Persistence of Brownian motion in a shear flow

Takikawa, Yuichi; Orihara, Hiroshi

doi:10.1103/physreve.88.062111

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…In this letter, we obtain this result analytically and investigate numerically related problems involving multiple maxima and diffusion in higher dimensions. Anomalous relaxation with nontrivial persistence exponents [7][8][9], enhanced transport due to disorder [10,11], and anomalous diffusion due to exclusion [12,13] are dynamical phenomena that were recently demonstrated in experiments involving Brownian particles. Understanding the nonequilibrium statistical physics of these diffusion processes is closely intertwined with the characteristic behavior of extreme fluctuations and the statistics of extreme values [14][15][16][17][18][19].…”

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confidence: 87%

Slow Kinetics of Brownian Maxima

Ben‐Naim

Krapivsky

2014

Phys. Rev. Lett.

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We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability P (t) that the two maxima remain ordered up to time t, and find the algebraic decay P ∼ t −β with exponent β = 1/4. When the two particles have diffusion constants D1 and D2, the exponent depends on the mobilities, β = 1 π arctan D2/D1. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.PACS numbers: 05.40. Jc, 05.40.Fb, 02.50.Cw, 02.50.Ey Consider a pair of particles undergoing independent Brownian motion in one dimension [1]. These two particles do not meet with probability that decays as t −1/2 in the long-time limit. This classical first-passage behavior holds for Brownian particles with arbitrary diffusion constants. It holds even for particles undergoing symmetric Lévy flights [2,3], and has numerous applications [3,4]. Here, we generalize this ubiquitous first-passage behavior to maxima of Brownian particles. Figure 1 shows that the maximal position of each particle forms a staircase and it illustrates that unlike the position, the maximum is a non-Markovian random variable [5,6]. We find that two such staircases do not intersect with probability P that is inversely proportional to the one-fourth power of time, P ∼ t −1/4 , in the long-time limit. If the particles move with diffusion constants D 1 and D 2 , the two maxima remain ordered during the time interval (0, t) with the slowly-decaying probabilityIn this letter, we obtain this result analytically and investigate numerically related problems involving multiple maxima and diffusion in higher dimensions. Anomalous relaxation with nontrivial persistence exponents [7][8][9], enhanced transport due to disorder [10,11], and anomalous diffusion due to exclusion [12,13] are dynamical phenomena that were recently demonstrated in experiments involving Brownian particles. Understanding the nonequilibrium statistical physics of these diffusion processes is closely intertwined with the characteristic behavior of extreme fluctuations and the statistics of extreme values [14][15][16][17][18][19].We first establish Eq. (1) for two Brownian particles having the same diffusion constant D. Let us denote the positions of the particles at time t by x 1 (t) and x 2 (t), and without loss of generality, we assume x 1 (0) > x 2 (0). We define the maximum of the first particle, m 1 (t), to be its rightmost position up to time t; similarly, m 2 (t) is the maximal position of the second particle. Our goal is to find the probability P (t) that the two maxima remain ordered m 1 (τ ) > m 2 (τ ) for all 0 ≤ τ ≤ t.The two maxima remain ordered if and only if m 1 (τ ) > x 2 (τ ) at all times 0 ≤ τ ≤ t. Hence, to find P , there is no need to keep track of the maximum m 2 , and it suffices to consider only the position x 2 . As a further simplificat...

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confidence: 87%

Slow Kinetics of Brownian Maxima

Ben‐Naim

Krapivsky

2014

Phys. Rev. Lett.

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“…When a shear flow exists in a system, colloids still exhibit Brownian motion . Particle pairs with a large distance r are likely to be separated by the symmetric axis of the shear flow.…”

Section: Resultsmentioning

confidence: 99%

“…The motion of particles in a fluid with shear flow is independently driven by two mechanisms: Brownian motion and shear flow. The displacement of each particle Δ s⃗ can be written as Δ s⃗ = Δ s⃗ random + Δ s⃗ shear .…”

Section: Theoretical Analysismentioning

confidence: 99%

Spatial Cross-Correlated Diffusion of Colloids under Shear Flow

Zhang

Jiang

et al. 2018

Langmuir

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The spatial cross-correlated diffusion of colloidal particles is often used as an essential tool to study the dynamic properties of a fluid because it directly describes the hydrodynamic interaction between two particles in a fluid. However, the experimental measurement of cross-correlated diffusion can be substantially modified by even a weak shear flow. In this work, the effect of a shear flow on spatial cross-correlated diffusion is demonstrated using experimental measurements that show a clear dependence on pair angles. An analytical solution is proposed to explain the experimental observations. A numerical simulation is performed to systemically demonstrate the influence of shear flow on spatial cross-correlated diffusion. The results of the experiment, theoretical analysis, and numerical simulation agree well with each other. Therefore, this research provides a sensitive experimental method to determine the weak shear flow in any quasi-two-dimensional fluid systems.

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“…Related work concerning passive Brownian particles under external flows, i.e., a spherical Brownian particle under a shear flow [19,20] and under a general linear flow [21], has already been reported. More recently, active particles (driven by an assumed internal mechanism) have been widely studied [7,[22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional motion of Brownian swimmers in linear flows

Sandoval

Jimenez

2015

J Biol Phys

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The motion of viruses and bacteria and even synthetic microswimmers can be affected by thermal fluctuations and by external flows. In this work, we study the effect of linear external flows and thermal fluctuations on the diffusion of those swimmers modeled as spherical active (self-propelled) particles moving in two dimensions. General formulae for their mean-square displacement under a general linear flow are presented. We also provide, at short and long times, explicit expressions for the mean-square displacement of a swimmer immersed in three canonical flows, namely, solid-body rotation, shear and extensional flows. These expressions can now be used to estimate the effect of external flows on the displacement of Brownian microswimmers. Finally, our theoretical results are validated by using Brownian dynamics simulations.

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Persistence of Brownian motion in a shear flow

Cited by 6 publications

References 15 publications

Slow Kinetics of Brownian Maxima

Slow Kinetics of Brownian Maxima

Spatial Cross-Correlated Diffusion of Colloids under Shear Flow

Two-dimensional motion of Brownian swimmers in linear flows

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