Abstract
We study the spectral properties of classical and quantum Markovian processes that are reset at random times to a specific configuration or state with a reset rate that is independent of the current state of the system. We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state. Based on this result, we provide expressions for the stationary state and probability current of the reset process in terms of weighted sums over dynamical modes of the reset-free process. We also discuss the effect of resets on processes that display metastability. We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability.
Highlights
The dynamics of stochastic processes, such as animals foraging for food in the wilderness or a person searching for car keys, often include random resets in time, taking the form of returns to past locations where food was successfully located or the last place a person remembers seeing their keys [1]
We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state
We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability
Summary
The dynamics of stochastic processes, such as animals foraging for food in the wilderness or a person searching for car keys, often include random resets in time, taking the form of returns to past locations where food was successfully located or the last place a person remembers seeing their keys [1]. Reset processes have been studied from a more physical point of view, as they provide a simple model of nonequilibrium processes breaking detailed balance [8,9,10,11], as well as of processes showing dynamical phase transitions in their relaxation dynamics [12], mean first-passage time [13], or large deviations [14,15,16] These studies follow many previous works in mathematics, in queuing theory and in population dynamics, in particular, on stochastic processes involving some form of random resets, variously referred to as failures, catastrophes, disasters or decimations; see, e.g., Refs. We illustrate this generalization by computing numerically the stationary state of a model of coherent hopping in one dimension realizing the reset quantum random walker, and by applying our method to a dissipative transverse field Ising model [33], known to display metastability [34]
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