Drift coefficient coordinates

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. 

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. 

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... 

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... 

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... 

Baird et al. [350]). In the following analysis, the functional forms, p(E), which have been proposed (see below) to represent the field-dependence of the drift mobility are used for electric fields up to 1010Vm 1. The diffusion coefficient of ions is related to the drift mobility. Mozumder [349] suggested that the escape probability of an ion-pair should be influenced by the electric field-dependence of both the drift mobility and diffusion coefficient. Baird et al. [350] pointed out that the Nernst— Einstein relationship is not strictly appropriate when the mobility is field-dependent instead, the diffusion coefficient is a tensor D [351]. Choosing one orthogonal coordinate to lie in the direction of the electric field forces the tensor to be diagonal, with two components perpendicular and one parallel to the electric field. 

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... 

The nonlinear friction coefficient y(V) thereby takes on the role of a confining potential while for y0 = y(0) the drift term Fy0, as mentioned before, is just the restoring force exerted by the harmonic Omstein-Uhlenbeck potential, the next higher-order contribution y2V3 corresponds to a quartic potential, and so forth. The fractional operator 0a/0 V a in Eq. (125) for the velocity coordinate for 1 < a < 2 is explicitly given by [20,64]... 

Appendix D The Calculation of Drift and Diffusion Coefficients in Curvilinear Coordinates... 

APPENDIX D THE CALCULATION OF DRIFT AND DIFFUSION COEFFICIENTS IN CURVILINEAR COORDINATES... 

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