Diffusion coefficient multiplicative noise

We have shown how to obtain the drift and diffusion coefficients for simple additive noise. A difference arises in the case of multiplicative noise, which we encounter in Gilbert s equation as augmented by a random noise field. In that case the system is governed by a Langevin equation of the form (we take the one-dimensional case for simplicity)... 

Having illustrated the problem associated with multiplicative noise we will now illustrate how the procedure is applied to obtain the drift and diffusion coefficients for the two-dimensional Fokker-Planck equation in phase space for a free Brownian particle and for the Brownian motion in a one-dimensional potential. This equation is often called the Kramers equation or Klein-Kramers equation [31]. 

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

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