# From Maximal Entropy Random Walk to Quantum Thermodynamics

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Brownian motion requires relatively large 'walker', which is constantly being pushed in random directions (no memory) - it's derived as infinitesimal limit of Generic Random Walk (GRW) on a graph(lattice), in which each outgoing edge is equally probable.

But let's imagine we want to estimate probability density of positions of some entity which doesn't just constantly 'stop and make new independent decision', but make some concrete trajectory, which for example could depend on the past (by e.g. velocity) - in such situations the safer than taking statistical ensemble among single edges as in GRW/Brownian motion, should be using statistical ensemble among whole possible paths.

The simplest such model is Maximal Entropy Random Walk (MERW) on graph, it can be defined in a few ways:

- stochastic process on given graph which maximizes average entropy production, or

- assuming uniform probability distribution among possible paths on graph, or

- for each two vertices, each path of given length between them is equally probable.

Obtained formulas are:

P(a->b ) = $\frac{M_{ab}}\lambda \frac {\psi_b}{\psi_a}$

where M is graph's adjacency matrix ($M_{ij}\in{0,1}$)

lambda is its dominant eigenvalue with psi eigenvector (real, positive because of Frobenius-Perron theorem)

$M \psi=\lambda \psi$

stationary probability distribution is:

P(a) is proportional to $(\psi_a)^2$

This stochastic process is Markovian - depends only on the last position, but to calculate these transition probabilities we just have to know the whole graph -

we should think about this probabilities not as that 'the walker' uses them directly, but that they are only used by us to propagate our knowledge while estimating probability density of his current position.

(Minus adjacency matrix) occurs to correspond to discrete Hamiltonian, so while GRW/Brownian motion spreads probability density almost uniformly, MERW has very similar localization properties as quantum mechanics.

While adding potential: changing statistical ensemble among paths into Boltzmann distribution and making infinitesimal limit of lattice constant, we get stationary probability density exactly as quantum mechanical ground state (similar to Feynman's euclidean path integrals).

Here is PRL paper about MERW localization properties: http://prl.aps.org/abstract/PRL/v102/i16/e160602

Here is my presentation with e.g. 2 intuitive derivations of MERW formulas and some connection to quantum chaos: http://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B7ppK4I%20yMhisYmI3YTAzNzYtMDkyNy00ZDAxLTg1NGEtOTg4NWNkYzU3M%20jQ1&hl=en

Here is simulator which allows to compare conductance models using GRW and MERW: http://demonstrations.wolfram.com/preview.html?draft/93373/000008/ElectronConductanceModelUsingMaximalEntropyRandomWalk

Here are more formal derivations: http://arxiv.org/abs/0710.3861

Here is a trial to expand this similarity to quantum mechanics: http://arxiv.org/abs/0910.2724

I'm currently working on my PhD thesis in physics on this subject and so I would really gladly discuss about it.

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If someone is interested, I have just finished large paper about MERW and its connections to quantum mechanics (e.g. to show these results on congress on emergent quantum mechanics this weekend) - a preliminary version of my current PhD:

"Surprisingly the looking natural random walk leading to Brownian occurs to be often biased in a very subtle way: usually refers to only approximate fulfillment of thermodynamical principles like maximizing uncertainty. Recently, a new philosophy of stochastic modeling was introduced, which by being mathematically similar to euclidean path integrals, finally fulfills these principles exactly. Its local behavior is usually similar, but may lead to drastically different global properties. In contrast to having practically no localization properties Brownian motion, this recent approach turns out in agreement with thermodynamical predictions of quantum mechanics, like thermalizing to the quantum ground state probability density: squares of coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian for single particle in discrete case or of the standard Schrodinger operator while including potential and making infinitesimal limit. It also provides a natural intuition of the amplitudes' squares relating to probabilities. The present paper gathers and formalizes these results. There are also introduced and discussed some new expansions, like considering multiple particles with thermodynamical analogue of Pauli exclusion principle or time dependent cases, which allowed to introduce thermodynamical analogues of momentum operator, Ehrenfest equation and Heisenberg uncertainty principle."

It should appear on arxiv soon, now it can be download here:

http://dl.dropbox.com/u/12405967/phd2.pdf