Diffusion of ionic penetrants in charged disordered media

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The physical argument is as follows. We consider quenched disorders that are sufficiently large, and we believe that the system is self-averaging. Therefore, the system can be divided up into several large subsystems, each of which is larger than any relevant length scale of the problem. The penetrant’s motion is averaged over all these subsystems. Since there are several subsystems, the penetrant will visit a representative ensemble of subsystems that includes configurations of the quenched disorder that are arbitrarily close to those that characterize the annealed case in the adiabatic limit. Thus the quenched and adiabatic annealed averages should be the same in the limit of large, self-averaging systems. There is one subtlety in the argument above that has been noted in the polymer statistics literature (Refs. 18 and 22). The argument is only correct if we do not restrict the penetrants to any specific initial position. This is to say that we must average over all possible initial center of mass positions of the penetrant. This average over initial positions is necessary because such an average is an ensemble average when many uncorrelated subsystems are contained in the system of interest. In our analysis to follow, we will average over initial positions. Note that any experimental measure of penetrant mobility (e.g., measurements using pulse field gradient NMR spectroscopy) is obtained by averaging over all possible initial positions.

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