Projection onto Diffusion Modes

The foregoing discussion of the dynamic processes contributing to the rate kernel was given in terms of phase-space propagators for the A and B 

In Section X.A we described the structure of the propagator [z-L ( 12 z)] and pointed out that it could be written in terms of the propagator for the correlated motion of the pair and a coupling term The simplest versions of the configuration space diffusion and Smoluchowski theories do not take into account such correlated motion. To make connection with these simpler theories, we therefore write 

The analysis can be carried out more easily if an abstract notation is first introduced. We write an arbitrary operator A(12) in the form 

We also introduce abstract basis functions, / , whose v-space matrix elements are related to Hermite polynomials H/(v) 

We also let / 17 = 177 . In this notation, the relaxing part of the rate kernel may be written as 

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We very briefly consider the effects of the terms that were neglected in the course of projecting onto the diffusion mode. Once again, the analysis closely parallels the derivation of Stokes law. To explicitly consider the effects of nonhydrodynamic (nondiffusion mode) states on the rate kernel, we use an analysis similar to that of van Beijeren and Dorfman and introduce a projection operator... 

The expression for the projection of the /th normal mode onto m of the mth rod is presented elsewhere/17 29) Besides these deformational normal modes, there is also a uniform axial spinning mode in which all of the rods rotate in phase like a speedometer cable. The contribution of that mode to is just 2k Tt/(N+ l)y, as expected for diffusion in one dimension with the friction factor (N + 1)7 of the entire filament. 

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