Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

Kalinay, Pavol; Percus, J. K.

doi:10.1007/s10955-012-0570-2

Cited by 10 publications

(9 citation statements)

References 38 publications

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“…With respect to previous attempts at incorporating finite-mass effects in the analysis of Brownian transport through corrugated narrow channels [31][32][33], we stress that the inertial effects reported here are not of mere academic interest [26].…”

Section: Discussionmentioning

confidence: 86%

Brownian transport in corrugated channels with inertia

et al. 2012

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The transport of suspended Brownian particles dc-driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on the the channel corrugation and the drive intensity. Being such a diffusion length inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility and diffusivity, markedly differ for smoothly and sharply corrugated channels.

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Section: Discussionmentioning

confidence: 86%

Brownian transport in corrugated channels with inertia

et al. 2012

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“…More complex systems, such as confined diffusion driven by an external force along the channel, 26 diffusion biased by a transverse force, 27 or motion of massive particles described by the Fokker-Planck (Kramers) equation 28,29 can be mapped onto the longitudinal coordinate x using the same reduction scheme (the small parameter is proportional to the mass of particles in the last case). Anyhow, the result has the form of Eq.…”

Section: ∂ T P(x T) = ∂ X A(x)d(x)∂ X P(x T) A(x)mentioning

confidence: 99%

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

Kalinay

2013

The Journal of Chemical Physics

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Applicability of the effective one-dimensional equations, such as Fick-Jacobs equation and its extensions, describing diffusion of particles in 2D or 3D channels with varying cross section A(x) along the longitudinal coordinate x, is studied. The leading nonstationary correction to Zwanzig-Reguera-Rubí equation [R. Zwanzig, J. Phys. Chem. 96, 3926 (1992); D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001)] is derived and tested on the exactly solvable model, diffusion in a 2D linear cone. The effects of such correction are demonstrated and discussed on elementary nonstationary processes, a time dependent perturbation of the stationary flow and calculation of the mean first passage time.

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“…where m, ζ , D = (βζ ) −1 , and β = (k B T ) −1 respectively denote the particle mass, the friction coefficient, the diffusion coefficient, and the inverse thermal energy (k B is the Boltzmann constant and T is the absolute temperature), V (x) and V (x) are the potential and its second spatial derivative, andĴ (x) is the current operator in the overdamped limit (the Smoluchowski current operator). The factor [1 + mV (x)/ζ 2 ] can be associated with the position dependence of the diffusion coefficient (or mobility) [32]. It would seem that one could use the results obtained in early studies of ratchet systems [33][34][35][36], with the position-dependent particle mobility (or the friction coefficient), to account for the inertial correction in the present case.…”

Section: Introductionmentioning

confidence: 99%

“…The most well known and commonly used is the high-friction expansion of the Klein-Kramers equation [28][29][30][31]. In this way, the Klein-Kramers equation can be reduced to a Smoluchowski-like one-dimensional form [32] containing a series of corrections, so that the mapped equation respects inertia, although working in coordinate space. In the lowest order of such approximate dimension reduction, the mapped equation takes the form of the Smoluchowski equation with the current operator [28][29][30][31][32]…”

Section: Introductionmentioning

confidence: 99%

Inertial effects in adiabatically driven flashing ratchets

et al. 2014

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We study analytically the effect of a small inertial correction on the properties of adiabatically driven flashing ratchets. Parrondo's lemma [J. M. R. Parrondo, Phys. Rev. E 57, 7297 (1998)] is generalized to include the inertial term so as to establish the symmetry conditions allowing directed motion (other than in the overdamped massless case) and to obtain a high-temperature expansion of the motion velocity for arbitrary potential profiles. The inertial correction is thus shown to enhance the ratchet effect at all temperatures for sawtooth potentials and at high temperatures for simple potentials described by the first two harmonics. With the special choice of potentials represented by at least the first three harmonics, the correction gives rise to the motion reversal in the high-temperature region. In the low-temperature region, inertia weakens the ratchet effect, with the exception of the on-off model, where diffusion is important. The directed motion adiabatically driven by potential sign fluctuations, though forbidden in the overdamped limit, becomes possible due to purely inertial effects in neither symmetric nor antisymmetric potentials, i.e., not for commonly used sawtooth and two-sinusoid profiles.

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Phase Space Reduction of the One-Dimensional Fokker-Planck (Kramers) Equation

Cited by 10 publications

References 38 publications

Brownian transport in corrugated channels with inertia

Brownian transport in corrugated channels with inertia

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

Inertial effects in adiabatically driven flashing ratchets

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