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Talk-Polywell.org • Anderson-like localization for electron currents in plasma?
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Anderson-like localization for electron currents in plasma?

Posted: Sat Apr 28, 2018 5:33 pm
by jarek
Electron currents in plasma are usually modeled using classical diffusion. However, semiconductor studies have shown that there is a problem with using classical diffusion for modeling electron conductance: it leads to nearly uniform probability distribution of electrons, what means that electric field would make them flow - making semiconductor a conductor.
In contrast, semiconductor is usually not a conductor due to : e.g. QM ground state has highly localized density, preventing conductance.

Maximal entropy random walk (MERW) allows to understand and repair this discrepancy: https://en.wikipedia.org/wiki/Maximal_e ... andom_walk
It shows that standard diffusion often only approximates the required by statistical physics modeling. MERW repairs this approximation, by the way repairing the discrepancy with QM prediction (and reality) - its stationary probability distribution is exactly as for quantum ground state: with Anderson-like localization property.
The agreement can be also seen through popular - there are some small differences but both assume uniform or Boltzmann distribution among possible paths.

Here is example of evolution of standard random walk/diffusion (GRW) and maximizing entropy MERW leading to quantum stationary probability distribution - while locally looking similar, they lead to drastically different stationary probability distribution. It is for defected lattice: all nodes but the marked ones have self loop (edge to itself):
Image

To see that such universal thermodynamical considerations are not restricted to nanoscale, imagine 1D potential well: V=0 at [0,1], infinite otherwise. Standard diffusion would obviously predict rho = 1. In contrast, both MERW and QM would predict rho ~ sin^2 stationary probability distribution - with avoidance of the boundaries - in any scale:
Image

Sure in practice we have no infinite potential barriers, but energy of electrons is comparable to local potential differences - of barrier slope. For complex inhomogeneous filed such thermodynamical effect can be huge - while small local differences of potential lead to small differences between local behavior of standard stochastic propagator and maximizing entropy QM-like ... globally they lead to completely different stationary probability distribution: while standard diffusion has nearly uniform distribution, MERW/QM ground state can have localization with density lower by many orders of magnitude.

Here is a simple Mathematica conductance simulator for standard diffusion (GRW) and MERW : http://demonstrations.wolfram.com/Elect ... ndomWalks/
Intuitively, while in standard diffusion electron flow is nearly uniform, in MERW it kinds of short-circuits through lowest resistance regions, like in formation of lightning.

Is there maybe experimental evidence suggesting that standard diffusion models for electrons currents in plasma predict insufficient localization?