Spurious drift

Exercise. Any equation of the form (4.5) with Ito prescription is equivalent with another equation with Stratonovich prescription. Find the connection between the coefficients in both equations. The difference between their coefficients A is sometimes called the spurious drift . 

Equations (6.305) and (6.306) are different because of the drift term V (PV D), which is sometimes called a spurious drift term. These diffusion equations have different equilibrium distributions and are two special cases of a more general diffusion equation. 

Equation (6.308) implies that the isotropic diffusive motion along the coordinate axes is independent. Here, V V/KT is the drift due to an external potential force field V, while Vcr/cr represents an internal drift caused by a concentration gradient of the traps. The term PV(e /a2) is the spurious drift term. Equation (6.308) allows spatial variations of all parameters T, V, T>. and a with inhomogeneous temperature. FromEq. (6.308), the diffusion coefficient becomes... 

This is the traditional diffusion model given in Eq. (6.305) with the diffusion coefficient D proportional to 1/er. For this case, the so-called spurious drift term vanishes because the effects of a and a cancel each other out in the stationary state. The stationary distribution is proportional to the Boltzmann distribution exp(-I 7/c7 ) and independent of D. 2. a = constant. Then, Eq. (6.308) becomes... 

Ito [51] obtains different expressions for the Kramers-Moyal coefficients in which the spurious drift term is absent. However use of Ito coefficients involves new rules for calculus and so Stratonovich s method will be used here since it is also in agreement with the original method of Brown [8] and is the correct definition to use in the case of a physical noise which always has a finite correlation time [58] (see B.2). 

The last term in Eq. (E.17) is as we have seen called the noise-induced or spurious drift [31]. 

The superscript identifies the conformation at the beginnin of the time-step. For small timesteps At this should be reasonable to do. Fj in the above equation is the force exerted on particle j. The so-called spurious drift , i.e., the third term in the r.h.s. of eq. (3.20) usually vanishes, since most diffusion tensors which have been used in the literature have zero divergence (this is directly related to the assumption of incompressible flow). p] At) is the random displacement by the coupling to the heat bath. The crucial difficulty comes from the connection of the displacement by the heat bath and the hydrodynamic interaction tensor Dy via the fluctuation dissipation theorem. This fixes the first two moments to be... 

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