Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

Guggenberger, Tobias; Pagnini, Gianni; Vojta, Thomas; Metzler, Ralf

doi:10.1088/1367-2630/ab075f

Cited by 60 publications

(74 citation statements)

References 75 publications

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“…We emphasize that the behavior of the FLE on a finite interval observed here is very different from the behavior of FBM on a finite interval reported in Ref. [24]. Whereas the probability density of the FLE reaches a uniform distribution for long times for all values of the correlation exponent α, the stationary probability density of FBM is not uniform and depends on the value of α.…”

Section: Finite Intervalcontrasting

confidence: 93%

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Probability density of the fractional Langevin equation with reflecting walls

2019

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We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while anti-persistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for non-thermal fractional Brownian motion with reflecting walls, and we discuss broader implications. arXiv:1907.08188v1 [cond-mat.stat-mech]

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Section: Finite Intervalcontrasting

confidence: 93%

“…stead, it develops singularities close to the walls and resembles the probability density of reflected FBM found in Ref. [24]. We have obtained similar results for α = 1.5.…”

Section: B Generalized Langevin Equation Without Fluctuation-dissipasupporting

confidence: 88%

Probability density of the fractional Langevin equation with reflecting walls

2019

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“…This phenomenon is consistent with previous work at thermodynamic equilibrium. Related work involved fraction Brownian motion with the reflecting boundary condition can be found in [69,70]. However, it should be noted that the observed diffusion gradients come from particle-wall effects or the preciseboundary structure, which has a different regime with the temperature gradient.…”

Section: Steady State Probability Density Function (Pdf)mentioning

confidence: 99%

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

Mei

et al. 2020

New J. Phys.

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This work focuses on the dynamics of particles in a confined geometry with position-dependent diffusivity, where the confinement is modelled by a periodic channel consisting of unit cells connected by narrow passage ways. We consider three functional forms for the diffusivity, corresponding to the scenarios of a constant (D 0 ), as well as a low (D m ) and a high (D d ) mobility diffusion in cell centre of the longitudinally symmetric cells. Due to the interaction among the diffusivity, channel shape and external force, the system exhibits complex and interesting phenomena. By calculating the probability density function, mean velocity and mean first exit time with the Itô calculus form, we find that in the absence of external forces the diffusivity D d will redistribute particles near the channel wall, while the diffusivity D m will trap them near the cell centre. The superposition of external forces will break their static distributions. Besides, our results demonstrate that for the diffusivity D d , a high dependence on the x coordinate (parallel with the central channel line) will improve the mean velocity of the particles. In contrast, for the diffusivity D m , a weak dependence on the x coordinate will dramatically accelerate the moving speed. In addition, it shows that a large external force can weaken the influences of different diffusivities; inversely, for a small external force, the types of diffusivity affect significantly the particle dynamics. In practice, one can apply these results to achieve a prominent enhancement of the particle transport in two-or three-dimensional channels by modulating the local tracer diffusivity via an engineered gel of varying porosity or by adding a cold tube to cool down the diffusivity along the central line, which may be a relevant effect in engineering applications. Effects of different stochastic calculi in the evaluation of the underlying multiplicative stochastic equation for different physical scenarios are discussed.New J. Phys. 22 (2020) 053016 Y Li et al confined spaces, transport of particles in nanopores, zeolites and gel networks [8-10], or protein diffusion in the mammalian cell cytoplasm [11]. In many cases, such structured environments can be viewed as confined channels with different boundaries and properties [12][13][14]. The behaviours of systems in these confined channels have been studied by virtue of the analytical Fick-Jacobs (FJ) equation and numerical simulations of the dynamic equations [15][16][17][18][19][20]. Results for symmetric periodic channels indicate that the mean transport velocity may be proportional to large external forces in the confined environment [19][20][21][22]. For asymmetric channels, it was found that the shape of the confinement contributes to the moving direction and mean displacement [20,23], which can be applied to fine separation of mixtures and also the design of many devices [24,25]. Time-dependent and deformable boundaries were discussed to explore the activity in living cells [26,27]. In addition, the interaction...

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“…Generally the formulation of non-local and/or correlated stochastic processes is not always an easy task and, in some cases, still not fully understood. Apart from LFs, we may allude to the debate on the formulation and solution of boundary value problems for fractional Brownian motion, a process fuelled with Gaussian yet longrange correlated noise [187,188,189]. For LFs, in addition to the results obtained here it will be interesting to generalise the results obtained for symmetric α-stable laws in the presence of an external drift in [113].…”

Section: Discussion and Unsolved Problemsmentioning

confidence: 75%

First-passage properties of asymmetric Lévy flights

Padash

Chechkin

Dybiec

et al. 2019

J. Phys. A: Math. Theor.

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Lévy Flights are paradigmatic generalised random walk processes, in which the independent stationary increments-the "jump lengths"-are drawn from an α-stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy Flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy Flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy Flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of firstpassage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index α and the skewness (asymmetry) parameter β. The other approach is based on the stochastic Langevin equation with α-stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.∞ 0 t℘(t)dt can capture some aspects of this dynamics, ‡ the full information encoded in ℘(t) provides significant additional insight [114,115,116]. Here we study the first-passage properties for a general class of α-stable Lévy laws. We go beyond previous approaches [113,117,118,119,120,121,122,123,124,125,126,127,128] focusing on symmetric and one-sided α-stable relocation distributions and consider α-stable laws with arbitrary asymmetry in semi-infinite and bounded domains. Our approach is based on the convenient formulation of LFs in terms of the space-fractional diffusion equation. We derive these integro-differential equations for LFs based on general asymmetric α-stable distributions of relocation lengths in finite domains, and thus go beyond studies of the exit time and escape probability in bounded domain for symmetric LFs [129,130,131]. An important aspect of this study is that we complement our results with numerical analyses of the (stochastic) Langevin equation for LFs and show how both approaches complement each other.The paper is organised as follows. In section 2 we define Lévy stable laws and the associated fractional diffusion equation. In section 3 we set up our numerical model for the fractional diffusion equation and the associated Langevin equation. Moreover, a comparison between the numerical method and α-stable distributions for symmetric and asymmetric den...

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Fractional Brownian motion in a finite interval: correlations effect depletion or accretion zones of particles near boundaries

Cited by 60 publications

References 75 publications

Probability density of the fractional Langevin equation with reflecting walls

Probability density of the fractional Langevin equation with reflecting walls

Particle dynamics and transport enhancement in a confined channel with position-dependent diffusivity

First-passage properties of asymmetric Lévy flights

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