Imagine a semiconductor lattice - a regular lattice (e.g. of Si or Ga) with a small fractions of a different atoms (like Mn).(adsbygoogle = window.adsbygoogle || []).push({});

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 [Broken]

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, thatthe 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 [Broken]

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, Schrodinger 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 anatural 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 [Broken]

Mathematica conductance simulator: https://dl.dropboxusercontent.com/u/12405967/conductance.nb [Broken]

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?

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# Anderson localization and electron flow/diffusion?

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