When is the next extending of Fick-Jacobs equation necessary?

Kalinay, Pavol

doi:10.1063/1.4817198

Cited by 7 publications

(8 citation statements)

References 49 publications

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“…and related theories [36,37]. However, as it appears clear from the above recollection, the need for further, systematic investigation of transport in crowded and confining media is evident.…”

Section: Introductionmentioning

confidence: 99%

Diffusion of tagged particles in a crowded medium

Galanti¹,

Fanelli²,

Maritan³

et al. 2014

EPL

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The influence of crowding on the diffusion of tagged particles in a dense medium is investigated in the framework of a mean-field model, derived in the continuum limit from a microscopic stochastic process with exclusion. The probability distribution function of the tagged particles obeys to a nonlinear Fokker-Planck equation, where the drift and diffusion terms are determined self-consistently by the concentration of crowders in the medium. Transient sub-diffusive or super-diffusive behaviours are observed, depending on the selected initial conditions, that bridge normal diffusion regimes characterized by different diffusion coefficients. These anomalous crossovers originate from the microscopic competition for space and reflect the peculiar form of the non-homogeneous advection term in the governing Fokker-Planck equation. Our results strongly warn against the overly simplistic identification of crowding with anomalous transport tout court.

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Section: Introductionmentioning

confidence: 99%

Diffusion of tagged particles in a crowded medium

Galanti¹,

Fanelli²,

Maritan³

et al. 2014

EPL

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“…In the last equation we have used definition of heat standard in stochastic thermodynamics [50,51,55]: According to the first law of thermodynamics and Eq. (5), the internal energy of the system, (20) whereQ x is the mean heat flow into the x reservoir. According to the definition (17), in the steady state we haveQ y =Q and the expression (19) follows directly from the last term on the right-hand side of Eq.…”

Section: Mean Velocity and Heat Currentmentioning

confidence: 99%

“…According to the definition (17), in the steady state we haveQ y =Q and the expression (19) follows directly from the last term on the right-hand side of Eq. (20). Our convention used in definition of heat is that heat is positive when it flows from the system to a heat bath.…”

Section: Mean Velocity and Heat Currentmentioning

confidence: 99%

“…It cannot be replaced by a pure elastic hard-wall repulsion without loss of the ratchet effect. For hard-wall repulsion the potential energy U(x, y) is constant everywhere in the channel and there is no heat flow between the two reservoirs (the expressions for heat flows (20) contain partial derivatives of the potential U). Thus for the hard-wall repulsion the two heat reservoirs decouple and the system cannot work as a ratchet.…”

Section: Conclusion and Outlooksmentioning

confidence: 99%

“…Thus Qx ( Qy ) denotes the average heat flow accepted by the longitudinal (transversal) bath. According to the conservation of energy (20) the following relation between energy flows holds: Ė = −( Qx + Qy ). In the steady state, the mean energy of the system is constant, hence we have Ė = 0 and Qx = − Qy .…”

Section: Mean Velocity and Heat Currentmentioning

confidence: 99%

See 2 more Smart Citations

Thermal Ratchet Effect in Confining Geometries

Holubec

Ryabov

Yaghoubi

et al. 2017

Entropy

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Abstract:The stochastic model of the Feynman-Smoluchowski ratchet is proposed and solved using generalization of the Fick-Jacobs theory. The theory fully captures nonlinear response of the ratchet to the difference of heat bath temperatures. The ratchet performance is discussed using the mean velocity, the average heat flow between the two heat reservoirs and the figure of merit, which quantifies energetic cost for attaining a certain mean velocity. Limits of the theory are tested comparing its predictions to numerics. We also demonstrate connection between the ratchet effect emerging in the model and rotations of the probability current and explain direction of the mean velocity using simple discrete analogue of the model.

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Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximation

Traytak

2014

The Journal of Chemical Physics

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The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.

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When is the next extending of Fick-Jacobs equation necessary?

Cited by 7 publications

References 49 publications

Diffusion of tagged particles in a crowded medium

Diffusion of tagged particles in a crowded medium

Thermal Ratchet Effect in Confining Geometries

Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximation

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