Aging arcsine law in Brownian motion and its generalization

Akimoto, Takuma; Sera, Toru; Yamato, Kosuke; Yano, Kouji

doi:10.1103/physreve.102.032103

Cited by 11 publications

(4 citation statements)

References 38 publications

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“…A counterintuitive aspect of the ∪-shaped distribution (1) is that its average value 〈V〉 = t/2 corresponds to the minimum of the distribution, i.e., the less probable outcome, whereas values close to the extrema V = 0 and V = t are much more likely. Recent studies led to many extensions of the law, such as constrained Brownian motions [4,5], random acceleration process [6,7], fractional Brownian motion [8,9], run-and-tumble motion [10], in the presence of reflective walls [11,12] and a permeable barrier [13], occupation times of a temporally heterogeneous diffusion in a bounded domain before random stopping times [14], and general stochastic processes [15][16][17][18][19][20][21][22][23][24][25]. The arcsine law and related distributions have also been explored in diverse fields.…”

Section: Introductionmentioning

confidence: 99%

Breakdown of arcsine law for resetting brownian motion

Yan,

Chen

2023

Phys. Scr.

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For a one-dimensional Brownian motion starting from the origin, the cumulative distribution of the occupation time V staying above the origin obeys the celebrated arcsine law. In this work, we show how the law is modified for a resetting Brownian motion, where the Brownian is reset to the position x r at random times but with a constant rate r. When x r is exactly equal to zero, we derive the exact expression of the probability distribution P r (V∣0, t) of V during time t, and the moments of V as functions of r and t. P r (V∣0, t) is always symmetric with respect to V = t/2 for arbitrary value of r, but the probability density of V at V = t/2 increases with the increase of r. Interestingly, P r (V∣0, t) at V = t/2 changes from a minimum to a local maximum at a critical value R * ≈ 0.742 338, where R = rt denotes the average number of resetting during time t. Moreover, we consider the case when x r is a random variable and is distributed by a function g(x r ), where g(x r ) is assumed to be symmetric with respect to zero and possesses its maximum at zero. We derive the general expressions of the moments of V when the variance of x r is low. The mean value of V is always equal to t/2, but the fluctuation in x r leads to an increase in the second and third moments of V. Our results provide a quantitative understanding of how stochastic resetting destroys the persistence of Brownian motion.

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Section: Introductionmentioning

confidence: 99%

Breakdown of arcsine law for resetting brownian motion

Yan,

Chen

2023

Phys. Scr.

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“…Apart from non-equilibrium systems, entirely disparate systems including the quantum state of a dressed photon in a fiber probe [8], the fluctuation of stock prices [9], waiting time distributions of human dynamics [10], leads in competitive sports like football or basketball [11] as well as various random walk models [12], are examples that follow arcsine distributions. These laws can even be modified to include the behavior of fractional Brownian motion [13], non-Markovian processes [14], and processes that display anomalous diffusion [15].…”

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confidence: 99%

“…On another note, interest regarding systems driven with colored noise [19,20] has been growing in the scientific community, primarily due to the fact that these serve as a tunable model in understanding real-world biological systems at the microscopic level -especially where phenomena such as non-Gaussian displacement statistics [21], active dissipative fluctuations [22], and flows [23] may become prominent. There is an obvious requirement to study the long-term statistics of stochastic micro-currents that drive these systems in a steady state, since any deviations from the arcsine laws will capture "aging" [15] and "non-markovian" [13] behavior that determine the efficiency of these micro-engines. For example, if a fluctuating current does not satisfy the Markov properties of a typical diffusing Wiener process, it will deviate from the arcsine behavior.…”

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confidence: 99%

Experimental verification of Arcsine laws in mesoscopic non-equilibrium and active systems

Dey¹,

Kundu²,

Das³

et al. 2021

Preprint

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A large number of processes in the mesoscopic world occur out of equilibrium, where the time course of the system evolution becomes immensely important -they being driven principally by dissipative effects. Non-equilibrium steady states (NESS) represent a crucial category in such systems -which are widely observed in biological domains -especially in chemical kinetics in cellular processes [1], and molecular motors [2]. In this study, we employ a model NESS stochastic system which comprises of an colloidal microparticle, optically trapped in a viscous fluid and externally driven by a temporally correlated colored noise, and show that the work done on the system and the work dissipated by it -both follow the three Lévy arcsine laws. These statistics remain unchanged even in the presence of a perturbation generated by a microbubble at close proximity to the trapped particle. We confirm our experimental findings with theoretical simulations of the systems. Our work provides an interesting insight into the NESS statistics of the meso-regime, where stochastic fluctuations play a pivotal role.

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“…In 1947 Ulam and von Neumann considered the logistic map as a candidate for generation of random numbers and obtained it as an invariant measure for the mapping function f (x) = 4x(1−x) [35,36]. This law is a cornerstone of extreme-value statistics and has been applied to describe inter alia conductance in disordered materials [37], mean magnetization in spin systems [38], currents in stochastic thermodynamics [39], fractional [40] and aging [41] Brownian motion, balistic Lèvy random walks [42], to name only a few.…”

mentioning

confidence: 99%

Arcsine Law and Multistable Brownian Dynamics in a Tilted Periodic Potential

Spiechowicz,

Łuczka

2021

Preprint

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Multistability is one of the most important phenomena in dynamical systems, e.g. bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to extreme value statistics in the form of arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and construct the phase diagram for the stability of the velocity dynamics. It suggests an efficient strategy to control the multistability by changing solely the force acting on the particle or temperature of the system. Our findings for the paradigmatic model of nonequilibrium statistical physics apply to, inter alia, Brownian motors, Josephson junctions, cold atoms dwelling in optical lattices and colloidal systems.

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Aging arcsine law in Brownian motion and its generalization

Cited by 11 publications

References 38 publications

Breakdown of arcsine law for resetting brownian motion

Breakdown of arcsine law for resetting brownian motion

Experimental verification of Arcsine laws in mesoscopic non-equilibrium and active systems

Arcsine Law and Multistable Brownian Dynamics in a Tilted Periodic Potential

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