# Maximal Entropy Random Walk - quantum corrections to stochastic models

1. Sep 16, 2013

### jarekd

Imagine there is a complex system and we are interested in its basic statistical properties, like the stationary probability distribution. For example for a single electron wandering in defected lattice of semiconductor.
Physics offers two basic ways of answering such question:
- from one side, quantum mechanics says that we need to take the operator describing energy: Hamiltonian (Bose-Hubbard in this case), take its dominant eigenvector/function and squares of its coordinates define the expected stationary probability distribution: the ground state,
- from the other side, stochastics says that we should choose stochastic operator (transition probabilities) and coordinates of its dominant eigenvector/function is the stationary probability distribution.

The problem is that these two answers are often in disagreement. For example accordingly to stochastic models, semiconductors should conduct electrons well, while in fact they don't. QM provided answer to this problem, what leaded e.g. to Nobel prize for Anderson for quantum localization properties - that while stochastic models usually lead to nearly uniform distribution, electrons in semiconductors are localized/prisoned e.g. in large defect-free areas, preventing conductance. Historically it was succeeding argument against classical intuitions about physics.

So cannot we imagine that single electron is in given place in given moment - then with some probability distribution jumps to a different place? (stochastic picture)
Heisenberg uncertainty principle says that this "place" cannot be too small, so let it be a few lattice sites - with some precision we should be able to determine its position and so asking about transition probabilities - stochastic models, should make sense.

So for some precision we should be able to use stochastic models - and they should be in agreement with quantum predictions.
The question is how to repair the disagreement? While quantum picture uses well defined energy density, in stochastic approaches we usually just guess the transition probabilities.
But there is a mathematical principle saying how to choose such probabilities - the maximal uncertainty principle: that when we have no information, the safest is to choose the maximizing entropy probability distribution.
If we denote the probability that while being in vertex $i$, the next jump will be to vertex $j$ by $S_{ij}$ and by $p_i:\ pS=p$ the stationary probability distribution of this stochastic matrix $S$, average entropy production of this stochastic process is $\sum_i p_i \sum_j S_{ij} \ln(1/S_{ij})$.
For a given graph, there is a unique stochastic process $S_{ij}$ maximizing this entropy production: the (recent) Maximal Entropy Random Walk (MERW):
$S^{MERW}_{ij}=\frac{M_{ij}}{\lambda} \frac{\psi_j}{\psi_i}$ where $M_{ij}$ is the adjacency matrix, $\psi M=\lambda \psi$ is the dominant eigenvector.
While standard way to choose the random walk (GRW), leading to currently used stochastic models, is : $S^{GRW}_{ij}=M_{ij}/\sum_k M_{ik}$.
GRW is kind of local approximation of entropy maximization.

GRW and MERW are in agreement for regular graphs/lattices, but generally MERW has much stronger localization properties. Its stationary probability distribution turns out to be the square of coordinates of the dominant eigenvector/function of the Hamiltonian (e.g. Bose-Hubbard or Schroedinger) - is exactly the quantum ground state probability distribution with its localization properties.
Here is example of comparison of their evolution in a defected lattice:
https://dl.dropboxusercontent.com/u/12405967/conf.jpg [Broken]

Some materials: PRL paper, (defended)PhD thesis, simulator, https://dl.dropboxusercontent.com/u/12405967/phdsem.pdf [Broken].

This disagreement has nearly completely split the realms of stochastic and quantum physicists, but MERW says where is the problem (approximation) and how to correct it.
Besides the conductance of semiconductor, what other "quantum" corrections to (usually guested) stochastic models should we expect?

Last edited by a moderator: May 6, 2017