On diffusions with stochastic resettings: noisy restarts, optimal rates and interaction modelling. (English) Zbl 1509.82089
Summary: In this paper we address a model of particle diffusion, introduced by M. R. Evans and S. N. Majumdar [J. Phys. A, Math. Theor. 44, No. 43, Article ID 435001, 15 p. (2011; Zbl 1465.60070)] where, at random times, at rate \(\beta\), a particle \(\xi\) resets to a position drawn from a probability distribution with density \(\phi(\xi)\). We study an invariance law for the standard error of first-passage time observed by S. Reuveni [“Optimal stochastic restart renders fluctuations in first passage times universal”, Phys. Rev. Lett. 116, No. 17, Article ID 170601, 6 p. (2016; doi:10.1103/PhysRevLett.116.170601)] and find that, when dealing with restarts, noise around the resetting position \(\xi_0\) breaches such a law. We show that the mean first-passage time is finite and can be minimized with respect to \(\beta\), for a rather general class of functions \(\phi\). Moreover, we propose a model of a particle system evolution where the particles independently diffuse and reset but also at random times interact. We exhibit the non-equilibrium stationary distribution for the case of two interacting particles.
MSC:
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 82C21 | Dynamic continuum models (systems of particles, etc.) in time-dependent statistical mechanics |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
Keywords:
optimal resetting; first-passage time; invariance; particle interaction; non-equilibrium stationary distributionCitations:
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