Langevin equation force bias

The results given above are essentially identical to those obtained by Hinch [10] by a similar method, except for the fact that Hinch did not retain any of the terms involving the force bias (tIv)o which he presumably assumed to vanish. An apparent contradiction in Hinch s results may be resolved by correcting his neglect of this bias. In a traditional interpretation of the Langevin equation as a limit of an underlying ODE, the bead velocities are rigorously independent of the hard components of the random forces, since the random forces in Eq. (2.291) appear contracted with K , which has nonzero components only in the soft subspace. Physically, the hard components of the random forces are instantaneously canceled by the constraint forces, and thus can have no effect... 

Equation (2.315) may be interpreted as a peculiar discretization of the Langevin equation, which is constructed so as to avoid the force bias that arises in the traditional interpretation. We identify random forces... 

Stratonovich interpretation of the Langevin equation, it is the use of a midstep value of C (X) that causes the unwanted bias in the random forces. 

Note that the term involving a derivative of In / in Eq. (2.331) is identical to the velocity arising from the second term on the RHS of Eq. (2.286) for the transformed force bias in the traditional interpretation of the Langevin equation. The traditional interpretation of the Langevin equation yields a simple tensor transformation rule for the drift coefficient A , but also yields a contribution to Eq. (2.282) for the drift velocity that is driven by the force bias. The kinetic interpretation yields an expression for the drift velocity from which the term involving the force bias is absent, but, correspondingly, yields a nontrivial transformation mle for the overall drift coefficient. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

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