The Dynamics of a Brownian Particle in a Hookean type Potential and its Effective Diffusion Coefficient in a Ratchet Potential
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Date
2007-07
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Addis Ababa University
Abstract
In the ¯rst part of this work, we use the one dimensional random walk model to
simulate the tracings, the autocorrelation function and the Gaussian distribution of
the positions of a Brownian particle (a bead of radius Rbead = 240 nm) tethered by
a DNA tether of persistence length, lp = 45 nm, whose other end is anchored to a
microscope slid. And in the second part, we derive the e®ective di®usion coe±cient
analytically and numerically for the one dimensional motion of the Brownian particle
in a region where there is a periodic (ratchet) potential pro¯le of period L. To derive
the e®ective di®usion coe±cient analytically, we use the known ¯rst passage time
(FPT) method. For the simulation part, we use the one dimensional random walk
model as in the ¯rst part of our work. The e®ective di®usion coe±cient of the bead
obtained both analytically and numerically vanishes at large potential barrier height.
However, at the low barrier limit the analytical solution takes the value 2
3D0 ( D0
is the free thermal di®usion of the bead), while the numerical solution takes the
expected value, D0. Understanding these physical situations is a practical necessity
in many experimental contexts, for instance in single biomolecules
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Keywords
Ratchet Potential