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Register a new userTransport properties and first arrival statistics of random searches with stochastic reset times
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Added by
Axel Mas\\'o-Puigdellosas
Publication date
2018
fields
Physics
and research's language is
English
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Stochastic resets have lately emerged as a mechanism able to generate finite equilibrium mean square displacement (MSD) when they are applied to diffusive motion. Furthermore, walkers with an infinite mean first arrival time (MFAT) to a given position $x$, may reach it in a finite time when they reset their position. In this work we study these emerging phenomena from a unified perspective. On one hand we study the existence of a finite equilibrium MSD when resets are applied to random motion with $langle x^2(t)rangle _msim t^p$ for $0<pleq2$. For exponentially distributed reset times, a compact formula is derived for the equilibrium MSD of the overall process in terms of the mean reset time and the motion MSD. On the other hand, we also test the robustness of the finiteness of the MFAT for different motion dynamics which are subject to stochastic resets. Finally, we study a biased Brownian oscillator with resets with the general formulas derived in this work, finding its equilibrium first moment and MSD, and its MFAT to the minimum of the harmonic potential.
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In this work we consider a stochastic movement process with random resets to the origin followed by a random residence time there before the walker restarts its motion. First, we study the transport properties of the walker, we derive an expression for the mean square displacement of the overall process and study its dependence with the statistical properties of the resets, the residence and the movement. From this general formula, we see that the inclusion of the residence after the resets is able to induce super-diffusive to sub-diffusive (or diffusive) regimes and it can also make a sub-diffusive walker reach a constant MSD or even collapse. Second, we study how the reset-and-residence mechanism affects the first arrival time of different search processes to a given position, showing that the long time behavior of the reset and residence time distributions determine the existence of the mean first arrival time.
Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport sub-diffusive and even reaching a stationary MSD., i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.
A hot Markovian system can cool down faster than a colder one: this is known as the Mpemba effect. Here, we show that a non-equilibrium driving via stochastic reset can induce this phenomenon, when absent. Moreover, we derive an optimal driving protocol simultaneously optimizing the appearance time of the Mpemba effect, and the total energy dissipation into the environment, revealing the existence of a Pareto front. Building upon previous experimental results, our findings open up the avenue of possible experimental realizations of optimal cooling protocols in Markovian systems.
We study a stochastic process $X_t$ related to the Bessel and the Rayleigh processes, with various applications in physics, chemistry, biology, economics, finance and other fields. The stochastic differential equation is $dX_t = (nD/X_t) dt + sqrt{2D} dW_t$, where $W_t$ is the Wiener process. Due to the singularity of the drift term for $X_t = 0$, different natures of boundary at the origin arise depending on the real parameter $n$: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behaviour is observed in the case of a regular boundary.
Explicit expressions for arrival times of particles moving in a one-dimensional Zero-Range Process (ZRP) are computed. Particles are fed into the ZRP from an injection site and can also evaporate from anywhere in the interior of the ZRP. Two dynamics are considered; bulk dynamics, where particle hopping and decay is proportional to the numqber of particles at each site, and surface dynamics, where only the top particle at each site can hop or evaporate. We find exact solutions in the bulk dynamics case and for a single-site ZRP obeying surface dynamics. For a multisite ZRP obeying surface dynamics, we compare simulations with approximations obtained from the steady-state limit, where mean interarrival times for both models are equivalent. Our results highlight the competition between injection and evaporation on the arrival times of particles to an absorbing site.
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