Brownian Particle under an External Force

Returning to the Langevin equation (Equation 6.31), for a Brownian particle under an external foree, let us consider the one-dimensional version with the 

It must be remarked that the coefficient of the noise term in Equation 6.46 was chosen so that the fluctuation-dissipation theorem for F(r) satisfies the equations. 

We shall encounter Equation 6.47 and its solution given in the following sections many times in later chapters in the context of translocation. The external force/(x) can arise generally from free energy barriers along the pore and the electric forces driving polymer translocation. 

The above two examples of a Brownian particle are representative of a large class of stochastic processes. Let us consider a general stochastic process (f), without any memory, where the random variable is time-dependent and the noise is white noise. The general Langevin equation describing the stochastic process (t) can be written as (Risken 1989) 

The angular brackets indicate averaging over a sufficiently long period of time or equivalently an equilibrium average. h, t) and g(f, t) are functions corresponding to the specific stochastic process. With the definition of the correlation function given in Equation 6.52, the noise strength is absorbed into g(, t). 

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As an example, for the Brownian particle under an external force discussed in Section 6.2.2,... 

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