Thermal Ratchet Effect in Confining Geometries

Viktor Holubec,Martin Varga,Ayub Khodaee,Artem Ryabov,Petr Chvosta,M Foulaadvand,Mohammad Yaghoubi

doi:10.3390/e19040119

Viktor Holubec, Martin Varga + Show 5 more

Open Access

https://doi.org/10.3390/e19040119

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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.

Highlights

Diffusion in narrow channels of varying cross-sections, e.g., through micropores of zeolites or channels in cell membranes, is essentially a three-dimensional (3D) problem with rather complex boundary conditions
The mean velocity of rotating wheels and the mean heat current between the reservoirs, Equation (25), and their ratio (the figure of merit, Equation (26)), are given in terms of scaling functions that depend on the fraction Tx /Ty only
These functions provide a compact description of the ratchet performance in its different working regimes

Summary

Introduction

Diffusion in narrow channels of varying cross-sections, e.g., through micropores of zeolites or channels in cell membranes, is essentially a three-dimensional (3D) problem with rather complex boundary conditions. The random motion is caused by molecules of a fluid from the second reservoir (small rectangle) at the temperature Ty. We model stochastic dynamics of the mechanical device by an overdamped diffusion (one can use a more fundamental and less tractable underdamped description, which, is expected to yield qualitatively similar results in the long-time limit [50,71,72]) of two coupled degrees of freedom denoted as x (t) and y(t). K( x ) can be a symmetric function, like k ( x ) = 2 − sin(2πx )/2, which corresponds to symmetric teeth In this second case, the device is not able to work as a heat engine: the mean velocity v vanishes and only heat flow can be nonzero. In contrast to entropic transport, our channel is “soft” since its walls are represented by the potential

Solution of the Fokker–Planck Equation

Mean Velocity and Heat Current

Current Circulation and Local Heat Transfer

Conclusions