Isothermal Langevin dynamics in systems with power-law spatially dependent friction

Regev, Shaked; Grønbech‐Jensen, Niels; Farago, Oded

doi:10.1103/physreve.94.012116

Cited by 25 publications

(29 citation statements)

References 17 publications

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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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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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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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“…In this section, we briefly outline the general theory on which our analysis of the ergodic properties of the system (1) and (2) in the next section is based. We are following in large parts the presentation in the review articles [22,23] and we refer the reader to these articles for a more detailed and comprehensive presentation.…”

Section: Stationary States Of Sdes and Their Stabilitymentioning

confidence: 99%

“…In this section we consider three systems that fit the form of (1) and (2). In the first, we consider a system of particles diffusing in a temperature gradient.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Second, we do not assume a conservative force field. Such generalized forms of Langevin dynamics can be used to model diffusion (or thermal) gradients by particle simulation [1][2][3], as well as a variety of models for flocking [4][5][6], protein folding [7] and bacterial suspensions [8]. Problems with nonconservative forces are considered often in the physics literature, but the precise characterization of the convergence to stationary conditions is rarely discussed.…”

Section: Introductionmentioning

confidence: 99%

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Langevin Dynamics with Variable Coefficients and Nonconservative Forces: From Stationary States to Numerical Methods

Sachs

Leimkuhler

Danos

2017

Entropy

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Langevin dynamics is a versatile stochastic model used in biology, chemistry, engineering, physics and computer science. Traditionally, in thermal equilibrium, one assumes (i) the forces are given as the gradient of a potential and (ii) a fluctuation-dissipation relation holds between stochastic and dissipative forces; these assumptions ensure that the system samples a prescribed invariant Gibbs-Boltzmann distribution for a specified target temperature. In this article, we relax these assumptions, incorporating variable friction and temperature parameters and allowing nonconservative force fields, for which the form of the stationary state is typically not known a priori. We examine theoretical issues such as stability of the steady state and ergodic properties, as well as practical aspects such as the design of numerical methods for stochastic particle models. Applications to nonequilibrium systems with thermal gradients and active particles are discussed.

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Heterogeneous diffusion in comb and fractal grid structures

Sandev

Schulz

Kantz

et al. 2018

Chaos, Solitons & Fractals

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We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior x 2 (t) ∼ t 1/(2−α) , where α is the powerlaw exponent of the position dependent diffusion coefficient D(x) ∼ |x| α . Depending on the value of α we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., x 2 (t) ∼ t (1+ν)/(2−α) , where 0 < ν < 1 is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the

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Isothermal Langevin dynamics in systems with power-law spatially dependent friction

Cited by 25 publications

References 17 publications

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

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

Langevin Dynamics with Variable Coefficients and Nonconservative Forces: From Stationary States to Numerical Methods

Heterogeneous diffusion in comb and fractal grid structures

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