Electron conductance model using Maximal Entropy Random Walk


	We use thermodynamical models while having incomplete information about a system. They rely on mathematical theorems like maximum uncertainty principle, which says that in such situations, we should assume maximizing entropy statistical ensemble among possible scenarios. To model behavior of particles on a graph this way, we usually use simplified models - maximizing entropy locally: for each vertex assume uniform probability among possible outgoing edges - Generic Random Walk (GRW). By considering a defected lattice as the graph and add potential gradient, we get classical electron conductance models this way. There were recently introduced models which really maximizes entropy among possible stochastic processes as mathematics expects: using uniform distribution among possible paths - Maximal Entropy Random Walk (MERW). It leads to completely different stationary probabilistic density - exactly as corresponding quantum ground state, expected also from thermodynamical behavior of quantum mechanics and so it could allow to better approximate and understand current flow in nanoscale. This demonstration allows to compare stationary and dynamical behavior of both models in 2D for different defect densities and potential gradients.


	Contributed by: Jarek Duda

To emphasize some scenarios in thermodynamical models, we assign energy to each of them and instead of assuming uniform probability distribution, we use Boltzmann distribution which is some compromise between maximizing entropy and minimizing energy.

To express stronger walker’s tendency for jumping in given direction for conductance models, we introduce a (fixed) difference of potential energy between succeeding layers of the defected lattice.

We can use Boltzmann distribution in generally two ways now: locally - for each vertex among its outgoing edges (natural expansion of GRW), or globally - among whole paths (MERW analogue).

The controls allows to choose applicated potential gradient (

), edge removal probability from regular 2D lattice

), the size of lattice (

) and the number of considered pseudorandom scenario (

). The ‘

’ button allows to turn on/off cyclical boundary conditions in vertical direction.

The ‘

’ button allows to switch between observing stationary probability density and dynamics of current flow. By clicking on the picture we can choose the point in which the current is injected.

To find more informations about characteristic for quantum mechanics localization properties finally observed in random walk, see http://link.aps.org/doi/10.1103/PhysRevLett.102.160602

More information about the extended models and their resemblence to quantum mechanics can be found in http://arxiv.org/abs/0910.2724

"Electron conductance model using Maximal Entropy Random Walk" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/ElectronConductanceModelUsingMaximalEntropyRandomWalk/

Contributed by: Jarek Duda

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