Oo Oo Oo I was looking for this. Transitioning from conducting to insulating – two distinct behaviors emerging as second-order effects that can’t be predicted in a linear fashion.
Basically, “how to boundary”. Not crazy about it being Brownian but hey, it’s workable and will tie to statistics well
“In fact, both systems display a second-order phase transition between “conducting” and “insulating” phases, controlled by the drift velocity in the Brownian system.”
Anytime I’ve seen these:
a) size matters
b) rate matters
But I’m always hoping for that magical “something” – not sure what.
Ok: that magic:
“The drift term is a control parameter, bringing about a
second-order phase transition as it changes sign. This transition
separates a regime in which diffusing particles (starting at
the origin) barely reach the distant boundary, from a regime
in which particles can reach the boundary”
It’s approximating random walks with diffusions, which allows them to go from brownian motion to to branching. (or stochastic to choice, which needs a root).
Clever. It’s called “Harris walk mapping” which I wasn’t aware of.
err, it stills strikes me as smearing over the problem of going from infinities to finites (continuous to discrete) and back again… ie – handwaving.Related to Schur multiplier. I hate being always on the cusp of ALMOST understanding this stuff but not quite.
So basically: Use all your smooth functions all you like but when you have some nice discretes to work with and some discrete time (change) steps, flip over to that but only at the boundary.
Then, you can flip back at the boundary back to infinites.
BUT, above the boundary, you got your continuous timeless Platonics to fuss with.
Below the boundary, you got your bits and pebbles and fractals and stuff operating in a generative (calculatable) fashion, all approximating their way to an eventual smooth boundary, SO LONG AS, you don’t try to smooth out ALL of the square pixels: Just ‘know that” they will map.
I guess this is logical. Once you have total covering or tiling or approximating or somothing, it doesn’t matter how you got there if you don’t need to work in that zoom realm anymore.Also explains how it goes from conductor (particles operating) to insulator (covering, soothing, barrier, partition).Thing with me is, there’s no perfect insulator.BUT… I guess there’s such a thing as “close enough”.
Found the Schur – Hadamard_transform connection in a nice book on AI I found while searching forSchur multiplier bullshit.Gotta love Google. In AI it’s used for cost calculating, which need _efficiency_. What’s efficient? Creating ELEMENTS out of parts (wholes, integers, discrete things) and glossing over precision.It maps but has ongoing flaws but it maps.
Back propagation in AI is similar to (actually, it *is* a form of) percolating that’s sort of controlled. It’s back percolating. Understand one and you understand both.
There’s some middle ground between a continuous power law and the discrete power law,” Newberry said. “In the discrete power law, everything is laid out in perfectly rigid proportions from the highest scale to the infinitesimally small. In the continuous power law, everything is perfectly randomly laid out. Almost everything self-similar in reality is a mix of these two.”
Thanks for being an audience for my grumping. Boundaries always get my ire because:
a) They’re ALWAYS breakable
b) They’re never smooth
c) We’re taught all this marvelous Platonic ideals just to watch them shred, as all things have a birth, life, death and a genealogy.
But, it’s too complicated to work with individual parts and even those parts always have parts. So we gotta pretend we have perfect gases in enclosed thermodynamic universes.
That’s fine but THEN folks go around thinking “noise” ISN’T constructed of lots of patterns or the flip, get caught up in the woo of the construction of noise, as I always do.
I have to accept their joint reality and pragmatic value of each.