The hardest sum aka the Ising model

[@Number_Cruncher]

The energy decreases after contact with the magnet, because ...

[@nice3294]

My guess is the potential energy of two aligned magnets is lower than unaligned magnets(since you can gain energy from aligning them), so aligning the spins should give a lower potential energy

[@insouciantFox]

@@nice3294 This is because symmetric wavefunctions have lower energies than anti-symmetric ones.

[@howdy832]

Not really magnetic reasoning, but you need energy to form cracks in a material. Aligning the poles in the needle is undoing the magnetic "cracks" inbetween domains, so it should be lower energy.

[@Icenri]

It loses entropy so...

[@kodirovsshik]

​@@insouciantFox🤓☝️

[@ARBB1]

Dear Lord man, this is a colossal amount of effort, and it looks perfect too! Congratulations on producing this.

[@Number_Cruncher]

Thank you. Yes it was a lot of work but a lot of fun too.

[@peterbenoit5886]

Very clear and the visuals are first rate. This should make the Ising model accessible to a much broader audience.

[@elishalangat3791]

This is excellent presentation, I have watched a couple of videos on Ising model but this is the best so far, Thanks for your immense effort and animation, Super super thanks!

[@5MinutesBlender]

Thank you so much for the mention 💖 A nice presentation for the people who love physics 👍 All the best 👍

[@lexinwonderland5741]

WOW! This was increibly well done!! Animations, content, delivery, all A-tier stuff; I'll have to rewatch a time or two to make sure I understand it all to proper depth, but just one pass through made me feel like I learned the deep inner workings of something almost fantastical. Great job, dude!!

[@Number_Cruncher]

Thank you for such a detailed and flattering response. It's a pleasant reimbursement for my efforts during the production.

[@Priya_dancelover]

loved it , clearly explained . Waiting for your next video

[@drozfarnyline4940]

Wow that was pretty fast upload I am here after the announcement of SoME3 Last time I enjoyed Complex Number Approach in Geometry from your channel This video seems interesting too btw

[@IshanBanerjee]

Amazing work sir ❤

[@felipea.barretto7503]

Loved the way you displayed the Mathematica code!

[@MATHsegnale]

Great video, and also a very good explaination of quite a complicated topic!

[@Number_Cruncher]

Thank you for your pleasant feedback.

[@Number_Cruncher]

The combinatorical solution sounds more interesting, because ...

[@krishnadasnair8765]

jeez what an explanation, well done!

[@alvargd6771]

woah, this channel bmaking cool math videos again

[@Number_Cruncher]

The solution in the book by Kerson Huang sounds more interesting, because ...

[@shubhamjat6926]

I'm downloading this video thank you ❤

[@johnchristian5027]

what a great video!

[@Number_Cruncher]

The energy increases after contact with the magnet, because ...

[@evgeniibarannik8996]

While watching for the first time, I believe that the internal energy of the needle has increased after magnetization. This is because I know that the magnetized state is temporary and will be relaxed to the normal soon. Usually such transitions lower the energy of the system, so I believe that the excited state has more energy.

[@ZestyMcLime]

The increased alignment of spins in the needle increases it's own demagnetizing field, which increases the energy of the system. This will tend to cause a reduction in the needle's magnetization over time (through domain wall motion, in the extreme that the needle was uniformly magnetized, through the formation of domains) because there's no longer the presence of the permanent magnet. Although the exchange energy does decrease from the alignment of neighboring spins, the energy associated with magnetostatic effects is of much larger magnitude on these scales. The new configuration might be quite stable (need a large amount of thermal energy to escape) but the total energy of a less magnetized state will be lower.

[@Elitekross]

It has to gain energy because it gains the ability to impart a magnetic force and the energy for that force must come from somewhere. Also, magnets will lose magnetism over time as the exert that force

[@ajs1998]

@NumberCruncher Awesome video! I have a question about the more/less energy question (disclaimer I haven't watched the full video yet, so apologizies if you answer it later in the video) Wouldn't the atomic structure of the magnet determine whether it loses or gains energy? It feels similar to making a sheet of bar magnets. A single line of NS-NS-NS magnets is stable, but when you try to make it a grid, the next row of magnets needs to be SN-SN-SN right? Or else be offset by 1/2 length of each bar magnet into a hexagonal structure. In either case, the stable configuration of the sheet of magnets is not magnetic overall. I imagine things are a little more complicated with 3D structures. Are we sure there aren't some types of metal crystal structures that are less stable when magnitized?

[@ajs1998]

I just realized I'm wrong about the hexagonal structure. It would be both stable and magnetic overall. But for the alternating rows of SN-SN-SN and NS-NS-NS magnets, are you saying every other row will snap to the hexagonal structure because that's the more stable configuration? it's hard to visualize how this works at such small scales

[@Number_Cruncher]

Thank you for your interesting question. I can only give an answer based on my intuition. Your analogy is interesting but in my opinion it is missing an important incredient. At the atomic level, the magnetic interaction is weaker than the electrostatic interaction caused by the charged nuclei and electrons. The magnetic interaction is of dipole nature and falls off more quickly than the interaction between electric charges. In your analogy with macroscopic magnets, the magnetic interaction is the only significant interaction. I don't think that a change in the magnetic configuration can alter the lattice structure. Otherwise the paper clips would at least deform at the Curie temperature. Finally, the configuration with alternating spins was purely hypothetical. In nature it would not be realized. The spins are either mostly aligned (below the Curie temperature) or fluctuating randomly (above the Curie temperature).

[@Number_Cruncher]

Errata: At 24:03 the energy values at the axis are too small by a factor of two. They should read 36kJ, 72kJ,...

[@phyarth8082]

Nice info about heat capacity cp. 👍 That happens when liquid bonds separated and liquid becomes water vapor with latent heat capacity constant r, MJ/kg and density of water goes from 980 kg/m^3 goes into water gas with 0.54kg/m^3. To calculate heat transfer also requires fancy sum additional with recursive iteration. Biggest problem material phase transition is discontinuous process, in graph curve has abrupt breaking point at 100oC point. Transition from liquid to vapor and back again is not applicable for math which deals with integrals which are continuous processes with smooth graph curves. fancy sum hardest sum because no other way but brutal force calculation.

[@NicholasWong-vv1nn]

Can you please do one for the simpler 1D Ising model with the transfer matrix method?

[@Number_Cruncher]

I'll keep it in the back of my head. But the 1D Ising model is less entertaining, since it doesn't possess phase transitions.

[@InquilineKea]

How do you label interactions JUST as nearest-neighbor interactions in the sum?

[@Number_Cruncher]

Assume that you have row indices r and s running from 0 to M-1 and column indices i,j running from 0 to N-1, than you can write underneath the sum sign: |(r-s)%M|+|(i-j)%N|=1. This way you are guaranteed that only the row index or the column index (but not both of them) differ by one.

[@YitzharVered]

Just a few months too late to help for my statistical mechanics course. Let's see if it'd have done me any good!

[@MrParry1976]

But why does the Ising model, with spins and such properties work for calculating specific heat? Is there an underlying reason?

[@Number_Cruncher]

Hey, this is a great question. Afterall it shows, I think, that such a simple model is a great tool to approximate interacting collections of particles or objects in general. In fact, the Ising model is not only used to model atoms or molecules. You can find it at various places, eg. biologists use it to model neurons. This universality is the reason, why the Ising model has become such a popular part of physics and mathematics. There are also a lot of ways to generalize this model, eg. change interactions, add dimensions, vary boundary conditions. Also, thermodynamic properties such as specific heat are very generic. They can be defined on the level of partition functions, i.e. they are derivable from statistical mechanics and they are rather robust with respect to the particular parameters of the model. What I want to say, whenever you have a large number of objects, interacting with each other, there is a good change that these objects will thermalize and in such a case you can always define quantities such as temperature, entropy, specific heat, etc. Therefore, any reliable physical model should account for these quantities iin one way or another.

[@parl8150]

oh god, I dodged a bullet when decided not to write my thesis on this model

[@gabitheancient7664]

I don't see how the energy of the needle would change tbh

[@Number_Cruncher]

If you have two magnets and you bring them close together such that they combine, the system of the two magnets has less energy than before. You need to apply work (energy) to separate them again. Conversely, if you bring two magnets close together that you can feel their repulsive force, now the system has more energy. Because, if you let it go the two magnets will accelerate and move away from each other (or more realistically: one of them will be rotated and then they combine to a state of lower energy). In the same way, the energy of one needle strongly depends on the neighboring needles.

[@Dark-tk9xu]

You are god..

[@theeraphatsunthornwit6266]

get a better microphone with better magnet

[@Number_Cruncher]

Thank you for your feedback. In fact, I bought a brand new mic for this video. But probably, I screwed up the settings. I'm also not very happy with my audio and my voice. Next time, I'll try to get someone else to tell the story.

[@eqwerewrqwerqre]

I personally thought it was just fine. In fact, i never even thought about the audio, which i think I'd a great thing! Your careful diction matching with the very well crafted presentation is amazing. I probably spent an hour to get through it just rewinding and stuff and i had a great time with it. I'm basically teaching myself basic stat mech before grad school and this video is a godsend. Also, I'm definitely interested in walking through one of those very long solutions, i kinda like the matrix one because I'm more familiar with that math. Though i bet both of them are packed full of interesting and useful nuggets of math analysis. Thanks for making the video! Someday i hope to be able to teach as well as you

[@theeraphatsunthornwit6266]

@@Number_Cruncher forgot to tell you. Are u using pop filter? You should use it . Good microphone is 50 % of the quality. Also audio interface is important...i feel old already .... long time since i was involved with those thing.

[@Number_Cruncher]

Hey, I only found your answer accidentally today. I must have somehow missed it. Thank you very much. I was very pleased to hear about the rewinding and that you really can get something useful out of it. This refills my batteries for future projects. Good luck with grad school.