The Dynamics Of A Brownian Particle In A Hookean Type Potential And Its Effective Diffusion Coefficient In A Ratchet Potential

In the ¯rst part of this work, we use the one dimensional random walk model tornsimulate the tracings, the autocorrelation function and the Gaussian distribution ofrnthe positions of a Brownian particle (a bead of radius Rbead = 240 nm) tethered byrna DNA tether of persistence length, lp = 45 nm, whose other end is anchored to arnmicroscope slid. And in the second part, we derive the e®ective di®usion coe±cientrnanalytically and numerically for the one dimensional motion of the Brownian particlernin a region where there is a periodic (ratchet) potential pro¯le of period L. To derivernthe e®ective di®usion coe±cient analytically, we use the known ¯rst passage timern(FPT) method. For the simulation part, we use the one dimensional random walkrnmodel as in the ¯rst part of our work. The e®ective di®usion coe±cient of the beadrnobtained both analytically and numerically vanishes at large potential barrier height.rnHowever, at the low barrier limit the analytical solution takes the value 2rn3D0 ( D0rnis the free thermal di®usion of the bead), while the numerical solution takes thernexpected value, D0. Understanding these physical situations is a practical necessityrnin many experimental contexts, for instance in single biomolecules

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