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So as you've seen, the coat basically works, it's helped to order the randomly generated structure

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and we got a structure that was pretty much ordered and neighboring spins are almost parallel to each

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other.

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But there are also some problems.

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You see, this is really not a ferromagnetic yet, even though we are getting closer and closer.

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So options would be to increase the number of steps or decrease the size of the of the sample.

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So on a number of the moments that we consider, but still, there's always the problem that we may

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get stuck in some local energy minimum.

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And the problem really is that we do not accept any increase in the energy.

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This means if we are trapped inside some local energy minimum, there's really no way to get out of

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it.

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For example, if you imagine you go hiking and you see a valley and you want to go to the closest valley

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and you are only allowed to go down.

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So this means you can never get out of your valley to get to a next valley, which may be even lower

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in height.

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And this the same problem here, so we must allow the codes to sometimes increase the energy to help

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us get out of such a valley.

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And this is one update one means.

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And this, in a physical sense, corresponds to considering finite temperatures.

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So this means or when you think about it, the temperature is just some thermal fluctuations, just

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some random motion of particles or some random or reorientation of our magnetic moments.

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So this means it has a particular energy.

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This is corresponding to this Boltzmann constant KB.

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And very often you see these Boltzmann factors here, such exponential functions where you compare one

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energy with another energy, which will be the thermal energy.

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So what we do is we still accept every update in the configuration when the energy is decreased and

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we sometimes even accept and change in the magnetization when the energy is increased.

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So let me copy this code here.

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We want to keep it just for completeness.

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I will write it here and call this routine step exchange tea for temperature.

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And we will give it another parameter, which will be the CCB kbe temperature.

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OK, and now we must implement that.

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We accept the change even if the energy increases with a chance that is given by this factor here.

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So basically, we must change this code here.

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So we need another if statement.

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We need if.

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A randomly generated number, if not random, thought rent is smaller than this exponential exponential

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factor here.

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So NPR dot expect.

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Minus energy, new minus energy holds divided by K B temp.

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So if this is true, if this random randomly generated number between zero and one is smaller than this

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term.

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Then we will still accept the change so that we accept the change and update the energy as before so

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we can copy this line of code and if not, so this means if the energy change is positive and if the

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random number is larger than this factor, which is basically always the case.

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If the energy change here is very large, then we will decline.

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So then we will do what we did before, and I think that's really all we have to change here.

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We just included this additional if statement with the random number that's smaller than this exponential

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factor.

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If it's true, we still accept, and if not, then we decline as before so we cannot run this.

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Or we will later rerun the whole code, I would say, and we can now change this one here.

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We can just say, OK, this one was our initial code, initial version, zero temperature, and now

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we write.

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That we use this function here and we need to temperature.

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So we could, for example, write KBE temp and KBE temp would be, for example, 0.01.

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So I think it works best if this value is pretty small.

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I tested this a bit before, so let's see if it works this time.

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And then we can comment here.

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This would be update one.

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Consider finite temperatures.

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OK, so now I will restart and rerun everything, and then we will see what will be the result.

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First of all, I hope that we do not encounter any error, but here it is again already.

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So no error yet.

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There's one looks also good.

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Let's wait a bit longer.

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So you see, we didn't really change that much.

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We just changed this one, basically, and we changed this one.

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The rest is really the same.

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And now we see still the energy decreases, even though sometimes it is allowed to increase.

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But it happens not very often.

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So we don't really see it here.

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And that's the whole trick.

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It should encounter only a few times, but it should help us to not get stuck in any energy minimum.

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Let's see.

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Oh, I think this looks pretty good.

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If we would just continue to simulate this, then I think we would get to the correct result because

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there is no really no rural yet.

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So I would say I will just crank it up here to five million.

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And when it's finished, I'll get back to you and show you the result.

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So I just tried this a few times and I noticed sometimes it got a bit better, but never really perfect.

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And so what I did now is I increased the temperature from 0.01 to 0.1.

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And you see in the energy now, that's really sometimes the energy now increase this and this is really

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what we want.

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So you see this nice fluctuation that here you see exactly why we added this, because if we would not

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have allowed for the increase of the energy, then maybe the system would have been stuck in this state

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for quite a long time.

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But then here, due to the increase of the energy, it allows it to get rid of this local minimum and

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go deeper in energy.

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So the total configuration now looks like this, which looks kind of like a ferromagnetic magnet.

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But you also see, of course, the effect of the thermal fluctuation here.

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So you see that now the temperature is maybe a bit too high.

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And so now you don't get the perfect ferromagnetic.

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But I think this is really what I wanted to show you here, and that's really just a good result.

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I would say it's it clearly resembles a thorough magnet.

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Another trick that we could do and that we will use later on is that we can, of course, also dynamically

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change the temperature.

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So we could see, for example, that we do not specify here a fixed temperature, but we see that the

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temperature updates in every step.

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So we could say we start with 0.01 and then we go one minus

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no steps.

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Basically, the index idolised by a number of steps.

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Exactly.

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So that when I is equal to zero, then we have a temperature of 0.1.

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And when I is equal to the number of steps, so this is the last step, the temperature is then zero.

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So maybe we should go here then to zero point two, but I will check and then show you the results later

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on.

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So really, decreasing the temperature from 0.2 to zero linearly did the trick.

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So you see the energy and you see here at the beginning, the temperature is still quite high.

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So you often see this increase, which helps to get to low energies very briefly and then the temperature

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decreases and you see the fluctuations become weaker and weaker.

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And in the end, there is no fluctuation left.

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And our result is really a very almost perfect ferromagnetic and a total energy, as I said.

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So the expected result would be minus 800.

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And so now we are really, really close to this result.

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And it worked out pretty well.

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So just to come and everything is, of course, based on randomness.

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So if you do this simulation, it could happen that you get a different result and that maybe sometimes

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it doesn't work so well.

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But I think if you rerun the calculations several times and then look at the solution, which gives

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you the lowest energy, then this should really give you such a nice ferromagnetic.