Stochastic Resetting and Applications (1910.07993v2)

Abstract: 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. L\'evy 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.

• The paper reviews the field of stochastic processes under resetting, discussing its impacts on various systems from simple diffusive particles to complex multi-particle dynamics.
• It details how stochastic resetting introduces non-equilibrium stationary states and how optimally tuning the resetting rate can minimize metrics like the mean first passage time (MFPT).
• The review highlights broad applications across search algorithms, biological processes, and complex system modeling, showing how resetting can significantly enhance performance measures like search efficiency.

Stochastic Resetting and Applications: A Review

The paper of stochastic processes under resetting has become a focal point of research, seeking to uncover both fundamental insights and practical implications. The paper "Stochastic Resetting and Applications," authored by Martin R. Evans, Satya N. Majumdar, and Grégory Schehr, provides a comprehensive topical review along these lines, discussing the impacts of stochastic resetting on various processes, from simple diffusive systems to more complex multi-particle dynamics.

Key Concepts and General Framework

Stochastic resetting generally corresponds to the idea of intermittently returning a process to a predetermined state or position at random times, which might follow specific statistical distributions. The review begins by examining the simplest case of a diffusive particle subject to Poissonian resetting—a scheme where resets occur at a constant rate $r$. This model sets the stage for exploring more sophisticated processes where resetting introduces unique features such as reaching a non-equilibrium stationary state or optimizing metrics like the mean first passage time (MFPT).

Diffusive Systems

For a diffusive particle, resetting to a point results in a non-trivial stationary state instead of a monotonous drift to equilibrium. The review highlights that the MFPT to an absorbing target can be minimized by choosing an optimal resetting rate, $r^*$. This principle of optimization is supported by presenting exact analytical methods to evaluate survivability probabilities and mean absorption times, elucidating when stochastic resetting provides an advantageous strategy compared to traditional diffusion.

Extensions to Multiparticle Systems and Beyond

The framework evolves from single to multiparticle systems, such as teams of independent searchers, offering insights into group search strategies where resetting impacts the collective success rate. These systems showcase scenarios where typical and average behaviors diverge due to large fluctuations in system variables—providing fertile ground for connections to areas like extreme value statistics.

Generalized and Non-Poissonian Resetting

Beyond the Poissonian paradigm, the paper extends to arbitrary stochastic processes and non-Poissonian resetting, where intervals follow non-exponential distributions. This generality allows the examination of practical contexts where different temporal dynamics are in play, such as enzymatic reactions or search problems in fluctuating media. One significant theoretical advancement discussed is the universal aspect of resetting, where the resetting scheme must be tuned in harmony with the intrinsic dynamics of the underlying processes to induce optimization phenomena.

Fluctuating Interfaces and Extended Systems

Resetting is further explored in the field of fluctuating interfaces, illustrating how an external reset can lead to non-equilibrium steady states even when the intrinsic dynamics don't have such states naturally. It notably highlights how the introduction of memory—via resetting protocols that use past trajectories—enriches possible outcomes and phase behaviors.

Implications and Future Directions

The implications of stochastic resetting resonate across numerous disciplines, from search algorithms and biological processes to complex system modeling. Practically, stochastic resetting techniques can significantly enhance performance measures like search efficiency and reachability, providing tools potentially useful in the design of algorithms and artificial systems operating in uncertain environments.

Moreover, the review speculates on future pathways, such as exploring the thermodynamics of resetting systems, better understanding large deviation functions, and blending quantum dynamics with reset protocols where history-dependent state resturations could yield novel quantum states.

Conclusion

The review effectively captures the versatile nature of stochastic resetting across a spectrum of applications and processes. It not only consolidates the rigorous mathematical foundation required to understand these systems but also paves the way for practical exploration and utilization in tangible real-world systems, setting a comprehensive agenda for subsequent theoretical and empirical research.