Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Jarek Duda on 03/09/2020 06:35:57
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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: https://en.wikipedia.org/wiki/Maximal_entropy_random_walk
It brings a crucial question which philosophy should we choose for various applications - I would like to discuss.
GRW
- uses approximation of (Jaynes) https://en.wikipedia.org/wiki/Principle_of_maximum_entropy
- 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.
MERW
- 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: https://demonstrations.wolfram.com/ElectronConductanceModelsUsingMaximalEntropyRandomWalks/
Diagram with example of evolution and stationary denstity, also some formulas (MERW uses dominant eigenvalue):
(https://i.imgur.com/dcYlkvL.png)