Talk:Ising model/Archive 1

Archive 1

Mathematical or physical model?

The introduction starts off by proposing a purely mathematical model: there's no mention of anything physical (such as particles or their spins.) Then it suddenly switches to talking about real particles and their spins, even mentioning that:

It's also possible to have an external magnetic field.

which makes zero sense in the context of the introduction. It needs to be consistant, either describing a physical system of particles with spin and magnetic moment, or an entirely abstract system where fermionic particles are later discussed as a special case. Mixing the two is confusing. --Starwed 02:04, 7 March 2006 (UTC)

Yes, the article needs a total re-write. However, "to have an external magnetic field" is a mathematical statement: you just add a term to the hamiltonian that corresponds to spins interacting with a magnetic field; its still a model. Let me put it this way: I have no clue what the difference is between a "mathematical" and "physical" model is. Aren't they the same thing? linas 14:29, 7 March 2006 (UTC)

I hope the difference is obvious when you look at how it's phrased:

It can be represented on a graph where its configuration space is the set of all possible assignments of +1 or -1 to each vertex of the graph. The graph can exhibit periodic boundary conditions or free space boundary conditions depending on the system being modelled.

This uses mathematical language to describe the system.

To complete the model, a function, E(e) must be defined, giving the difference between the energy of the "bond" associated with the edge when the spins on both ends of the bond are opposite and the energy when they are aligned. It's also possible include an external magnetic field.

Suddenly we're talking about energy and magnetic fields. This isn't mathematical language, it's how a physicist would describe the situation. Of course the whole point is that the mathematical model describes a physical situation, so in that sense there is "no difference." But the article shouldn't mix up the language used to describe it like it does, anymore than it should switch to from english to german half way through. --Starwed 20:01, 12 March 2006 (UTC)

Atomistic Solvents

Does anyone have any idea what this means:

In fact due to the computational efficiency the Ising model stands to render atomistic solvents obsolete by early 2007.

If it means something I think it needs some explaination and a reference, otherwise it should be removed! Also a reference is needed for the proof of intractibility of the 3d model.Notjim 10:35, 12 October 2006 (UTC)

By early 2007??? I think what it is trying to say is that using a simple lattice representation one can accurately simulate atomistic behaviour given a fine enough lattice thus eliminating the need for molecular dynamics (I could be wrong here). If you ask me it's a big load of @#%$.

Finite-size scaling

This article should contain more information about finite-size scaling. For a quick primer, see these lecture notes. 82.139.85.119 17:23, 23 October 2007 (UTC)

On a related note, the article should contain the critical exponents. 82.139.85.119 08:02, 24 October 2007 (UTC)

Off-criticality Green function

I don't understand the statement about the off-critical Green function. The prefactor should be of order $r^{-(d-1)/2}$, not $r^{2-d}$ as indicated. This is the well-known Ornstein-Zernike behaviour, which has been established rigorously (see Campanino, Ioffe, Velenik: Ornstein-Zernike theory for finite range Ising models above $T_{c}$, Probab. Theory Related Fields 125 (2003), no. 3, 305--349) for finite-range, d-dimensional Ising model above the critical temperature. The only exception to this behaviour is when d=2 and T is below critical; in that case the prefactor is of order $r^{-2}$, for well-understood reasons, as can be obtained by explicit computations (see, e.g., the book by McCoy and Wu). —Preceding unsigned comment added by 129.194.8.73 (talk) 13:55, 9 November 2007 (UTC)

The thing I put down there is the free field propagator, and this is the right scaling for the free field. I don't know what this other scaling is. The two-point free field function in 2d is logarithmic. Could you explain? I'll look up the references.Likebox 19:03, 9 November 2007 (UTC)

The 2-point function of the Ising model (truncated 2-point function if in the phase coexistence regime, or if a magnetic field is present) has the behaviour I mention above (as have almost all simple fluids, actually) when they are away from criticality, and the distance between the 2 sites is large compared to the correlation length. I am not talking about approximation via field theory, just the plain 2-point function of the Ising model (I agree that the 2-point function of the free field has logarithmic divergence in 2d, but this is not relevant when discussing non-critical systems). A very general proof (but valid only in perturbative regimes - high temperature, low densities, etc.) is [Abraham, Kunz: Ornstein-Zernike theory of classical fluids at low density. Phys. Rev. Lett. 39 (1977), no. 16, 1011-1014]. You can also have a look at, e.g., the even older reference [Camp, Fisher: Behavior of Two-Point Correlation Functions at High Temperatures. Phys. Rev. Lett. 26, 73 - 77 (1971)]. Hope this helps. —Preceding unsigned comment added by 129.194.8.73 (talk) 08:27, 12 November 2007 (UTC)

You can also have a look at equ. (2.3.13) in [Fisher: The theory of equilibrium critical phenomena. Rep. Prog. Phys. 30, 615 (1967)], after which he states that this result does indeed go back to [Zernike. Proc. Acad. Sci. Amst., 18, 1520 (1916)]... —Preceding unsigned comment added by 129.194.8.73 (talk) 08:45, 12 November 2007 (UTC)

I guess that what you are describing is the asymptotic behaviour of the 2-point function in a massive limit, i.e., you look at two points separated by a distance large compared to the correlation length, then take the limit as T goes to T_c (notice that the correlation length then diverges). The result you state might indeed describe such a quantity. But this is really the 2-point function "asymptotically close to the critical point" (actually, to give a precise mathematical definition, you would probably have to take the limit). On the other hand, the behaviour of the 2-point function away from the critical point (i.e, you fix T>T_c (for example) and then look at the correlation between any two points x,y at a distance larger than the (finite) correlation length) is given by the Ornstein-Zernike asymptotics I stated above. This should be made much clearer than it is at the moment. One cannot expect the potential reader to know these things (otherwise, why would he been reading the article). The scope of the whole section (description of the continuum limit of the Ising model at and "around" the critical point) should be described precisely, and the corresponding results (such as this Ornstein-Zernike behaviour) for non-critical systems should also be stated somewhere for comparison. —Preceding unsigned comment added by 129.194.8.73 (talk) 10:01, 12 November 2007 (UTC)

You seem right to me, but I don't know anything about the off-critical behavior because I was only interested in the field theory limits. Please, if you have time, put in the discussion of the lattice-scale scaling behavior, because I don't know how.Likebox 19:13, 12 November 2007 (UTC)

Onsager's Solution

If Onsager found an analytic solution, wouldn't it make sense to state the solution in the article? The explanation of the 2-D solution stops making sense once the transfer matrix is defined. The link to transfer matrix seems to be irrelevant, and no explicit definition is given. I'm assuming that T_C1C2 is e^(-E/(kt)) where E is the sum of the interactions that arise in the slice between C₁ and C₂. But this doesn't agree with the expression in terms of the Pauli matrices given later. That latter formula, which is apparently an expression of T, doesn't make any sense. For example, let's call a basis vector odd if it has an odd number of spins pointing up and even otherwise. Then T sends odd basis vectors to even ones and vice versa. If N is odd, so does T^N, so trace(T^N) = 0. And then we get some random psi operators that aren't defined anywhere (on this page at least).--128.208.87.76 (talk) 21:22, 20 November 2007 (UTC)

I got lazy about some things. Sorry. I guess I should have diagonlized the matrix at the end, but I didn't. It makes sense as a sketch, but there's some perhaps bad notation and its incomplete--- you can exponentiate the transfer matrix or you can leave it in the unexponentiated form, for example. But the point you are bringing up is not a mistake. The expression in terms of Pauli matrices does agree with the slicing transfer matrix, and its published in old texbooks. I don't remember which ones, though.

The essential point of what Onsager did is what I wrote--- the transfer matrix is a sum of two terms, one which is the product of sigma-x at two neighboring sites, and the other a sigma-z. This form, if you express in terms of spin flip creation/annihilation operators, becomes a free field lattice fermion. I apologize for it being so sketchy, but I expected someone else to come in and clean it up and put down the integral for the specific heat.Likebox (talk) 22:43, 20 November 2007 (UTC)

Rigor?

Which of the results in this article are rigorous? As an passerby in the field, my impression was that some of the results (e.g. exponents and their universality) are not rigorously established. If that is indeed so, it has to said in the entry. If that is not so, then references to the proofs need to be added. Boris Bukh (talk) 03:25, 1 May 2008 (UTC)

Importance rating

I still feel that this is a mid-importance rating.

I mean I have never even heard of the Ising model before stumbling on this article, and I'm a masters student of physics. And looking at it, it seems to be a pretty damn specialized topic. Applications seems to be in the study of ferro-magnetic modelization, and that's about it. While I would tend to agree that the Ising model is a highly important topic within material science, I'm not so sure about its importance within physics as a whole.

However, since BCS theory is rated at High, and that it's possible that the Ising model is comparable in importance, I won't change the rating for now. The Top/High/Mid/Low assessment scale is pretty new and subject to change, so unless you think it's a bad idea, I'll rate this article in the same way I rate BCS theory. Headbomb {^{— Write so you cannot be misunderstood.}
_{— ταλκ / Wikiproject Physics: Projects of the Week} 11:42, 16 June 2008 (UTC)

The importance rating should remain high, because it is such an important toy model in statistical physics. You cannot compare this to the BCS model as far as applications are concerned, but the pedagogical value is huge here. The two dimensional Ising model is probably the only non-trivial statistical physics models most physics students will ever see the exact solution of. Also, the model has had a huge historical influence in mathematical physics. Count Iblis (talk) 15:49, 16 June 2008 (UTC)

Of course, I second that. This model is by far the most studied in Statistical Mechanics, and is known to virtually all physics graduates (at least, if they ever had a decent course in Statistical Physics). The number of publications on this model and its extensions, only during the period 1969-1997, is estimated to be over 12000! No way to qualify that as a marginal topic, right? Concerning applications: the main interest of the Ising model is not to describe some specific physical situation (although it does describe decently some aspects of the paramagnetic/ferromagnetic phase transition, the liquid/vapour phase transition, binary mixtures, etc.), but to be (one of) the simplest model with a phase transition that can be studied in many details, including the critical behaviour, and even at a mathematically rigorous level. As such, it serves as a testing ground for most of the new ideas in the field. And, as mentioned above, its historical importance is tremendous (its solution by Onsager triggered the whole modern theory of critical phenomena, and most of modern day statistical mechanics).--129.194.8.73 (talk) 07:25, 17 June 2008 (UTC)

Alright, high importance it is then. Tho I still find it weird that I haven't heard of it... But then again, my formation didn't really cover phase transitions, and paramagnetism/ferromagnetism were only treated at the intro level. Headbomb {^{— Write so you cannot be misunderstood.}
_{— ταλκ / Wikiproject Physics: Projects of the Week} 14:28, 17 June 2008 (UTC)

Historically speaking, the Ising model is, in my opinion, the single most important model in physics. I tried to explain the context in a history section. It was instrumental for establishing that statistical mechanics describes phase transitions, for constructing fermionic determinants, and for the equivalence of statistical and quantum field theory. If your department does not expose you to it, you should complain.Likebox (talk) 04:35, 18 June 2008 (UTC)

Yeah, well looking back, in my 2nd thermodynamics class we stopped at chapter 10 right before the Ising model. I think the teacher wanted to go deeper, but he had to spend a few weeks making us unlearn what the incompetent teacher taught us in the first class. Oh well, at least we're trying to get her fired so I guess that's that. Damned red tape. Headbomb {^{— Write so you cannot be misunderstood.}
_{— ταλκ / Wikiproject Physics: Projects of the Week} 06:13, 18 June 2008 (UTC)

Yeah, I'd vote for 'high' too. Its a door-opener for things like lattice QCD, and to go in a completely different direction, its a special case of a Markov network, which are hot in artificial intelligence, genomics and linguistics(!!) at the moment. (e.g. conditional random fields are used to model lexis in corpus linguistics. Just in case you were looking for a career change ... ) linas (talk) 03:25, 30 August 2008 (UTC)

Mixed Ising + Neural activity

On a different topic altogether. Rather than discussing the seemingly rather marginal "mixed Ising model", wouldn't it be better to cite the famous Edwards-Anderson model? In the latter, coupling constants are random, distributed according to an arbitrary fixed probability measure. For example, one can take the probability measure giving probability 1/2 to two opposite values $\pm J$, which would result in something apparently equivalent to what you describe as the mixed Ising model. But maybe I misunderstood your description: One possibility is that in the mixed Ising model the coupling constants are regularly distributed (say, in a translation invariant way). This would be different form the E-A model, in which each coupling constant is randomly chosen. If this is the case, could you maybe make it more apparent. Notice also that such models with competing interactions (with regularly placed ferro and antiferro couplings) have been used by many people in Stat. Phys., see, e.g., the well-known ANNNI model (which has its own page on wikipedia).

In any case, it seems to me that the E-A model is a much more classical piece of statistical mechanics than this "mixed Ising model", the original paper by Edwards and Anderson (Theory of spin glasses, J. Phys. F: Met. Phys. 5 965-974 (1975) doi: 10.1088/0305-4608/5/5/017) being cited approx. 2400 times! It is actually the standard short-range spin glass model.

So, my question would be: could you explain what makes the "mixed Ising model" worthy of inclusion here? I am ready to believe that it is relevant, but I fail to see that based on the text.--129.194.8.73 (talk) 10:22, 24 June 2008 (UTC)

Someone tacked that at the end, I just moved it to the beginning. I don't think it's at all relevant or interesting, and if you want to replace it with E-A go ahead.Likebox (talk) 11:33, 24 June 2008 (UTC)

Good. Removed it. I also think that the part on neural activity is somehow irrelevant, so I'll remove it too. We can't just list all applications/interpretations of the Ising model, as there are dozens of them (including in economy, history or geography!). Of course, it might make sense to add a section on various possible extensions (random coupling and/or random magnetic field, other spin space (-> Potts, XY, etc.), quantum version (XXZ, etc.)... but it seems to me that a whole section on neural activity is strange. Also, concerning application of Ising model to neural networks, a discussion of the Hopfield model would be more relevant...--129.194.8.73 (talk) 12:02, 24 June 2008 (UTC)

That's a judgement call. The Hopfield model applies Ising model to neural nets, but the application is not the main point of that section. The main point is to explain that the Ising model is the maximal entropy configuration given a collection of bits with given pairwise correlations. That's a fundamental point of view regarding the Ising model, which explains why it should be expected to show up in many contexts. It is related to the general reappraisal of the Boltzmann distribution in the 1960's as a maximum entropy distribution. This stuff seems pedagogically fundamental to me, and where I first heard about is in the recent papers by Bialek et. al. Maybe there's a better source, but I really think that its very important to say that the Ising model is a maximal entropy configuration, even if, like many fundamental insights, it is obvious in hindsight.Likebox (talk) 01:44, 25 June 2008 (UTC)

OK, I see your point. But then what I fail to see is why you focus on neural nets; it seems to me that what you just wrote conveys your meaning in a much clearer way. The particular application to neural nets does not seem very fundamental to me, and should at most be used as an illustration (in particular, it shouldn't appear in the section title). Regarding refs, I'd say that everything is already in Jaynes' famous 1957 paper. Of course, he does not discuss specifically the Ising model, but it is a completely trivial exercise to do so...--129.194.8.73 (talk) 07:24, 25 June 2008 (UTC)

I put in the citation to Jaynes, and while I agree that it is obvious in hindsight that Jaynes work leads to a reappraisal of the Ising model, somebody had to say it explicitly. The shift in perspective seems big to me. It changes the way in which you look for an Ising model. Before, you expect some microscopic dynamics corresponding to magnetism like in the lattice gas, now you just look for bits with constrained one and two point functions. That's a huge difference the way I see it--- it takes the Ising model out of physics and makes it a fundamental construct of pure mathematics, with applications to many fields. I hope that the current phrasing is to your liking.Likebox (talk) 01:04, 27 June 2008 (UTC)

I prefer this formulation, yes. I still believe that everything is already contained (really explicitly, but not specifically for the Ising model) in the huge litterature on MaxEnt, esp. in Jaynes' many papers. But I agree that it makes sense to explain here (as you do) why the Ising model is doomed to appear in so many different areas (in and out of Physics).--129.194.8.73 (talk) 07:25, 27 June 2008 (UTC)

Just, FYI, do be aware that the Ising model is a special case of a Markov network, and that Markov networks, plus variants such as conditional random fields are hot in artificial intelligence, genomics, and linguistics, where they, as well as Hopfield nets, as used to extract structure and data. So, for example, the article on belief propagation has a real crappy discussion of entropy, free energy, partition functions and the like that needs to be re-written ... bleeding edge are Markov logic networks, which try to attach thermodynamics to first-order logic in an attempt to solve problems in probabilistic logic. The Ising model is the progenitor of these ideas, and kind of a touch-stone you can get to. linas (talk) 03:41, 30 August 2008 (UTC)

Dozens of Applications

I was struck by your comment about the applications is economy, history, geography etc. I think it would be nice to add a section with a brief discussion of these applications, whatever they are.Likebox (talk) 01:55, 25 June 2008 (UTC)

They are mostly ridiculous in my opinion. Nearly as pathetic as the reversed case highlighted by Sokal: here it's physicists making abusive use of concepts from the social sciences. When I have time, I'll try to find out a few of them, so that you can make up your mind, but I feel that advertizing this kind of crackpot applications would be a bad idea. Of course, one might just list fields in which the Ising model has been applied, just to show how widespread its use has become, but I wouldn't like the article to cite explicitly any of these works (except if you can find a good one, of course!).--129.194.8.73 (talk) 07:32, 25 June 2008 (UTC)

Ok, here are a few. But you can find many, many others by googling for sociophysics, econophysics, etc. (Some models that have been introduced in the latter category are interesting, and might even have some relevance.) In any case, here's a short list (random refs I just found), which you can easily increase by looking at their bibliographies:

One with many refs, and a discussion of relevance: "Around the gap between the sociophysics and the sociology", http://arxiv.org/pdf/0711.2880.pdf

Some recent refs:

"Social applications of two-dimensional Ising models", American Journal of Physics 76, 470-473 (2008)

"Three-body interactions in sociophysics and their role in coalition forming", Physica A 379, 226-234 (2007)

(for a good laugh, just look at the end of the abstract)

"Ising model of society", http://www2.dse.unibo.it/wehia/paper/parallel session_3/session_3.4/yegorov_3.4.pdf

An older one:

"Systems analysis of socio-political conflict processes using spin models. I", Cybernetics and Systems Analysis 33, 51-57 (1997)

You can easily find many other ones, and even more easily convince yourselves that there's not much content there...--129.194.8.73 (talk) 08:25, 25 June 2008 (UTC)

Perhaps it is true that these papers are not the greatest contributions to their respective fields, but the interpretation of the Boltzmann distribution and the Ising model in particular as a maximal entropy distribution makes it reasonable that whenever you have links which you label "0" and "1" with a constraint on the total number of "0"s and a constraint on the pairwise correlations, but otherwise things are as random as possible, you naturally get the Ising model. So this means that an Ising model will describe any situation where there are random bits with only pairwise interactions. I still think that the first people who make this point explicitly about the Ising model are Bialek et al., and this is related stuff, so perhaps these other applications could be mentioned at the end of the "neural applications" section, retitled to "applications outside of physics".Likebox (talk) 00:24, 27 June 2008 (UTC)

Yes, such applications can be mentionned (if only to show how far the use of this model has gone), but I would refrain from citing any specific references, except possibly good review papers. The best might be to list general fields of applications, and refer to general wikipedia pages (I just checked: there is one on econophysics, but none yet on sociophysics, although links to the latter are already present in the former).--129.194.8.73 (talk) 07:28, 27 June 2008 (UTC)

I just posted above, before reading this, please see above. No doubt the application of the Ising model to sociology is due to a lack of understanding and maturity of ahem, "more visionary" authors (perhaps there's social status in pretending to understand the Ising model when you don't actually?). None-the-less, there is a very real application of Boltzman-distribution type statistical physics of graphs and networks of pair-wise interacting nodes. The Ising model has nodes on a square grid; but when the grid is instead some general graph, then, yeah, there are applications galore; I know for sure in AI, genomics, and linguistics; I guess economics doesn't surprise me. Anyone sitting bored in some humanities office, reading the latest linguistics research (where such topics are *hot*), may well get some itch to apply their poorly-grasped understanding of the so-called 'Ising model' to sociology, however inappropriate that may be. linas (talk) 03:56, 30 August 2008 (UTC)

Style cleanup too

Wikipedia:Manual of Style (mathematics) exists.

I suppose for people who specify variables in a preceeding formula by putting a period at the end of the formula and then writing "Where" with a capital "W" as if it's the beginning of a new sentence, it's a bit much to ask for standard spelling and punctuation, let alone following Wikipedia's standard style conventions. A huge amount of cleaning up after these people is still to be done.

5-3 (incorrect)

5 - 3 (incorrect)

5 − 3 (correct)

5−3 (incorrect)

Are such differences as those above invisible? Michael Hardy (talk) 20:18, 3 December 2008 (UTC)

Questions

What is the point of the "questions" part, these are not fundamental unsolved questions and therefore don't belong in a encyclopedia I think. I propose they are removed or placed in some relevant context. --Jaapkroe (talk) 17:30, 15 December 2008 (UTC)

Sorry--- that was me. I wrote them with the following point of view. The previous paragraph defines the Ising model by showing you how to write a program to simulate it. But then what do you output? The "questions" are just what you would be interested in printing out. I think "quantities of interest" might be more appropriate.Likebox (talk) 17:41, 15 December 2008 (UTC)

Needs a cleanup

In my opinion, the Ising model could be explained way better. As an addition to the semi-mathematical definition one could have a text-only explanation too. And there are way too many examples and applications. We don't bring up shopping as an application of addition now do we? —Preceding unsigned comment added by 213.64.27.237 (talk) 09:41, 9 October 2008 (UTC)

Yeah, it could be clearer--- it jumps straight to the mathematical definition. Maybe somebody knows how to put in a movie of a simulation to give a picture? As far as examples--- magnetism is necessary for history, lattice gas for history of universality (b.c. it was the mysterious equality of the corrections to the 3d fluid-vapor critical point and the 3d Curie point that historically motivated universality) and the general correlated bits/neurons/graph stuff is necessary to explain why its of interest in pure mathematics today. Those are the only three applications, they were chosen to be minimal. It's not the "kitchen sink". But what's your idea?Likebox (talk) 21:33, 9 October 2008 (UTC)

Also, the 2d exact solution stuff is sketchy, outdated and unfinished, there's a better treatment in Polyakov's book. I'll put it in at some point.Likebox (talk) 21:41, 9 October 2008 (UTC)

1-dim. case - Cite "If we designate the number of sign changes in a configuration as k, the difference in energy from the lowest energy state is 2k", seems wrong to me, only valid if the flips do not occur next to each other, e.g., state ++-++ has same energy as state ++--+ although it contains one flip more

The flips are sign changes. If you scan from left to right, the sign changes twice in both examples: once from + to -, then again from - to plus.Likebox (talk) 23:59, 19 February 2009 (UTC)

Finite-volume magnetization

From the text : As before, this only proves that the magnetization is zero at any finite volume. For an infinite system, fluctuations might not be able to push the system from a mostly-plus state to a mostly minus with any nonzero probability.

This is actually a bit misleading. The point is not that one has to consider the infinite-volume limit. Indeed, consider a 2D Ising model in big square box of sidelength L (with free, or periodic, boundary conditions, say) at subcritical temperatures. Then one can prove that a typical configuration will consist of a uniform "sea" of one species of spins with "excitations" that are at most of size $K\log L$, where K is some constant depending on the temperature, finite for all subcritical temperatures. Moreover, in any typical configuration (i.e. with a probability going to 1, as L goes to infinity), the average magnetization in the box (i.e. the density of +1 spins minus the density of -1 spins in this particular configuration) will be close to $\pm m^{*}(T)$, where $m^{*}(T)\neq 0$ is the spontaneous magnetization of the model.

Notice, however, that the average of the magnetization at a given site (the average being now taken over many configurations) will be 0 for all L by symmetry, so that taking the infinite volume limit does not help. Of course, if you now consider the evolution of the system under a stochastic dynamics, things would be different, but that's a different issue, which is not needed in order to discuss equilibrium properties of the system.--129.194.8.73 (talk) 11:28, 20 June 2008 (UTC)

The point of that argument was to show that in infinite volume the theorem that the magnetization is zero on average is wrong for a real physical system--- below the critical temperature there are two separated phases and a system stays in one phase and will be observed to be magnetized. While this is not exactly an equilibrium statement, as you say it requires some assumption on the dynamics being local, it does address the confusion people had. In terms of equilibrium properties, the magnetization is the limit of $\scriptstyle \langle S(x)S(y)\rangle$ for x,y well separated. In finite volume, it smoothly rises below T_c and is exponentially small above T_c, but it is always analytic in T. In the limit of infinite volume, $\scriptstyle \langle S(x)S(y)\rangle$ when x and y are infinitely far apart is zero for T greater than T_c and gets a fractional-power shoulder exactly at T_c, so it isn't analytic. Perhaps the section should be rewritten with the correct two-point function definition of the magnetization? Or maybe that makes it harder to follow.Likebox (talk) 04:38, 21 June 2008 (UTC)

Of course, non-analyticity only occurs in the infinite-volume limit. However, what makes the transition relevant for physics is that it has a clear manifestation in finite volumes: the spatially averaged magnetization in a box is, for a typical configuration, very close to $\pm m^{*}$, the thermodynamic (infinite-volume) magnetization. So, for all practical purposes, the transition is seen in finite volumes, not as the apparition of a non-analytic behavior, but as an abrupt change of behavior of the typical configurations (which in turns implies "smoothed singularities" for associated thermodynamical quantities). Probably, what I dislike in the current version of the text is the strong emphasis on the thermodynamic limit, and the fact that it is nowhere discussed that taking it is only necessary in order to have clear cut "zero/one" type results (in particular, an unambiguous, definition of the critical point and other thermodynamical quantities), but that the corresponding phenomena can be observed (of course, they are smoothed) in finite volumes. I feel that such a discussion is important, since real physical systems are finite (and undergo phase transitions). Maybe in a section on finite-size scaling?

Concerning the definition of the magnetization as the limit of (the square root of) 2-point functions as the distance between sites diverge, I think it would be too abstract for most people. I feel that the more natural definition of the magnetization as the derivative of the free energy w.r.t the magnetic field would be more appropriate (it is exactly what happens in thermodynamics). It is very easy to check (in finite systems, slightly bit more tricky in the thermodynamic limit) that the latter is given by the spatially averaged magnetization. It would then be possible to discuss the fact that, in the thermodynamic limit, the free energy is not differentiable at 0 magnetic field, but has well-defined right- and left-derivatives, that are precisely given by $\pm m^{*}$, making again connection with thermodynamics.

Finally, on a different note, here are a few things that might be considered worthwhile discussing in the article: 1) The (fully rigorous, and rather complete) description of phase separation in Ising systems with fixed magnetization (i.e., in the lattice gas interpretation, a fixed number of molecules): apparition of a (unique) macroscopic droplet of one of the equilibrium phases inside the other equilibrium phase, the shape of which becomes deterministic in large systems and is solution to a variational problem (minimization of the surface tension at fixed volume). 2) The (again, rigorous) existence of an essential singularity of the free energy at zero magnetic field, which implies that the standard, thermodynamical picture of metastable phases as analytic continuation of the equilibrium phases beyond the transition point is wrong for short-range systems.

I have unfortunately no time to do that, but I can give references if somebody finds it interesting. For the first point, there is a review paper by Bodineau et al, in J.Math.Phys. 41, 1033-1098 (2000). For the second, one can look at the original work by Isakov in Comm.Math.Phys. 95, 427-443 (1984), or in the recent extensions by Friedli and Pfister, described, e.g., in the lecture notes by Pfister in Ensaios Matematicos 9, 1-90 (2005).--84.73.61.129 (talk) 09:28, 21 June 2008 (UTC)

You are right that nowadays it is trivially obvious that there are no singularities in a finite volume, but nevertheless the finite volume remnant of a phase transition is still obvious from the gross change in behavior. But this point was historically very confusing. In the early twentieth century some people maintained that the partition function did not describe phase transitions because it could never get nonanalytic behavior, in any limit. It's dumb, but what can you do. They thought that.

This is the reason I put emphasis on that sort of thing, because I remember that it was historically emphasized in the old literature. The Yang Lee zeros gets at this, Yang and Lee were trying to determine exactly how nonanalyticity develops from a sum of exponentials in the thermodynamic limit. I might be wrong about how many people held this unfortunate opinion, but I think there were many, until Peierls.

While this does place a lot of emphasis on things that are nowadays pretty clear, the section is about the history. Please fix it if you have a better idea of what to write.Likebox (talk) 00:40, 22 June 2008 (UTC)

Of course, I agree that historically this was much debated. Actually, during a 1937 congress in Amsterdam commemorating the birth of van der Waals, the question of whether the formalism of Statistical Physics was sufficient to describe phase transitions was put to the vote by Kramers, who was chairman. The result of the vote turned out to be inconclusive, with a tiny majority for the affirmative answer. (Notice that this congress was held one year after publication of Peierls' argument, which shows that its relevance was not at all perceived by people at the time. It took Onsager's exact solution to convince people that the issue was settled!)

Nevertheless, my complaint is not that this should not be discussed, but rather that it should be put in perspective, at least in some other part of the article. In its present state, all the first part of the paper insists on the fact that the thermodynamic limit is necessary to describe phase transitions, and then, all the last part of the paper deals with the even more extreme case of the field theoretic approach to phase transition. There should be a discussion somewhere of the fact that phase transition have clear consequences on large but finite systems. The only reason to take the thermodynamic limit (other than getting rid of surface effects) is to have sharp definition and signatures of phase transitions. This is completely analogous to what happens in the simpler (and very much related) problem of the law of large numbers in Probability Theory: you need infinite sequences of random variables in order to have almost sure convergence (the strong law of large numbers), but in practice concentration around the mean is easily observed for sufficiently large, finite samples (and is actually the content of some versions of the weak law of large numbers and the Central Limit Theorem).--84.73.61.129 (talk) 17:22, 22 June 2008 (UTC)

I think one of your points is that the article needs to say that the convergence to the thermodynamic limit is fast, so that even a 10 by 10 Ising lattice might as well already have a phase transition. If there's something in there about the speed of convergence, maybe that will adress your concern.

But I think you have a second more serious complaint, with the general tone of the article, which emphasizes the continuum limit near criticality. This is not the whole Ising model, and there are many properties which are mathematically interesting outside of this regime, but for physical applications, any Ising model result which is going to be quantitatively exact must be in some continuum limit, because the model is so artificial. While it is not strictly necessary to write down continuous Lagrangians and Feynman diagrams to get the physics of the model right, the continuum description makes universality obvious and this is the link to real physical systems.Likebox (talk) 01:27, 23 June 2008 (UTC)

(There's too much indentation in this thread now, so I am reducing it. Sorry.) Yes, on both accounts. First, I do believe that a presentation of some results about finite systems would be very useful. Second, I do believe that making quantitative connections to real physical systems is not necessary! Such connections are not the main point of the Ising model. It is not meant to describe quantitatively anything. Its main interest (as far as I am concerned) is as a toy model, in which you can derive in full details (and often, with mathematical rigour) non-trivial properties that are expected to hold in more realistic situations, but are (possibly forever) out of reach of serious mathematical investigations. The two topics I proposed above are but two examples of that: it is expected that the shape of an equilibrium crystal (or droplet) of one phase inside another is given by the so-called Wulff construction, as the minimizer of the integrated surface tension among all sets of fixed volume. This is a classical statement in Thermodynamics, which can dealt with rigorously ony in a handful of models. The other point, about the existence of an essential sigularity of the free energy at h=0 preventing its analytic continuation beyond the phase transition point is another instance: it is expected to be true for all short-range models, but can be addressed only in simple classes of lattice models. There are dozens of other such examples, showing that the importance of the Ising model comes from its being a great (mathematical) laboratory for testing ideas and principles of statistical mechanics. Actually, just look at the trend in those many thousands of papers on this model: only a very small percentage of them deal with the critical behaviour. And that's understandable: universality of critical behaviour is a remarkable fact, certainly worthy of much study, but most of everyday systems are outside criticality, and many, many issues about such systems (even at equilibrium) are still unsettled. The main point of studying critical behaviour of systems is only that it's basically the only place where stat. mech. can make nontrivial predictions that can be quantitatively checked experimentally. But stat. mech. makes many more qualitative claims, that have much more important consequences for the description of physical systems around us. (My aim here is not to start a flame war. I am a professional mathematical physicist, having worked in this area for years, and I have also spent time on various problems related to critical behaviour. I just want to emphasize that there's plenty of other topics that are at least as relevant.)--129.194.8.73 (talk) 07:22, 23 June 2008 (UTC)

Yeah-- I see your point.Likebox (talk) 23:36, 23 June 2008 (UTC)

Just want to add something to this thread-- I'm not even a physicist, and I noticed that the section "A bogus... probability" is wrong (since the expected magnetization is still zero even for infinite volume, it's just that the probability distribution over magnetization is a sum of two delta-functions at plus or minus some constant). The symmetry argument that seems to be referred to as "bogus" would still hold for infinite volume. I recommend to just remove this whole section since it's not essential to the whole argument anyway. Danpovey (talk) 02:42, 28 January 2010 (UTC)

Please don't erase--- although you understand the issue, it was historically confusing. The two delta functions are separated in configuration space--- you can't move from one to the other by finitely many local moves. So you are stuck in one phase or the other. This means that the actual observed magnetization in a real system described by the Ising model is not going to average out to zero, even in a not-so-large finite volume. This is a real-world effect, although mathematically it is still true that the magnetization is zero when you sum over all configurations, that's the wrong thing to do in terms of physics.Likebox (talk) 03:15, 28 January 2010 (UTC)

Lattice QCD and Ising

I removed this subsection:

Ising Lattice QCD

It is not common knownledge, not even to specialists, that there is another relation to particle theory, namely to the lattice QCD: in fact, the S-variables can be interpreted as "quarks", and the J_i,k as "gluons", if both  quantities are allowed to fluctuate (which is in contrast to the preceding section). Additionally gluon-gluon couplings of the form $J_{i,k}\,J_{k,l}\,J_{l,m}\,J_{m,i}$, known as Wilson Loop quantities, must be added to the Hamiltonian.

Now one obtains gauge invariant models by the following simple set of local transformations, which depend on binary quantities $\epsilon _{i}=\pm 1$ and $\epsilon _{k}=\pm 1\,.$ The gauge transformations are

$s_{i}\to s_{i}\epsilon _{i}$ ,   $s_{k}\to s_{k}\epsilon _{k}$ ,   $J_{i,k}\to \epsilon _{i}J_{i,k}\epsilon _{k}$

The Hamiltonian energy, as all measurable quantities, is left unchanged by these transformationes, similarly as the Lagrangian of the quantum chromodynamics remains invariant against transformation with the group elements of the SU(3), which are here replaced by the ε variables. A relevant paper to these generalized Ising model originates from the german physicist Franz Wegner. 

reasons

"It is not common knowledge..." should be left out. There is a known equivalence between 3d Ising model and 3d Z2 gauge theory. This is not like QCD because the gauge group is smaller and discrete. In particular, it has a first order phase transition in 4d (I read this somewhere--- I never simulated it) and no continuum limit.

It might be good to add a section on the equivalence of lattice Ising and lattice Z2 gauge theory in 3d.Likebox (talk) 17:42, 1 February 2010 (UTC)

Perhaps this material can be linked in briefly on the article on discrete gauge theory? The addition of plaquette terms makes it different in fundamental ways (except in 3d) from the nearest neighbor Ising model.Likebox (talk) 17:51, 1 February 2010 (UTC)

This is essentially contained in Wegner's seminal paper. Besides: the 2nd-order phase transition to the Quark-Gluon plasma around T_c ~ 200 GeV/k_B (if it exists) seems to belong to the universality class of precisely the 3d-Ising model (the fourth coordinate, time, becomes noncritical). - Regards, 132.199.38.104 (talk) 11:08, 5 February 2010 (UTC)

It's not clear the quark gluon phase transition is Ising-like, even if it is three dimensional. The question is what the relevant order parameter is, and it is probably the pion field, so I would think it's like a O(N) model with either N=3 or N=1 (Ising) depending on whether you can neglect up and down masses. Other than that, I would retitle this section to "Z2 gauge theory", and actually show the equivalence to the three dimensional Ising model.Likebox (talk) 13:38, 5 February 2010 (UTC)

Istrail's result

This is an interesting result, but it must not be oversold. What it shows is that the free-energy of arbitrary sublattices of three and higher dimensional lattices are hard to compute. The proof is by making very complex chains of restricted spins which correspond to extremely high order correlation functions (if you imagine that the restriction to the sublattice is done by a chemical potential, to reconstruct the behavior of an arbitrary sublattice requires knowing essentially all the correlation functions and all resummations of these).

So there isn't going to be a simple formula which gives every single one of the high order correlation functions, that's true. But that's not what people mean when they say they "solved" a model. What they want is a good description of the model which allows some understanding of simple correlation functions. For example, in dimensions 5 or more, mean field theory essentially solves the model, but Istrail's result still applies.

In three dimensions, Polyakov's string description is very similar to Onsager's point description. The essential difference is that the properties of strings are harder than the properties of points. But Polyakov's representation reduces the computation required to find two-point functions, so it should be thought of as some sort of solution.Likebox (talk) 20:34, 23 November 2009 (UTC)

I agree with that. I think that the description of his work should not appear at the top of the article as is the case at the moment (fourth paragraph!). It might be mentioned later on in the article, if it even really needs to be mentioned at all: It does not seem to me that it says much about the homogeneous, ferromagnetic model that is the main focus of the article. Moreover, completeness would not be an argument either, since there are many other, some much more important, results about the Ising model that do not appear in the article.--YVelenik 15:58, 15 February 2010 (UTC) —Preceding unsigned comment added by Velenik (talk • contribs)

It used to be more prominent, I edited it down, and it reappeared. It's a significant result that at least one editor cares about, so what's the harm? I agree it's not the most significant result ever, but it does show a real difference between planar and nonplanar Ising graphs that suggests that every planar Ising model has a simple formulation.Likebox (talk) 19:11, 15 February 2010 (UTC)

Ok, first I don't think that the fact that one editor cares about a given result should be a sufficient to give it undue weight in the article. But in any case, the claim should then be correct. It is claimed in the article that

Istrail (2000) showed that computing the free energy of an arbitrary subgraph of an Ising model on a lattice of dimension three or more is computationally intractable.

I think that this is simply not true. He does not treat at all (as far as I remember) the case of homogeneous, ferromagnetic Ising model (with fixed coupling constant J throughout the system). Note that this is the model the article focuses upon, and moreover the model that is solved in dimension 2 (except for very partial generalizations, such as different coupling constant for vertical and horizontal edges, or some very specific geometries). If I am no mistaken then that means that the relevance of Istrail's result (at least for this article) should be played down, since it does not imply the fact that the free energy of the homogeneous 3d Ising model is computationally intractable. I am not saying that his result is uninteresting, but it certainly has been suggested (here and in many other places) that it contains much more than it actually does. I wonder whether the editor that cares so much about this result has actually read the paper...--YVelenik 13:50, 16 February 2010 (UTC) —Preceding unsigned comment added by Velenik (talk • contribs)

One is tempted to suspect that the editor who cares so much wrote the paper! I agree with all your comments, and I did read the paper, and I made sure that the caveats you mention are in there. If you want to bury the referemce deeper, go right ahead. I think you are wrong about the general significance of the result: while it does not imply anything about computing the free energy of the homogenous Ising model, it does suggest a real difference between 2d and 3d models.

Anyway, the current ugly administrative climate has put me off of this place for now. Hope you find some good compromise.Likebox (talk) 00:19, 17 February 2010 (UTC)

In my memory, Istrail does consider subgraphs of a 3d homogenous coupling lattice. To take subgraphs can be thought of as changing "J", but it can also be thought of as considering an arbitrarily varying chemical potential/magnetic field. This is why I chose the wording "computing the free energy of an arbitrary subgraph of a 3d Ising model", instead of "solving the 3d Ising model".Likebox (talk) 02:26, 17 February 2010 (UTC)

One-dimensional Ising model, math problems

I cannot make head nor tail of the following formula:

The partition function is the volume of configurations, each configuration weighted by its Boltzmann weight. Since each configuration is described by the sign-changes, the Partition function factorizes:

$Z=\sum _{\mathrm {configs} }e^{\sum _{k}S_{i}}=\prod _{k}(1+p)=(1+p)^{L}\,.$

What does $\sum _{k}S_{i}$ mean? How do the indices i and k relate to each other? What are the intermediate steps in this derivation? (E.g. I imagine beta should appear in the early stages of the equation, and I would expect to see some combinatorial stuff relating to the number of configurations with a particular k). I can't even begin to make sense of this. Danpovey (talk) 03:25, 28 January 2010 (UTC)

This is a mathematical statement of the previous sentence: the energy is the sum of an independent contribution from each spin-flip, so the partition function factorizes:

S_i is a variable which is 0 or 1 at site i, it is 0 if sigma_i = sigma_i+1, and it is one otherwise. In other words, S_i tells you if there is a spin flip between site i and site i+1. The sum on k is the sum over all positions in the one-dimensional lattice, the "i" in the interior is a typo--- it should be a k.

Note that if you make an assignment of the two values 1,0 to each S_i, you get a unique configuration of spins (assume that the leftmost spin, the spin at the left-edge of the line, is down--- if you don't assume this, there is a factor of 2 which is irrelevant). This means that summing over configurations is the same as summing over each S_i, plus or minus one, and since the energy is a sum over the S_i's, the probability of each S_i is completely independent of all the others.

The rfactorization of the partition function is a standard result in statistical physics: the partition function of a bunch of independent systems is the product of the partition functions for the systems. "Independent" means that the energy of the system described by variables "x" and "y" is the sum of two terms, F(x)+G(y), with no terms that depend on both x and y. Exponentiating the energy, you see that this means that the statistical probability of x and y is the product of a probability of x times the probability of y.

In this case, the total partition function is the prodcut of the partition function of x and y considered separately. There are no intermediate steps. Here is the statement for any two systems, whose configurations are indexed by "x" and "y":

$\sum _{x,y}e^{F(x)+G(y)}=(\sum _{x}e^{F(x)})(\sum _{y}e^{G(y})\,$

This is called Fubini's theorem in calculus, when the sum is an integral. For finite sums, it doesn't have a name, but its the same thing. You should be able to see now that it works for any number of systems.

So in this special case, the sum over all spin-flips is a product of the sum over each S_i separately. For any one S_i, the sum is of two terms:

1 (when S_i =0)

$e^{J}$ (when S_i = 1)

So that if you define p=e^{J}, the sum is (1+p)^L, where L is the number of sites. All this explanation belongs at partition function, not here.Likebox (talk) 04:58, 28 January 2010 (UTC)

Thanks for taking the time to reply to this. So there seem to be various aspects of unclear notation here. In the section in question, up to this point at least $S_{i}$ means the actual spin at position i which is 1 or -1. In the equation it suddenly changes to mean 0, or 1 if there is a spin flip. I wonder if it would be possible to define a variable, say $D_{i}$ which is 0 if $S_{i}=S_{i}+1$ and 1 otherwise. Also in the text prior to the equation I mentioned, the variable k was used to mean the total number of spin flips so it might be good to change k in the equation to i in order to avoid any confusion on this point. The article does not seem to be clear whether, when we define the partition function Z, this is factoring out the base energy -L. I don't know what the convention is here. I will assume it excludes the -L. So the change in energy, given the definition of the energy at the top of the section, would be 2 if there is a change in spin and zero otherwise, so in place of the factor (1+p) we would actually have, as far as I can see it, $(1+e^{-2\mathrm {B} })$. I don't see how this equals $(1+p)$. Using the stated expression for $e^{-2\mathrm {B} }$, this should equal $1/(1-p)$, which while quite similar for small p, is not the same. This would also affect the next few lines. Danpovey (talk) 01:18, 1 February 2010 (UTC)

I thought I defined S_i as the spin-flip function, otherwise the whole thing would be incorrect. Perhaps it got changed later, or maybe I screwed up. Please fix anything that is unclear. The most standard way is to define S_i to be the product sigma_i sigma_i+1, so that it takes the value +/- 1 according to whether there is a spin flip or not.Likebox (talk) 02:11, 1 February 2010 (UTC)

No, it's not defined that way in the text just above, the energy is written as $-\sum _{i}S_{i}S_{i+1}$. Throughout the document, $S_{i}$ is used for the actual spin, so I think we would need new notation for the spin-change. Also the definition you suggested for $S_{i}$ would not I can't really fix this because I don't know what is the normal notation. Also, could you have a look at my other point about the expression $(1+p)$ not being right? If I were to edit it, it might look like this:

Since each configuration is described by the sign-changes, the Partition function factorizes. Defining $C_{i}$ as $S_{i}S_{i+1}$, which is $-1$ if there is a sign change and $1$ otherwise, and ignoring end effects,

$Z=\sum _{\mathrm {configs} }e^{\beta \sum _{i}S_{i}S_{i+1}}=\sum _{\mathrm {configs} }e^{\sum _{i}\beta C_{i}}=\prod _{k}(e^{\beta }+e^{-\beta })=(e^{\beta }+e^{-\beta })^{L}$

I don't know where to go from here because I don't know what is meant by "free energy density". I can see a few hits online for "free-energy density" but without explanation. Neither the Gibbs nor Helmholtz free energy seems relevant here, unless I am missing something. Danpovey (talk) 23:18, 1 February 2010 (UTC)

I'll fix it--- thanks for bringing it up. I don't know if there is a standard notation.

Free energy just means the log of Z (that's not exactly true, it's logZ times the temperature, but same difference). The free energy density is the free energy divided by the length, so that it is independent of length. It's the "Helmholtz free energy" in a more physical context. The best way is just to take this as a definition: define F/T = log Z. The point of this is that derivatives of Z give you expected values of quantities like forces, so derivatives of F at constant T give you thermodynamic forces. I can explain it if you don't find a good source. Like most of thermodynamics, this is explained in awful ways in the literature.Likebox (talk) 00:25, 2 February 2010 (UTC)

yep, the text is wrong. up to a factor of 2, Z is correct in the discussion above. check out goldenfeld's book on phase transitions and the renormalization group. the point is that spin flips are a better way to specify the configuration of the system. the p/(1-p) equation in the text is quite spurious. —Preceding unsigned comment added by 131.111.20.38 (talk) 08:48, 1 June 2010 (UTC)

Graphs

This article should contain graphs of M & E, showing the phase transition that occurs in the multi-dimentsional model. The M graph should clearly show how the "fork" like quality of the graph. 82.139.85.143 21:44, 14 October 2007 (UTC)

I agree, one of the interesting features of e.g. the Magnetization of a =>2D Ising model w/o external magnetic field is its symmetry breaking nature below Tc. There is a lots of nice mathematical solutions on this page, but the spontaneous symmetry breakdown of the model should be discussed qualitatively somewhere in the introduction.82.170.211.237 (talk) 11:55, 21 October 2011 (UTC)

Cleanup?

Hello Colleagues,

the article is very long and hard to read. I propose the following:

1. leave only nearest neighbour interaction (perhaps it is a good idea to move long-range interaction to a separate article Long-range Ising model)
2. Move most of the details pertaining to the explicit solution in 2D to Square-lattice Ising model
3. remove most of the proofs (they are unref-d, so it will be easy to do so). Again, if someone prefers to keep the proofs, they can be moved to Square-lattice Ising model, Ising model in 1D, et cet.
4. "Historical significance" has no references at all right now. I would make it twice shorter, add ref-s, and move it to the end of the article.
5. Expand "basic properties and history" so that it becomes a guide to the rest of the article.
6. MoS.

Please comment. Sasha (talk) 18:45, 23 November 2011 (UTC)

sounds good — Preceding unsigned comment added by 98.201.116.241 (talk) 06:15, 4 December 2012 (UTC)

Block spins

The section entitled "block spins" doesn't contain any connection to the term 'block spins'.It would be instructive to, for instance, have a definition of 'block spins' in the paragraph. 71.139.179.193 (talk) 20:41, 23 December 2012 (UTC)

"log" in Onsager's exact solution

Should it be $\ln$ or $\log _{2}$? — Mikhail Ryazanov (talk) 22:18, 19 January 2013 (UTC)

Nearest-Neighbor Definition, Problem with J_{ij} in One Dimension section

I don't understand the insistence, both in this article as well as in other articles defining well known nearest-neighbor models in statistical mechanics, for giving the most general requirements for a finite phase transition to exist based on an very general interaction energies/potential. By this I mean the section immediately under the heading "One dimension". The interaction energy J_{ij} is only defined for Ising when i,j are next to each other - see the article section titled "Definition" for a definition of the model. If someone wants to make J_{ij} be valid for two spins which are not nearest-neighbors, then please update the Definition section, but in practice anyone who is calling a model an Ising model almost always uses it to refer to the nearest-neighbor case, unless otherwise stated.

So, since J_{ij} is clearly only defined for nearest-neighbors, there is no need to confuse the reader by suddenly giving general theorems on the existence of finite temperature phase transitions for interaction energies which don't apply to Ising anyway. It is sufficient to only give the nearest-neighbor result and refer to another article in statistical mechanics on why its phase transition happens only at T=0. — Preceding unsigned comment added by 50.198.121.235 (talk) 01:21, 5 October 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Ising model/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

I've rated this article as: Top importance. Because this model is almost standard canon of physics. Even people who branch off in entirely different parts of physics (or indeed life) should know at least something about it. Stub class. No graphs; almost completely unreferenced; inaccessible; poses "Questions" without providing answers; no discussion of qualitative / intuititive properties of the model. Needs finite size scaling discussion and values for the critical exponents.

Last edited at 08:03, 24 October 2007 (UTC). Substituted at 19:09, 29 April 2016 (UTC)

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Periodic conditions

The Ising model is said to be solved exact for 1D using periodic boundary conditions, but the proof is not using periodic boundaries.

$H(\sigma )=-J\sum _{i=1,\ldots ,L-1}\sigma _{i}\sigma _{i+1}-h\sum _{i}\sigma _{i}$

If periodic condiction is used it should be

$H(\sigma )=-J(\sum _{i=1,\ldots ,L-1}\sigma _{i}\sigma _{i+1}+\sigma _{L}\sigma _{1})-h\sum _{i}\sigma _{i}$

However I beleive that the first is correct and periodic boundaries are not to be used. — Preceding unsigned comment added by 212.60.125.156 (talk) 14:23, 14 March 2018 (UTC)