
The Ising model: a cartoon picture of magnets that became ubiquitous in science - pseudolus
https://www.quantamagazine.org/the-cartoon-picture-of-magnets-that-has-transformed-science-20200624/
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julian37
That's very interesting. I found this simulator, try setting "field" to a
value just a little bit off zero:
[https://mattbierbaum.github.io/ising.js/](https://mattbierbaum.github.io/ising.js/)

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abetusk
For others wanting a very "computer science friendly" introduction to the
Ising model, Rice piles and other critical and complex systems (in the physics
sense), I would recommend "Complexity and Criticality" by Christensen and
Moloney.

[1] [https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-
Colle...](https://www.amazon.com/COMPLEXITY-CRITICALITY-Imperial-College-
Advanced/dp/1860945171)

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andbberger
What's not touched on in this article and arguably explains the ubiquity of
the Ising model: it's the maximum entropy model given a mean and pairwise
correlations

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kgwgk
That doesn’t make much sense to me. The model is a model (coupled spins in a
heat bath). The “solution” (the equilibrium states) can be found using maximum
entropy considerations but that’s not unusual. What makes the Ising model
popular is that’s simple and interesting.

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andbberger
Information entropy, not statistical mechanical entropy [1], which is a sort
of Occam's razor for models.

I would not describe the Ising model as simple, not even in the 2d case and
especially not the general spin glass (where W_ij can be anything).

[1]
[https://en.wikipedia.org/wiki/Principle_of_maximum_entropy](https://en.wikipedia.org/wiki/Principle_of_maximum_entropy)

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kgwgk
That link has a couple of sentences about “maximum entropy models” and no
references. From a quick search it seems that this is a terminology used in
some areas thought. It’s not yet completely clear to me what does it mean.
(Seems to be about the Ising model which is selected from a large set of
‘similar” Ising models; I would think of “Ising model” as the general thing.)

Statistical mechanical entropy is information entropy (and related to
thermodynamical entropy). From the link: “The principle was first expounded by
E. T. Jaynes in two papers in 1957[1][2] where he emphasized a natural
correspondence between statistical mechanics and information theory. In
particular, Jaynes offered a new and very general rationale why the Gibbsian
method of statistical mechanics works. He argued that the entropy of
statistical mechanics and the information entropy of information theory are
basically the same thing. Consequently, statistical mechanics should be seen
just as a particular application of a general tool of logical inference and
information theory.”

Simplicity is a relative thing, I guess. Let’s say the
interestingness/complexity ratio is high.

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andbberger
Statistical mechanical (Boltzmann) entropy and informational (Shannon) entropy
are not the same thing, but they are closely related. Subtleties are
important.

The Shannon entropy of the distribution of the positions and velocities is
precisely the Boltzmann entropy of the same, but there is no statistical
mechanical analog to the Shannon entropy of a (non-Boltzmann) distribution, to
my knowledge.

~~~
kgwgk
> Statistical mechanical (Boltzmann) entropy

While Boltzmann entropy was first, and an important breakthrough, it has some
limitations. Statistical mechanics is actually based on Gibbs/von Neumann
entropy, which is equivalent to Shannon entropy. There are many subtleties
indeed but I think the relationship between statistical mechanics and
information theory may be closer than you think.

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photonemitter
Something I found interesting when running analysis on Ising models; if you do
an FFT (2 dimensional) on it, you get what amounts to a wavey star-like blob
in the middle (assuming you center the lowest frequencies)

If that blob is a small dot, or even nothing at all; you have a stable system,
and if you have a sprawling kind of wavey response-pattern, then your lattice
is gone critical.

(Also it looks cool, and it’s quick to diagnose by a glance.)

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dang
An APL implementation showed up in 2018 but didn't get attention:
[https://news.ycombinator.com/item?id=16597207](https://news.ycombinator.com/item?id=16597207)

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noobermin
I mostly scanned the article as it's getting late but it doesn't look like
they discussed the funniest part of the story. Ernst Ising himself after grad
school essentially become an obscure teacher and not a researcher, and I
believe only until decades later did he become aware that his model was now
the quintessential simple model for classical spins. It's funny especially
given his name was attached to the model.

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labster
Looked up Ising’s story on Wikipedia. It doesn’t seem funny to me that he was
unaware of the progress in the literature after being forced out of his job,
then his country, then finally doing forced labor for the Wehrmacht.

[https://en.m.wikipedia.org/wiki/Ernst_Ising](https://en.m.wikipedia.org/wiki/Ernst_Ising)

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ganzuul
This model is very useful for understanding magnetic refrigeration, which in
turn gives deep insight into how entropy works.

