Random forces, constrained Brownian motion

In what follows, we distinguish between the drift velocity V associated with a random variable A , which is defined by Eq. (2.223), and the corresponding drift coefficient that appears explicitly in a corresponding SDE for A , which will be denoted by A (A), and which is found to be equal to V only in the case of an Ito SDE. The values of the generalized and Cartesian drift velocities required to force each type of SDE to mimic constrained Brownian motion are determined in what follows by requiring that the resulting drift velocities have the values obtained in Section VII. 

Due to the ubiquitous nature of Brownian motion in microfluidics, the ability to constrain it has wide-ranging applications in sensing and directed assembly. The constraint is provided by imposing an appropriately shaped external potential on the particle s motion. For example, if the purpose is to trap a colloidal particle positionally, then the potential well should be deep enough so that the particle cannot escape the well due to the random Brownian perturbations experienced by it. Such potentials typically employ forces that depend on the particle s material properties, for example, optical and electrical. An alternate approach that does not rely on the particle s material properties but instead involves shaping the potential well by altering the flow surrounding... 

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