Signalled Diffusion Book III: Drift-Diffusion

Book III of my "Signalled Diffusion" project can be downloaded here. Books I and II can be downloaded from here and here respectively. Below I reproduce the summing  up section that appears at the end of Book III

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Summing up and Interpretations

Our final equation, equation 108.0 was:

110.0

…where

111.0

What does this equation mean? Going through the terms on the right hand side we have:

1. The first term is the diffusion term for randomly walking agents.
2. The second term, the drift term, results of the random walk having a systematic bias in the same direction as the diffusion.

3. Because a positive sign in front of the third term only makes sense if the walk agents are multiplying then we interpret the quantity Y  not as probability but as a count of stepping agents at a point in time and space. If the sign in front of the third term is negative then it is possible for Y to be a decaying probability.

The coefficients on the right hand side of 110.0 depend on k. This is a consequence of the constraint which sets the drift current equal to the diffusion current. The drift has the effect of moving the Y envelope to the left or right depending on the direction of the slope; in fact in 110.0 the drift is always in the opposite direction to the slope. Therefore changes in the slope cause differential drifting resulting in the distortion and dispersion of a localised envelope in Y. The consequence is that a spatially limited envelope in Y will be pulled apart by differential drifting. Hence the idea of a moving frame defined by the systematic drift of a spatially localised envelope in Y is not found in equation 110.0. 

The highly disruptive dispersion in real number drift-diffusion is ameliorated when we move over to complex number diffusion.  In complex number diffusion the analogue of k is the wave number of a corkscrew wave and the vector wave number can be uni-directional and still allow a localised envelope in Y . Moreover the wave envelope can have a packet profile that moves with an identifiable velocity. Even so, as we know, a measure of dispersion also occurs in wave theory.

It is hoped that a study of real number drift-diffusion will assist in the understanding of complex number drift-diffusion. As we saw in chapter 6 equation 110.0 implies a kind of relativity of time and space in that standards of measurement change with drift value. However, in the case of real number diffusion this is not likely to lead to any kind of elegant frame invariance as it does with complex number diffusion, but it nevertheless shows how the complementary nature of drift and diffusion entail a relativity of time and space measurements; in the case of complex number diffusion this relativity has the effect of masking the existence of an absolute frame. The other feature that we begin to see in real number drift-diffusion (and also true of complex number diffusion) is how it disguises the asymmetries in the construction of space and time by compressing the node structure as slopes increase.