Anderson localization and electron flow/diffusion?

[jarekduda]

Imagine a semiconductor lattice - a regular lattice (e.g. of Si or Ga) with a small fractions of a different atoms (like Mn).
The natural question is: how electrons flow through it?
It can be measured experimentally: put a potential and use scanning tunneling microscope to map electron flow from the surface.
Here are some nice pictures of such experiment for two different concentrations of Mn from (Science)
http://chair.itp.ac.ru/biblio/papers/studLiteratureSeminar/Huse.full.pdf

https://dl.dropboxusercontent.com/u/12405967/local.jpg

We can see some strong localization properties - generally called Anderson localization.
The problem is that standard diffusion leads to nearly uniform probability distribution instead. Hence, if attaching a potential gradient, electrons would flow - semiconductor would be a conductor.
In contrast, it often isn't - as in the pictures, electrons are imprisoned (in local potential/entropic wells), what makes conductance/flow more difficult.

Hence Anderson localization is seen as a quantum phenomena, requiring to see electron as waves.
So cannot we see electrons (charge carriers) from stochastic perspective: probabilities of traveling between regions, flows?

I would like to argue/discuss that we can.
Specifically, that the problem with standard diffusion models is that they only approximate the (Jaynes) maximal entropy principle - which is crucial for statistical physics models.
We can maximize entropy in the space of random walks (transition probabilities) instead, getting Maximal Entropy Random Walk (MERW) - and diffusion models based on it.
While it has similar local behavior as standard random walk (GRW), it can have very different global behavior for nonhomogeneous space - for example here are densities after 10, 100, 1000 steps in a defected lattice: all nodes but the defects (squares) have additional self-loop (edge to itself):

https://dl.dropboxusercontent.com/u/12405967/conf.jpg

It turns out that MERW leads to exactly the same stationary probability distribution as QM: squares of coordinates of the dominant eigenvector of adjacency matrix, which corresponds to minus hamiltonian (Bose-Hubbard in discrete case, Schrödinger in continuous limit).
So in contrast to standard diffusion, MERW-based diffusion is no longer in disagreement with thermodynamical predictions of QM, like Anderson localization for semi-conductor.

Basically MERW is uniform probability distribution among paths - becomes Boltzmann distribution when adding potential.
This is very similar to euclidean path integrals - the differences are:
- motivation - here we just repair diffusion, path internals are "Wick rotation" of QM to imaginary time,
- normalization - path integral propagator is not yet stochastic,
- here we start with better understood: discrete system, with continuous in path integrals.

It also brings a natural intuition for the Born rules: squares relating amplitudes and probabilities.
So we ask about probability in fixed time cut of ensemble of infinite paths.
Amplitudes corresponds to probability at the end of half-paths: toward past, or alternatively toward the future (and they are equal).
To get a given random value, we need to get it from both half-paths, so the probability is multiplication of both amplitudes.

Materials about MERW:
Our PRL paper: http://prl.aps.org/abstract/PRL/v102/i16/e160602
My PhD thesis: http://www.fais.uj.edu.pl/documents/41628/d63bc0b7-cb71-4eba-8a5a-d974256fd065
Slides: https://dl.dropboxusercontent.com/u/12405967/MERWsem.pdf
Mathematica conductance simulator: https://dl.dropboxusercontent.com/u/12405967/conductance.nb

Are we restricted to see electrons from quantum perspective here - as waves?
Can we ask about flow of electrons - transition probabilities, diffusion models?
Is MERW the proper way for quantum corrections of diffusion models?
Beside semiconductor, in what other situations (like molecular dynamics) such corrections seem crucial?

 

[jarekduda]

Sadly there is zero interest for non-standard questions/topics in this forum, like is synchrotron radiation CPT symmetric?, or forgotten dual Lorentz force: for magnetic dipole traveling in electric field (classical spin-orbit interaction).

This thread is what has turned me a few years ago from super-mainstream physics PhD student into someone not satisfied with "shut up and calculate", searching for real understanding of what quantum mechanics is describing.
MERW shows that in contrast to a popular belief, diffusion and QM doesn't lead to disagreeing predictions (e.g. semiconductor) - that the problem was using approximated diffusion, not the one required by statistical physics: maximizing entropy.
Doing it right, we get the same statistics as in QM, stationary Schrödinger equation to get the stationary probability distribution.

However, diffusion specialists and QM specialists are currently at the opposite ends of the corridor ... but they often want to describe the same system and MERW says why and how they can find a common language, for example to understand quantum corrections to diffusion in micro-scale like in biology.

And this is only one of many arguments to start asking for what is happening behind the quantum (probabilistic) description, like Couders's walking droplets: classical wave-particle objects undergoing interference, tunneling, orbit quantization etc. ...