Standard random walk on a graph – that for every point each outgoing edge is equally probable, doesn’t really maximize entropy as mathematics expects from thermodynamical models, but do it only locally.

Such models lead to Brownian motion in continuous limit, which is good enough approximation to model diffusion in fluids, but isn’t longer appropriate within fixed structure of solids, like for recently measured electron stationary probability density on a defected lattice of potential wells of semiconductor surface:

http://physicsworld.com/cws/article/news/41659

We would rather say that this probability density is quantum mechanical ground state ... but this sample is macroscopic, so we should expect some current flow behind – some thermodynamical behavior of these 'quants of charge'.

It occurs that when we use stochastic model which finally do what mathematics expects from us – really maximize entropy, we get going to exactly quantum mechanical ground state stationary probability density, like we would thermodynamically expect from quantum mechanics.

So maybe starting from such models we could better understand dynamics of current flow in quantum scale...

I’ve just made Mathematica demonstration which allow to compare electron conductance through defected lattice using both – based on standard Generic Random Walk (classical) and these new models based on Maximal Entropy Random Walk.

It allows to observe both stationary probability distribution and dynamics of current flow for different defect densities and applied potential gradient:

http://demonstrations.wolfram.com/pr...ropyRandomWalk or

https://docs.google.com/leaf?id=0B7p...ZTkzMTJk&hl=en

They give completely different qualitative picture – I would like to ask which of them better correspond to conductance at quantum level?

For example in standard model for even the smallest potential applied, we immediately get almost uniform current flow through the whole sample, while in this new models we usually require some nonzero minimal potential gradient to 'soak' out of entropy wells through some complicated entropic landscape.

And generally I would be grateful for any remarks and comments about the demonstration.