Rotational friction tensor

Let us suppose that the liquid system is described by a MFPKE in N + 1 rigid bodies (the solute, or body 1 and N rotational solvent modes or bodies ), each characterized by inertia and friction tensors I and a set of Euler angles ft , and an angular momentum vector L (n = 1,..., N -I-1) plus K fields, each defined by a generalized mass tensor and friction tensor and a position vector and the conjugate linear momentum k = 1,..., K). The time evolution of the joint conditional probability x", L , P° 11, X, L, P, t) (where ft, X, etc. stand for the collection of Euler angles, field coordinates etc.) for the system is governed by the multivariate Fokker-Planck-Kramers equation... 

Since the expressions (8.5) and (8.15) are identical, they can be combined, by introducing so-called global tensors of friction f and mobility V, including translational and rotational components. In the Stokes flow, these tensors have some universal properties [2], of which the most important are dependence on instant configuration and independence of velocity, as well as symmetry and positive definiteness of matrixes fj and V. ... 

As structured fluids such as liquid crystals are at least partially fluid, we also need to consider the forces and torques produced by friction. The frictional forces are given by a dissipative stress tensor, which is most conveniently derived from the dissipative function (j)F It is a homogeneous positive definite quadratic function of the time derivatives of the strains and rotations (the time derivatives of the torsions can be generally ignored) giving ... 

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