Langevin equation Cartesian coordinates

The present analysis builds directly on three previous analyses of SDEs for constrained systems by Fixman [9], Hinch [10], and Ottinger [11]. Fixman and Hinch both considered an interpretation of the inertialess Langevin equation as a limit of an ordinary differential equation with a finite, continuous random force. Both authors found that, to obtain the correct drift velocity and equilibrium distribution, it was necessary to supplement forces arising from derivatives of C/eff = U — kT n by an additional corrective pseudoforce, but obtained inconsistent results for the form of the required correction force. Ottinger [11] based his analysis on an Ito interpretation of SDEs for both generalized and Cartesian coordinates, and thereby obtained results that... 

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. 

Cartesian coordinates by a canonical point transformation. The Langevin equations, or the equivalent FPEs, for the N coupled atoms are most conveniently written in the 3Af-element Cartesian coordinate vector x = (Xj,..., with conjugate momentum vector p = (p,...,Pjjy),... 

Equations (D.6) and (D.7) are Gilbert s equation in spherical polar coordinates. To obtain the Gilbert-Langevin equation in such coordinates we augment the field components //, and with random field terms and h. By graphical comparison of the Cartesian and spherical, polar coordinate systems, we find these to be... 

In this equation x, is the Cartesian coordinate x for atom i, /3, is the frictional drag on atom i and/, is the Langevin random force on atom i obtained from a Gaussian random distribution of zero mean and variance... 

The Langevin equation of motion, Eq. 72, can be easily solved by means of a transformation from the Cartesian coordinates R to the normal coordinates Q, Eq. 9. The transformation reads ... 

To solve the Langevin equations of motion, one can use the following transformations from Cartesian coordinates R to normal coordinates Q ... 

In order to obtain the intrachain phase shift as well as the transformation from the Cartesian coordinates to the normal coordinates, seven constants, Ai, A2, A3, Bo, Bi, B2, and Bs, have to be determined, see Eqs. 84 and 85. For these purposes one can use the Langevin equations of motion for the network junctions (cross-links), Eq. 80. Formally we also add the following six conditions at the junction points ... 

To summarize, the relaxation times (or eigenvalues) of a rather complex system such as a 3-D topologically-regular network end-Unked from Rouse chains were determined analytically. In fact, one can do even better it is possible to construct all of the eigenfunctions of the network analytically (which amounts to the transformation from Cartesian coordinates to normal coordinates). Briefly, to construct the normal mode transformation, see Eqs. 84 and 85, one has to combine the Langevin equations of motion of a network jimction, Eq. 80, and the boundary conditions in the network junctions, Eqs. 87 to 92. After some algebra one finds [25,66] ... 

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