# Electron Conductance Models Using Maximal Entropy Random Walks

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We use thermodynamic models for systems for which we have incomplete information. These models are based on theorems such as the maximum uncertainty principle, which states that we should choose the scenario which maximizes the entropy of the statistical ensemble.

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Contributed by: Jarek Duda (August 2016)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

To emphasize some scenarios in thermodynamical models, we fix the energy instead of assuming a uniform probability distribution. The Boltzmann distribution is a compromise between maximizing entropy and minimizing energy.

We can use the Boltzmann distribution in two different ways: locally, for each vertex of the outgoing edges (natural expansion of GRW); or globally, considering entire paths (MERW analog).

The controls let you choose the potential gradient , the edge removal probability from the regular 2D lattice , the size of the lattice and the number of a pseudorandom "seed". The "bound" button lets you toggle the cyclical boundary conditions in the vertical direction. The "eig" slider lets you use some other eigenvector than the dominant one.

The "flowing" button lets you switch between stationary probability density and dynamic current flow. By clicking the graphic you can choose the point at which the current is injected.

For more information about localization properties in quantum mechanics observed in a random walk see [1].

More information about the extended models and their relation to quantum mechanics can be found in [2].

References

[1] Z. Burda, J. Duda, J. M. Luck and B. Waclaw, "Localization of the Maximal Entropy Random Walk," *Physical Review Letters*, 102(16), 2009 160602. doi:10.1103/PhysRevLett.102.160602.

[2] J. Duda, "From Maximal Entropy Random Walk to Quantum Thermodynamics." arxiv.org/abs/1111.2253.

## Permanent Citation

"Electron Conductance Models Using Maximal Entropy Random Walks"

http://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/

Wolfram Demonstrations Project

Published: August 22 2016