Stochastic resetting and applications Evans, Martin R; Majumdar, Satya N; Schehr, Grégory
Journal of physics. A, Mathematical and theoretical,
05/2020, Letnik:
53, Številka:
19
Journal Article
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In this topical review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose ...position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.
We consider an active run-and-tumble particle (RTP) in d dimensions and compute exactly the probability S(t) that the x component of the position of the RTP does not change sign up to time t. When ...the tumblings occur at a constant rate, we show that S(t) is independent of d for any finite time t (and not just for large t), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed v of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.
We study one-dimensional fluctuating interfaces of length L, where the interface stochastically resets to a fixed initial profile at a constant rate r. For finite r in the limit L→∞, the system ...settles into a nonequilibrium stationary state with non-Gaussian interface fluctuations, which we characterize analytically for the Kardar-Parisi-Zhang and Edwards-Wilkinson universality class. Our results are corroborated by numerical simulations. We also discuss the generality of our results for a fluctuating interface in a generic universality class.
In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence
{
x
0
=
0
,
x
1
,
x
2
,
…
,
x
n
}
up to
n
steps where
x
i
represents the ...position at step
i
of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek–Spitzer formula and the associated Sparre Andersen theorem.
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}≥0, where ...successive jumps are drawn independently from an arbitrary jump distribution f(η). In addition, with a probability 0≤r<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(η), 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<μ<2, we show that, for any given x_{0}, the MFPT has a global minimum in the (μ,r) plane at (μ^{*}(x_{0}),r^{*}(x_{0})). We find a remarkable first-order phase transition as x_{0} crosses a critical value x_{0}^{*} at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.
We study nonequilibrium dynamics of integrable and nonintegrable closed quantum systems whose unitary evolution is interrupted with stochastic resets, characterized by a reset rate r, that project ...the system to its initial state. We show that the steady-state density matrix of a nonintegrable system, averaged over the reset distribution, retains its off-diagonal elements for any finite r. Consequently a generic observable Ô, whose expectation value receives a contribution from these off-diagonal elements, never thermalizes under such dynamics for any finite r. We demonstrate this phenomenon by exact numerical studies of experimentally realizable models of ultracold bosonic atoms in a tilted optical lattice. For integrable Dirac-like fermionic models driven periodically between such resets, the reset-averaged steady state is found to be described by a family of generalized Gibbs ensembles characterized by r. We also study the spread of particle density of a noninteracting one-dimensional fermionic chain, starting from an initial state where all fermions occupy the left half of the sample, while the right half is empty. When driven by resetting dynamics, the density profile approaches at long times to a nonequilibrium stationary profile that we compute exactly. We suggest concrete experiments that can possibly test our theory.
In this review, we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such ...as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spin models undergoing phase-ordering dynamics, diffusion equation, fluctuating interfaces, etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalizations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.
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