Equations for the Non-Equilibrium Moments

The situation is getting considerably complicated, if one takes local anisotropy into account. The mobility of a particle along the axis of a macromolecule is considered to be bigger than that in the perpendicular direction, so that the entire macromolecule can move easier along its contour. Introduction of the local anisotropy of mobility in Chapter 3 (equations (3.12) and (3.13)) allows us to specify the extra forces of external and internal resistance and define the quantities (7.5) as 

While introducing of the global anisotropy, the equation for the macro-molecular dynamics remains linear in co-ordinates and velocities, the introduction of the local anisotropy makes it non-linear in co-ordinates. Both global and local anisotropy are needed to describe the non-linear effects of the relaxation phenomena in the mesoscopic approximation. 

The dynamics of the macromolecule in the form of a set of differential equations of the first order is convenient for derivation of relaxation equations (Volkov and Vinogradov 1984, 1985 Volkov 1990). As a starting point, we use equations (7.6) and consider m = 0 in this system, so that the second equation allows us to define 

These equations allow one to find relaxation equation for different moments, if the quantities ) and Q°f are given. We consider here that the quantities are independent on the co-ordinates the more complicated case, when these quantities depend on the co-ordinates of particles, is left for other researchers. It is convenient in this section to omit the mode label and write down the above equations in the form 

Further on we shall use the following symbols for the moments of the considered variables 

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