stochastically gated targets with diffusive particles under resetting
• Stationary state of harmonic chains driven by boundary resetting
• A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes
• Evanescent random walker on networks: Hitting times, budget renewal, and survival dynamics
• Thermodynamics of quantum-jump trajectories of open quantum systems subject to stochastic resetting
• Accumulation time of diffusion in a 2D singularly perturbed domain
• Local large deviation principle for Wiener process with random resetting
• Chaotic resetting: A resetting strategy for deterministic chaotic systems