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I am looking for some literature on the Ising model, but I'm having a hard time doing so. All the documentation I seem to find is way over my knowledge.

Can you direct me to some documentation on it that can be parsed by my puny undergrad brain? If the answer is negative, can you explain it right there, on the answer form?

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Ising ?? or Icing ?? –  ja72 Dec 1 '10 at 18:38
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It's correctly written, Ising. –  Francisco P. Dec 1 '10 at 18:47
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@jalexiou: Ising is the surname of a physicist. –  KennyTM Dec 1 '10 at 19:37
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Ising model is probably weirdest name of a famous model there is. It was first proposed by Lenz who assigned it as a problem to his student Ising. Ising then concluded that the model is uninteresting in one dimension and didn't even bother to investigate other cases. So he actually hadn't done anything useful but, nevertheless, became a famous figure in statistical physics and condensed matter physics forever :-) –  Marek Dec 1 '10 at 20:02
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This is good question actually. The sort of thing I liked to see on this site. :) –  Noldorin Dec 2 '10 at 0:54
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2 Answers 2

up vote 12 down vote accepted

The Ising model is a model, originally developed to describe ferromagnetism, but subsequently extended to more problems.

Basically, it is an interaction model for spins. Imagine you have a system which is a collection of $N$ spins. Each spin $S_i$ has two possible states $+1$ or $-1$. Here you can imagine already a possible extension to more states. You can also imagine a different interpretation to the spins: $-1$ is a box containing no gas particle, $+1$ is a box containing a gas particle.

But I'm getting ahead of myself. Let's stick to spins for now.

The next step is to define the energy of the system.

$$E= - \sum_i h_i S_i - \sum_{i \neq j} J_{ij} S_i S_j$$

The first term can be interpreted as the contribution to the energy of the interaction of a spin with a local magnetic field. If the magnetic field is the same for all spins, then $h_i=h$ for all $i$. You see that aligning with the field will means a lower energy than going against the field.

The second term represents interactions between spins within the system. If $J_{ij}>0$ spins which align will contribute negatively to the energy of the system, thus lowering the total energy. If $J_{ij}<0$ then anti-alignment will contribute.

What I still haven't specified is how the spins are structured. By choosing the coefficients $J_{ij}$ appropriately, I can introduce this structure. Suppose I want a 1-dimensional system, which is the one Ising originally solved. Then, you have a countable infinity of spins arranged along the real line with equal spacing. Ising imposed an interaction only allong neighboring spins. So spin $S_n$ can interact with spin $S_{n-1}$ and spin $S_{n+1}$. The energy formula I gave above becomes:

$$E= - J \sum_{n} S_n S_{n+1} \; ,$$

if all spins can interact equally strongly.

Now, to each configuration of the system corresponds a certain energy. In statistical mechanics, we know that the probability of a certain configuration is

$$P(\{S\}) \sim e^{-E(\{S\})/kT}$$

where $T$ is the temperature. Or, if we compute the partition sum

$$Z=\sum_{\{S\}} e^{-E(\{S\})/kT}$$

we can deduce the complete equilibrium thermodynamic properties of the system. In particular, we can see if there are phase transitions for certain values of the parameters. (There is none in the 1D case, which at the time, combined with the lack of attention his model was getting, made Ising abandon physics.)

Of course, there are many more generalizations of this model. You can also make dynamical versions of the model where you are not just interested in the equilibrium configurations.

Here's the wikipedia page for Ising models. Some references:

  • H.E. Stanley 'Introduction to Phase Transitions and Critical Phenomena' Clarendon Press Oxford

  • J.M. Yeomans 'Statistical Mechanics of Phase Transitions' Clarendon Press Oxford

  • R.J. Baxter 'Exactly Solved Models in Statistical Mechanics' Academic New York

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Accepted and upvoted. Thanks a bunch. If you can find that note I'd be grateful. –  Francisco P. Dec 1 '10 at 19:12
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I found some books I used back in the day when I had to do an undergrad project about the subject. I can't remember what book was most useful, but I'll just add the ones I think are relevant. Note, they are pretty old, so there might be some recent books with nicer approaches. –  Raskolnikov Dec 1 '10 at 19:27
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I would add Chandler's Introduction to Modern Statistical Mechanics. I don't think there is any literature directly for Ising model. Rather, it is introduced basically in every textbook on statistical physics because it is (along with ideal gases) one of the very few easily approachable (and even solvable) models. –  Marek Dec 1 '10 at 20:08
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There's a recent mathematical physics book by John Palmer called "Planar Ising Correlations" which takes a super-high-tech look at the Ising model. I can't recommend it to someone learning the subject for the first time, though. –  j.c. Dec 2 '10 at 16:40
    
Maybe something for Marek, since he's trying to understand the 'Onsager solution' better. –  Raskolnikov Dec 2 '10 at 23:07
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Probably the best book in the field of spin models is the classic book by Baxter "Exactly Solved Models in Statistical Mechanics", it is very well written and contains both the basics and some more advanced topics.

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Not only that, but you can get a PDF of the scans from Baxter's webpage. tpsrv.anu.edu.au/Members/baxter/book –  j.c. Dec 2 '10 at 16:43
    
thanks for that reference @j.c. I've been looking for an error free copy of Baxter for a while now ! –  user346 Dec 3 '10 at 7:27
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The link to the book has changed physics.anu.edu.au/theophys/baxter_book.php –  j.c. Jun 10 '13 at 18:01
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