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The thermal ratchets model for transport of diffusive particles


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Average displacement of a diffusing particle from its starting point is zero at all times. Hence, a population of such particles may only be transported in a particular direction by applying a drift-inducing external potential. However, a potential that exerts a force whose space-average is zero is not expected to result in transport (i.e. a flux whose space-average is non-zero). One such potential is the 'ratchet' potential, named so because its asymmetric sawtooth shape resembles the mechanical rectifying device called a ratchet. Surprisingly, if such a potential is turned on and off periodically or even stochastically, it results in a transport of the particles in the direction of its gentler slope. This 'thermal ratchets model' explains several transport phenomena observed in nature, including the transport of vital materials by molecular walkers along the cytoskeleton inside the living cell. Several papers in the 1990s studied this model and the resultant flux via analytical, computational and experimental approaches. In this thesis we employ a new approach of study, via a numerical solution of the drift-diffusion equation, which offers several advantages and the promise of new insights over the older methods. With this method we have been able to probe the previously unexplored regime of high alternating rate that does not allow adiabatic relaxation. One of the more surprising results is that even in such a case the system settles into periodic dynamics asymptotically.
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... By Eq. (2), the steady state condition c t = 0 enforces uniform flux over space. It may be shown that the flux in fact vanishes everywhere [11], unless there exists an additional external force, in which case there is a uniform flux in its direction. However, switching the potential at finite intervals prevents the attainment of the steady state, but achieves transport even in the absence of any further external force. ...
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The thermal ratchets model toggles a spatially periodic asymmetric potential to rectify random walks and achieve transport of diffusing particles. We numerically solve the governing equation for the full dynamics of an infinite 1D ratchet model in response to periodic switching. Transient aperiodic behavior is observed that converges asymptotically to the period of the switching. We study measures of the transport rate, the transient lifetime, and an equivalent of `amplitude', then investigate their dependence on various properties of the system, along with other features of the transient and asymptotic dynamics.
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