Naked Science Forum

Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Jarek Duda on 03/09/2020 06:35:57

Title: How to choose random walk, diffusion? (local vs global entropy maximization)
Post by: Jarek Duda on 03/09/2020 06:35:57
To choose random walk on a graph, it seems natural to to assume that the walker jumps using each possible edge with the same probability (1/degree) - such GRW (generic random walk) maximizes entropy locally (for each step).
Discretizing continuous space and taking infinitesimal limit we get various used diffusion models.

However, looking at mean entropy production: averaged over stationary probability distribution of nodes, its maximization leads to usually a bit different MERW:

It brings a crucial question which philosophy should we choose for various applications - I would like to discuss.

- uses approximation of (Jaynes)
- has no localization property (nearly uniform stationary probability distribution),
- has characteristic length of one step - this way e.g. depends on chosen discretization of a continuous system.

- is the one maximizing mean entropy, "most random among random walks",
- has strong localization property - stationary probability distribution exactly as quantum ground state,
- is limit of characteristic step to infinity - is discretization independent.

Simulator of both for electron conductance:
Diagram with example of evolution and stationary denstity, also some formulas (MERW uses dominant eigenvalue):