4 ms·

Magnetism Simulations: Three Months in Monte Carlo

77 points by tempodox 5y ago

dr_zoidberg 5y ago
I think it's been HN-hugged to death, so cached link: https://web.archive.org/web/20210717103315/http://bit-player.org/2021/three-months-in-monte-carlo https://web.archive.org/web/20210717103315/http://bit-player... Edit: I glimpsed at it... it's long and full of details, plus the simulations run in the archived version!
BenoitEssiambre 5y ago
That was amazing.
mikewave 5y ago
Surprised to see an entire article about Ising model problems without any mention of the machines designed to solve them! This is exactly the kind of problems that the D-Wave quantum computers are designed to run, and in fact simulations of physical systems are one place where we've been able to demonstrate solid progress, e.g. https://science.sciencemag.org/content/361/6398/162 https://science.sciencemag.org/content/361/6398/162 and several other papers. I'd highly encourage anyone interested in the Ising Hamiltonian and its uses for optimization problems to check out what we're doing and sign up for D-Wave Leap at www.dwavesys.com - we have demos, an in-browser IDE, Jupyter notebooks, a comprehensive Python SDK with excellent docs, and we'll give you free QPU time to run your problems in realtime on a quantum annealer.
- anniegarbage 5y ago
  I recently had to read up on D-Wave for my job. Very cool stuff to an ex-computational-physicist.
- kgwgk 5y ago
  As far as I can see it's about the Ising model and doesn't mention spin glasses at all.
pops8 5y ago
So...if things break down at high heat how is the earth producing its magnetic field? Do we have a frozen iron core?
- _Microft 5y ago
  Check this: https://en.wikipedia.org/wiki/Dynamo_theory https://en.wikipedia.org/wiki/Dynamo_theory
- kadoban 5y ago
  Good question. The article is only talking about so-called permanent magnets. Hunks of usually metal that exhibit this property under the right conditions. Sibling comment seems like it already covers the rest.
sampo 5y ago
> How did the method get the name “Monte Carlo”? > The name, of course, is an allusion to the famous casino, a prodigious producer and consumer of randomness. Nicholas Metropolis claimed credit for coming up with the term. In a 1987 retrospective he wrote: ... Here is the 1987 retrospective by N. Metropolis (6 pages): https://library.lanl.gov/cgi-bin/getfile?00326866.pdf https://library.lanl.gov/cgi-bin/getfile?00326866.pdf
bjornsing 5y ago
The author seems fascinated with the moving patterns produced by the M-rule but doesn’t go into its similarity with Conway’s game of life, and the fact that it’s been proven Turing complete.
honie 5y ago
I have only skimmed through the article but spent a bit of time on the "simultaneous update" part because it caught my eyes. If I'm not mistaken the author made a mistake in assuming that it's okay not modifying the spins in the first pass and then doing the update in the second pass, which would cause the system to approach a strange equilibrium with that flips between two states. For example, if you set the simulation temperature to well below that of the critical temperature of the system, a robust algorithm should eventually cause all of the spins align (to take the same sign). Also, and if I'm not mistaken, the author may have misunderstood that these simulations show the evolution of a system over time -- I think they are meant to show the possible states that a system can be in under a set of conditions, trying to rationalise whether or not it's sensible that spins should or shouldn't is perhaps not quite the right approach. The robustness of these algorithms are usually tested by carefully collecting many samples, and at different temperatures, and use them to estimate known properties of the systems. If an algorithm fails to estimate those quantities reasonably, then there is a good chance that it's not correctly implemented. If you're interested in this topic, one paper that I can immediately remember and is easy to read is this: https://arxiv.org/pdf/cond-mat/9703179.pdf https://arxiv.org/pdf/cond-mat/9703179.pdf. The section on Wolff algorithm, in particular, should solve the "mystery" of the simultaneous update. Here is a demo I have played with a few years ago that has the Wolff algorithm correctly implemented: https://mattbierbaum.github.io/ising.js https://mattbierbaum.github.io/ising.js (make sure you change sweep skip to an odd number for simulations at lower temperatures).
- spekcular 5y ago
  I have not read the article you linked, but the book by the same authors (Barkema and Newman, Monte Carlo Methods in Statistical Physics) is fantastic. It's the best introduction to these methods for the mathematically-minded that I've seen, in the sense that it gives quasi-rigorous justifications for a lot of claims, but at the same time doesn't get bogged down in rigor. They mention in the book that simultaneous updates with a checkboard are in fact OK. One just has to make the checkboard out of large squares of spins instead of the single spins the author of this article uses, and occasionally move the squares around to prevent boundary effects.
evanb 5y ago
Even though the Ising model is a model of discrete variables, it is still possible to apply HMC to perform global updates, as we explain in https://arxiv.org/abs/1912.03278 https://arxiv.org/abs/1912.03278 However, without further improvement, HMC’s performance is similar to the ‘obvious’ algorithms and poor compared to the Swendsen-Wang or Wolff cluster algorithms or the worm algorithm of Prokof’ev and Svistunov near criticality.

77 points by tempodox 5y ago

I think it's been HN-hugged to death, so cached link: https://web.archive.org/web/20210717103315/http://bit-player.org/2021/three-months-in-monte-carlo https://web.archive.org/web/20210717103315/http://bit-player... Edit: I glimpsed at it... it's long and full of details, plus the simulations run in the archived version!

BenoitEssiambre 5y ago

That was amazing.

mikewave 5y ago

Surprised to see an entire article about Ising model problems without any mention of the machines designed to solve them! This is exactly the kind of problems that the D-Wave quantum computers are designed to run, and in fact simulations of physical systems are one place where we've been able to demonstrate solid progress, e.g. https://science.sciencemag.org/content/361/6398/162 https://science.sciencemag.org/content/361/6398/162 and several other papers. I'd highly encourage anyone interested in the Ising Hamiltonian and its uses for optimization problems to check out what we're doing and sign up for D-Wave Leap at www.dwavesys.com - we have demos, an in-browser IDE, Jupyter notebooks, a comprehensive Python SDK with excellent docs, and we'll give you free QPU time to run your problems in realtime on a quantum annealer.

anniegarbage 5y ago
I recently had to read up on D-Wave for my job. Very cool stuff to an ex-computational-physicist.
kgwgk 5y ago
As far as I can see it's about the Ising model and doesn't mention spin glasses at all.

pops8 5y ago

So...if things break down at high heat how is the earth producing its magnetic field? Do we have a frozen iron core?

_Microft 5y ago
Check this: https://en.wikipedia.org/wiki/Dynamo_theory https://en.wikipedia.org/wiki/Dynamo_theory
kadoban 5y ago
Good question. The article is only talking about so-called permanent magnets. Hunks of usually metal that exhibit this property under the right conditions. Sibling comment seems like it already covers the rest.

sampo 5y ago

> How did the method get the name “Monte Carlo”? > The name, of course, is an allusion to the famous casino, a prodigious producer and consumer of randomness. Nicholas Metropolis claimed credit for coming up with the term. In a 1987 retrospective he wrote: ... Here is the 1987 retrospective by N. Metropolis (6 pages): https://library.lanl.gov/cgi-bin/getfile?00326866.pdf https://library.lanl.gov/cgi-bin/getfile?00326866.pdf

bjornsing 5y ago

The author seems fascinated with the moving patterns produced by the M-rule but doesn’t go into its similarity with Conway’s game of life, and the fact that it’s been proven Turing complete.

honie 5y ago

I have only skimmed through the article but spent a bit of time on the "simultaneous update" part because it caught my eyes. If I'm not mistaken the author made a mistake in assuming that it's okay not modifying the spins in the first pass and then doing the update in the second pass, which would cause the system to approach a strange equilibrium with that flips between two states. For example, if you set the simulation temperature to well below that of the critical temperature of the system, a robust algorithm should eventually cause all of the spins align (to take the same sign). Also, and if I'm not mistaken, the author may have misunderstood that these simulations show the evolution of a system over time -- I think they are meant to show the possible states that a system can be in under a set of conditions, trying to rationalise whether or not it's sensible that spins should or shouldn't is perhaps not quite the right approach. The robustness of these algorithms are usually tested by carefully collecting many samples, and at different temperatures, and use them to estimate known properties of the systems. If an algorithm fails to estimate those quantities reasonably, then there is a good chance that it's not correctly implemented. If you're interested in this topic, one paper that I can immediately remember and is easy to read is this: https://arxiv.org/pdf/cond-mat/9703179.pdf https://arxiv.org/pdf/cond-mat/9703179.pdf. The section on Wolff algorithm, in particular, should solve the "mystery" of the simultaneous update. Here is a demo I have played with a few years ago that has the Wolff algorithm correctly implemented: https://mattbierbaum.github.io/ising.js https://mattbierbaum.github.io/ising.js (make sure you change sweep skip to an odd number for simulations at lower temperatures).

spekcular 5y ago
I have not read the article you linked, but the book by the same authors (Barkema and Newman, Monte Carlo Methods in Statistical Physics) is fantastic. It's the best introduction to these methods for the mathematically-minded that I've seen, in the sense that it gives quasi-rigorous justifications for a lot of claims, but at the same time doesn't get bogged down in rigor. They mention in the book that simultaneous updates with a checkboard are in fact OK. One just has to make the checkboard out of large squares of spins instead of the single spins the author of this article uses, and occasionally move the squares around to prevent boundary effects.

evanb 5y ago

Even though the Ising model is a model of discrete variables, it is still possible to apply HMC to perform global updates, as we explain in https://arxiv.org/abs/1912.03278 https://arxiv.org/abs/1912.03278 However, without further improvement, HMC’s performance is similar to the ‘obvious’ algorithms and poor compared to the Swendsen-Wang or Wolff cluster algorithms or the worm algorithm of Prokof’ev and Svistunov near criticality.