Maximal entropy random walk and quantum mechanics

[Duda Jarek]

While introducing random walk on given graph, we usually assume that for each vertex, each outgoing edge has equal probability. This random walk usually emphasize some path. If we work on the space of all possible paths, we would like to have uniform distribution among them to maximize entropy. It occurs that we can introduce random walk which fulfills this condition: in which for each two vertexes, each path between them of given length has the same probability.

Probability of going from a to b in MERW is

S_ab= (1/lambda) (psi_b/psi_a)

where lambda is the dominant eigenvalue of adjacency matrix, psi is corresponding eigenvector. Now stationary probability distribution is

p_a is proportional to psi_a^2

 

We can generalize uniform distribution among paths into Boltzmann distribution and finally while making infinitesimal limit for such lattices covering R^n, we get behavior similar to quantum mechanics. This similarity can be understand that QM is just a natural result of four-dimensional nature of our world

http://arxiv.org/abs/0910.2724

In this paper further generalizations are made in classical field of ellipsoids as its topological excitations. Their structure occurs to be very similar to known from physics with the same spin, charge, number of generations, mass gradation, decay modes, electromagnetic and gravitational interaction.

Here is for example presented behavior of the simplest topological excitations of direction field - spin 1/2 fermions:

http://demonstrations.wolfram.com/SeparationOfTopologicalSingularities/

 

What do you think about it?

Edited October 18, 2009 by Duda Jarek