Drift coefficients

The function a(x, t) appearing in the FPE is called the drift coefficient, which, due to Stratonovich s definition of stochastic integral, has the form [2]... 

The first approach to obtain exact time characteristics of Markov processes with nonlinear drift coefficients was proposed in 1933 by Pontryagin, Andronov, and Vitt [19]. This approach allows one to obtain exact values of moments of the first passage time for arbitrary time constant potentials and arbitrary noise intensity moreover, the diffusion coefficient may be nonlinear function of coordinate. The only disadvantage of this method is that it requires an artificial introducing of absorbing boundaries, which change the process of diffusion in real smooth potentials. 

Choosing the drift coefficient to be linear in U and the diffusion matrix to be independent of U ensures that the Lagrangian velocity PDF will be Gaussian in homogeneous turbulence. Many other choices will yield a Gaussian PDF however, none have been studied to the same extent as the LGLM. 

The transformation rule given in Eq. (2.166) is instead an example to the so-called Ito formula for the transformation of the drift coefficients in Ito stochastic differential equations [16]. ft is shown in Section IX that V q) and V ( ) are equal to the drift coefficients that appear in the Ito formulation of the stochastic differential equations for the generalized and Cartesian coordinates, respectively. 

This equation is equivalent to the result given by Ottinger for the Ito drift velocity of an arbitrary rigid system, generalized here by the use of an unspecified value of W so as to also apply to stiff systems. This drift velocity appears as the drift coefficient (i.e., the prefactor of dt) in the Ito SDE given in Eq. (29) of Ref. 11 andEq. (5.61) of Ref. 12. 

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. 

Expressions for the drift velocity and diffusivity of the solution of Ito SDE (2.226) may be obtained by calculating the first and second moments of AZ for the underlying discrete process. Taking the average of AX in Eq. (2.227) immediately yields a drift velocity, as defined by Eq. (2.223), that is equal to the drift coefficient A (X) that appears in the Ito SDE ... 

The nontrivial transformation rule of Eq. (2.231) for the Ito drift coefficient (or the drift velocity) is sometimes referred to as the Ito formula. Note that Eq. (2.166) is a special case of the Ito formula, as applied to a transformation from generalized coordinates to Cartesian bead coordinates. The method used above to derive Eq. (2.166) thus constitutes a poor person s derivation of the Ito formula, which is readily generalized to obtain the general transformation formula of Eq. (2.231). 

The coefficients and may be chosen so as to yield the correct Cartesian diffusivity, or may be obtained by applying the Ito transformation formulae to the transformation from generalized to Cartesian coordinates. Either method yields a drift coefficient... 

Note that for a Stratonovich SDE, unlike an Ito SDE, the drift velocity generally differs from the drift coefficient that appears in the SDE. 

Fokker-Planck Equation. A Stratonovich SDE obeys a Fokker-Planck equation of the form given in Eq. (2.222) with the drift velocity V (X) given in Eq. (2.243), and the diffusivity given in Eq. (2.229). The resulting diffusion equation may be written in terms of the drift coefficient... 

A set of Stratonovich SDEs for a constrained mechanical system may be formulated either as a set off SDEs for the soft coordinates or as a corresponding set of 3N SDEs for the Cartesian bead positions. The Stratonovich SDEs for the generalized coordinates are of the form given in Eq. (2.238), with a drift coefficient... 

Note that, in either system of coordinates, the value of the Stratonovich drift coefficient required to produce a given drift velocity depends on derivatives of... 

When supplemented by Eq. (2.269) for t p(r), the Langevin equation becomes equivalent to a standard Stratonovich SDE with a drift coefficient and a coefficient... 

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. 

The value of the drift coefficients required by each type of SDE may be obtained by comparing the value of the drift velocity generated by the SDE to that required by statistical mechanics. The desired drift velocity vector for a system of coordinates X, ..., X may be expressed in generic form as a sum... 

Expressions for the drift velocity produced by Ito (1), Stratonovich (S), Eangevin (L), and kinetic (K) SDEs, and for the drift coefficient required for each type of SDE to produce the predicted drift velocity of Eq. (2.344), in a generic system of coordinates. VL is as defined in... 

This is the reactive force on the first system as a result of its interaction with the second system. In order to obtain a proper equation in the SRLS case, SF modify the irreversible term in Xj, that is, Dj. In the present notation this is merely equivalent to substituting Eq. (1.59) by Eq. (1.48). In the FT case SF modify y, that is, they add a term -o) p fto the reversible drift coefficient in Xj, which was previously zero, and leave unmodified. In both cases these are the minimal modifications required to achieve detailed balance. No changes in the diffusion tensor elements are introduced, although such possibilities exist. This minimum effort choice yields equations derivable from a MFPKE in which the full set of variable (xj, pj, X2, P2) is considered. 

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