Infinite ergodic theory for heterogeneous diffusion processes

Leibovich, N.; Barkai, Eli

doi:10.1103/physreve.99.042138

Cited by 85 publications

(68 citation statements)

References 48 publications

(98 reference statements)

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“…This happens, for example, if x n is symmetric with respect to the starting point. Indeed, we could obtain the same from renewal theory, see [15][16][17][18] and references therein. It is worth observing that the parameter characterizing the distribution of the occupation time of the origin must be the same parameter of the Lamperti distribution, which in our setting describes the occupation time of the positive(negative) axis for a symmetric process.…”

Section: Number Of Visits At the Originmentioning

confidence: 99%

Statistics of occupation times and connection to local properties of nonhomogeneous random walks

et al. 2020

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We consider the statistics of occupation times, the number of visits at the origin and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single parameter, that is connected to a local property of the probability density function (PDF) of the process, viz., the probability of occupying the origin at time t, P (t). We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: when P (t) shows the same long-time behavior, each observable follows indeed the same distribution.

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Section: Number Of Visits At the Originmentioning

confidence: 99%

Statistics of occupation times and connection to local properties of nonhomogeneous random walks

et al. 2020

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“…Smyshlyaev and Chen applied a position-dependent diffusion coefficient to the boundary feedback control of a diffusive system and proved the Mittag-Leffler stability of the system [45,46]. Several other works have since focussed on position-dependent diffusive systems [47][48][49][50], to name but a few.Differing from a constant diffusion coefficient, the presence of a position-dependent diffusivity involves the problem of how to interpret multiplicative noise in a stochastic equation, particularly, a noise-induced drift, which varies by choosing different integral forms, such as Itô, Stratonovich and isothermal integrals [44,51]. It is found that the probability distribution generated within the isothermal integral formulation effects the required Boltzmann distribution, which is correct in thermal equilibrium state [48][49][50], while the Itô and Stratonovich integrals lead to 'athermal' forms.…”

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confidence: 99%

“…Several other works have since focussed on position-dependent diffusive systems [47][48][49][50], to name but a few.Differing from a constant diffusion coefficient, the presence of a position-dependent diffusivity involves the problem of how to interpret multiplicative noise in a stochastic equation, particularly, a noise-induced drift, which varies by choosing different integral forms, such as Itô, Stratonovich and isothermal integrals [44,51]. It is found that the probability distribution generated within the isothermal integral formulation effects the required Boltzmann distribution, which is correct in thermal equilibrium state [48][49][50], while the Itô and Stratonovich integrals lead to 'athermal' forms. Generally, the Itô interpretation is employed in economics and biology due to their features of being 'only related to the latest past'; the Stratonovich integral finds applications in physical systems, such as electrical circuits driven by multiplicative noises (see [51], and references therein).…”

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confidence: 99%

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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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“…A possible explanation is that the more general approach of Refs. [21][22][23][24] employing a position-dependent diffusivity that does not depend on the position through the concentration.…”

Section: Discussionmentioning

confidence: 99%

Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity: Exact Solutions and Particle Simulations

2020

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We explore the anomalous diffusion that may arise as a result of a concentration dependent diffusivity. The diffusivity is taken to be a power law in the concentration, and from exact analytical solutions we show that the diffusion may be anomalous, or not, depending on the nature of the initial condition. The diffusion exponent has the value of normal diffusion when the initial condition is a step profile, but takes on anomalous values when the initial condition is a spike. Depending on the sign of the exponent in the diffusivity the diffusive behavior will then be either sub-diffusive or super-diffusive. We introduce a particle model that behaves according to the non-linear diffusion equation in the macroscopic limit. This correspondence is demonstrated via kinetic theory, i.e. by means of Chapman-Kolmogorov equation, as well as by direct simulations.

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Infinite ergodic theory for heterogeneous diffusion processes

Cited by 85 publications

References 48 publications

Statistics of occupation times and connection to local properties of nonhomogeneous random walks

Statistics of occupation times and connection to local properties of nonhomogeneous random walks

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

Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity: Exact Solutions and Particle Simulations

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