1 00:00:00,240 --> 00:00:07,450 So now that we have our function for calculating the energy, according to this formula here, we can 2 00:00:07,470 --> 00:00:13,860 calculate it really for every configuration and we can just call this function whenever we slightly 3 00:00:13,860 --> 00:00:18,720 change this configuration of the magnetic moments, and it will update the energy immediately. 4 00:00:19,350 --> 00:00:26,460 So we know if the energy decreases or increases, this means now we are ready to implement the actual 5 00:00:26,460 --> 00:00:27,880 Monte Carlo algorithm. 6 00:00:28,620 --> 00:00:31,170 And I was teasing this already to you. 7 00:00:31,680 --> 00:00:35,190 This is basically just a loop over several metropolis steps. 8 00:00:35,370 --> 00:00:41,520 So I think this lecture is really the most important one because here you really learn the core feature 9 00:00:41,550 --> 00:00:42,750 of the Monte Carlo. 10 00:00:43,800 --> 00:00:48,840 So I was reading this already briefly to you, but here it's in detail. 11 00:00:49,140 --> 00:00:54,090 So what happens is at first we will randomly select a magnetic moment. 12 00:00:54,960 --> 00:01:01,170 So right now we have a grid of 40 times 40 more moons, so 1600. 13 00:01:01,650 --> 00:01:07,890 So we will generate two random indices between zero and 40 and then select a single one of these. 14 00:01:08,190 --> 00:01:09,300 For example, this one. 15 00:01:10,380 --> 00:01:16,410 And then we go to the second step, which will be this particular moment is reoriented along just some 16 00:01:16,410 --> 00:01:17,430 random direction. 17 00:01:17,910 --> 00:01:24,270 So as an as in the beginning, we will once again generate a random and Delphi and the random angle 18 00:01:24,270 --> 00:01:24,810 Fita. 19 00:01:25,200 --> 00:01:31,080 And then we will just change the orientation of this moment completely and we will do this randomly. 20 00:01:32,490 --> 00:01:40,290 And what's very important is we must still store the old orientation for now, because now we will calculate 21 00:01:40,380 --> 00:01:44,880 the new energy and we will compare the energy with the old energy. 22 00:01:45,330 --> 00:01:51,330 So with the new moment and the old moment, and now we must use an if statement and if the energy is 23 00:01:51,330 --> 00:01:56,640 decreased by those random change, then the change of the magnetic moment is accepted. 24 00:01:56,880 --> 00:02:05,670 So we accept it and we can just delete the old magnetic moment and when the energy is increased. 25 00:02:06,060 --> 00:02:13,860 This is considered a bat metropolis step, so we will just ignore it and we will restore the old magnetic 26 00:02:13,860 --> 00:02:14,280 moment. 27 00:02:15,810 --> 00:02:20,700 OK, so we know from this algorithm that two things can happen. 28 00:02:21,270 --> 00:02:27,270 Either the energy is decreased, which is good, then the magnetization is updated or the magnetic moments 29 00:02:27,270 --> 00:02:33,030 are updated, or we try to change it and we see the energy is increasing. 30 00:02:33,030 --> 00:02:38,910 And then basically nothing happens because the old Sputnik moment is just restored. 31 00:02:40,200 --> 00:02:42,390 So let's go ahead and implement this. 32 00:02:43,350 --> 00:02:45,750 So we implement a single step. 33 00:02:46,500 --> 00:02:54,630 And since we will add later on more energy terms here, I will call this function here step exchange 34 00:02:56,550 --> 00:02:59,820 so that we know that we only consider the exchange interaction for now. 35 00:03:01,560 --> 00:03:08,190 So as an argument, we only need the magnetization and we begin with the first step here. 36 00:03:08,430 --> 00:03:10,260 A random magnetic moment is selected. 37 00:03:10,980 --> 00:03:13,680 That's of course, pretty similar to what we have done previously. 38 00:03:14,820 --> 00:03:23,210 First of all, we just generate two indices X and Y and P Dot Random Dot brand. 39 00:03:23,580 --> 00:03:29,400 And since it's an integer, we write around it and when we write length, then it will be an integer 40 00:03:29,400 --> 00:03:32,220 between zero and thirty nine in our case. 41 00:03:33,270 --> 00:03:37,950 Then we do the same thing for why, because it must be a different random number. 42 00:03:38,520 --> 00:03:45,340 And now we can write an energy old is equal to. 43 00:03:45,600 --> 00:03:51,420 And now we could, of course, just write energy exchange magnetization. 44 00:03:53,370 --> 00:03:56,220 So we could do this, as I said. 45 00:03:56,550 --> 00:04:00,060 But it turns out that this is not really the ideal thing to do. 46 00:04:00,180 --> 00:04:01,530 I will come back to this later. 47 00:04:01,530 --> 00:04:02,730 We will change this later on. 48 00:04:02,820 --> 00:04:05,280 It would work, but it's pretty slow to do it that way. 49 00:04:06,630 --> 00:04:08,820 So now we come to the second step. 50 00:04:09,330 --> 00:04:12,180 So this would be first and this would be second. 51 00:04:12,810 --> 00:04:15,450 And this is exactly as we did previously. 52 00:04:15,600 --> 00:04:17,160 So I can scroll up. 53 00:04:18,120 --> 00:04:21,180 We will load these random values here. 54 00:04:23,040 --> 00:04:31,260 So here in this case, we were generating an array of 40 times 40 random angles. 55 00:04:31,590 --> 00:04:33,330 And here we only need a single one. 56 00:04:33,780 --> 00:04:38,970 So basically, we can just say, delete all of this stuff and do it like this. 57 00:04:40,260 --> 00:04:44,640 So now we have just a single random value phi and a single random all your theta. 58 00:04:45,870 --> 00:04:50,640 And now we generate the magnetization. 59 00:04:52,140 --> 00:04:54,420 So this we can also do as we did before. 60 00:04:59,850 --> 00:05:04,800 And we can say magnetization. 61 00:05:05,940 --> 00:05:12,120 And now we must specify which of these moments we will change because we will not randomly recreate 62 00:05:12,120 --> 00:05:13,020 the whole era. 63 00:05:13,200 --> 00:05:21,120 We would just change a single one of them and we will address here the moment x y zero. 64 00:05:21,390 --> 00:05:28,410 So the moment with the coordinates x, y and zero and this will be the X, Y and Z coordinates, according 65 00:05:28,410 --> 00:05:36,480 to the X, Y and Z coordinates, according to our two angles random fi and random Fita that we have 66 00:05:36,480 --> 00:05:38,790 just randomly generated. 67 00:05:40,500 --> 00:05:48,030 So one more thing that is really important that I didn't wrote explicitly here, but you see we had 68 00:05:48,030 --> 00:05:53,730 to calculate our old energy and likewise we have to store our old magnetic moment. 69 00:05:54,990 --> 00:05:58,530 So I will write here, save mark. 70 00:06:00,570 --> 00:06:03,660 Is equal to and p dot array. 71 00:06:06,090 --> 00:06:08,640 And then the magnetization. 72 00:06:11,720 --> 00:06:14,060 X my zero. 73 00:06:15,260 --> 00:06:20,660 So basically, we take this one, we store it in a different variable save mark. 74 00:06:21,650 --> 00:06:29,000 And now you may wonder, why don't you just write the following Why don't you just write safe markers 75 00:06:29,000 --> 00:06:31,550 equal to magnetization? 76 00:06:32,270 --> 00:06:37,250 And to be fair to you, this is what I did in the beginning, but I noticed it doesn't work. 77 00:06:37,340 --> 00:06:39,950 And this is because of some pointer issue. 78 00:06:40,640 --> 00:06:49,430 So if I would ride it like this, then it would assign the same pointer to save mark as this one or 79 00:06:49,430 --> 00:06:50,470 the same idea. 80 00:06:50,480 --> 00:06:53,030 I saw the same ID in the storage. 81 00:06:53,750 --> 00:06:57,880 And what happens then when we change this value here? 82 00:06:57,890 --> 00:07:03,500 So basically, once we run this line of code, it also changes the safe mark variable. 83 00:07:03,920 --> 00:07:05,750 This is exactly what we don't want. 84 00:07:06,110 --> 00:07:08,780 We don't want to overwrite to save Mach variable. 85 00:07:08,780 --> 00:07:11,150 We want to have it as an independent variable. 86 00:07:11,960 --> 00:07:14,300 We want the right and pitot array of this. 87 00:07:14,300 --> 00:07:16,850 Then it saves it as an independent variable. 88 00:07:17,510 --> 00:07:24,260 So this was a bit surprising to me, to be fair, but it makes sense because this is how Python works 89 00:07:24,260 --> 00:07:27,140 with the IDs and the pointer, you could say. 90 00:07:27,890 --> 00:07:29,570 So let me right here. 91 00:07:29,570 --> 00:07:35,300 Just as a reminder problem with the IDs save macro. 92 00:07:35,870 --> 00:07:36,890 This would lead to. 93 00:07:38,700 --> 00:07:43,620 Save mark will be changed once. 94 00:07:46,480 --> 00:07:49,750 This change in the next line. 95 00:07:51,380 --> 00:07:53,120 OK, so that's just a comment. 96 00:07:53,990 --> 00:07:56,690 All right, so what we have now is we have the old energy. 97 00:07:56,750 --> 00:08:04,970 We have the old magnetic moment saved and we have generated a new magnetization, a new magnetic magnetic 98 00:08:04,970 --> 00:08:07,610 moment at at this particular point in space. 99 00:08:08,900 --> 00:08:10,820 Now we can calculate the new energy. 100 00:08:13,340 --> 00:08:14,330 So that's pretty easy. 101 00:08:14,660 --> 00:08:21,800 We just write energy new is equal to and then we can just copy this because we have updated Mac. 102 00:08:22,310 --> 00:08:24,350 So this will now be our new magnetization. 103 00:08:24,980 --> 00:08:27,770 So as I said, this would work, but it's pretty slow. 104 00:08:27,770 --> 00:08:30,050 So we will change this in a few minutes. 105 00:08:31,010 --> 00:08:37,309 But let's continue for now, because now comes the if statement, the fourth part where we must compare 106 00:08:37,309 --> 00:08:38,030 the energies. 107 00:08:38,909 --> 00:08:50,210 So as I said, if and then rewrite if the energy new is smaller than the energy old, then do the following. 108 00:08:51,620 --> 00:09:04,310 So we in this case accept the change and we update the energy and we can write basically energy 109 00:09:07,280 --> 00:09:07,730 change. 110 00:09:07,760 --> 00:09:10,490 Or we could say we will. 111 00:09:10,580 --> 00:09:13,580 We will come to this in a second because you are on to do something special. 112 00:09:14,000 --> 00:09:18,620 So I will write your energy just to remind us that we have to do something here still. 113 00:09:20,510 --> 00:09:23,450 So just comment this and. 114 00:09:25,470 --> 00:09:28,740 If this is not true, then we will use ours. 115 00:09:30,060 --> 00:09:37,740 This would mean we decline and restore hold moment. 116 00:09:42,870 --> 00:09:46,230 And we would then say magnetization. 117 00:09:47,990 --> 00:09:54,500 I have the position X-Y Zero is equal to our safe mark magnetization or the magnetic moment that we 118 00:09:54,500 --> 00:09:55,730 have saved previously. 119 00:09:57,410 --> 00:09:57,820 OK. 120 00:09:58,940 --> 00:10:06,890 And then we basically return the compact to magnetization. 121 00:10:06,890 --> 00:10:16,460 We could say, OK, so the thing is now in the next step, we will loop over this many, many times, 122 00:10:16,700 --> 00:10:18,500 maybe one million times. 123 00:10:19,310 --> 00:10:26,120 And you see, we calculate here the energy once we calculate it here another time and we do this one 124 00:10:26,120 --> 00:10:26,940 million times. 125 00:10:27,830 --> 00:10:34,220 And the thing is, it takes a lot of time to do this, especially if the array or if the number of moments 126 00:10:34,220 --> 00:10:35,810 that we consider is very large. 127 00:10:36,620 --> 00:10:44,570 And it doesn't really make sense to do this because in each of these steps, we only change one single 128 00:10:44,570 --> 00:10:45,470 magnetic moment. 129 00:10:45,620 --> 00:10:51,380 And if you look at it here, we just change one particular moment, and the only energy terms that are 130 00:10:51,380 --> 00:10:54,650 affected are the ones interacting with the four neighbors. 131 00:10:55,100 --> 00:11:00,290 So why should we calculate the energy of the whole remaining thing over and over again? 132 00:11:00,620 --> 00:11:04,580 This loses us so much time and slows down our coat so much. 133 00:11:05,330 --> 00:11:10,700 So at this point, I think it makes much more sense to not calculate the energy every time, but to 134 00:11:10,730 --> 00:11:13,400 calculate the change of the energy every time. 135 00:11:14,390 --> 00:11:21,500 So we will only store here the energy that corresponds to this particular moment that we will change 136 00:11:21,860 --> 00:11:24,020 and the interaction with the neighbors. 137 00:11:25,250 --> 00:11:27,050 So let me show you how this works. 138 00:11:27,620 --> 00:11:34,310 So we will store here energy old and now we will not store total energy, but we will only store the 139 00:11:34,310 --> 00:11:35,210 contribution. 140 00:11:36,500 --> 00:11:43,880 This is why I programmed this here in these two individual functions, so we can now say the old energy 141 00:11:43,880 --> 00:11:46,160 would be energy exchange contribution. 142 00:11:46,940 --> 00:11:52,100 And to be honest, or maybe this is a bit difficult now, but you see, we have here. 143 00:11:52,760 --> 00:11:53,180 Where is it? 144 00:11:53,180 --> 00:11:53,600 Sorry. 145 00:11:53,930 --> 00:11:56,960 Here it is, the sum over Paris. 146 00:11:57,290 --> 00:12:04,880 So, for example, one moment interacts with its right neighbor, but then in the sum, there is also 147 00:12:04,880 --> 00:12:08,630 determined occluded where the right neighbor interacts with the left neighbor again. 148 00:12:09,230 --> 00:12:12,200 So basically, all of these terms occur twice. 149 00:12:12,980 --> 00:12:19,670 So this is why when we change one particular moment, it will not just be the four terms with the neighbors, 150 00:12:20,030 --> 00:12:24,470 but also determine where the neighbors interact with the moment itself. 151 00:12:24,860 --> 00:12:27,740 And this will, of course, just give us a factor of two. 152 00:12:29,930 --> 00:12:36,710 So the old energy we should actually may be called as energy or will change, but let's let's leave 153 00:12:36,710 --> 00:12:40,670 it at that is energy exchange contribution times two. 154 00:12:41,450 --> 00:12:45,860 And here the same thing, of course, two times energy exchange. 155 00:12:46,400 --> 00:12:47,540 And that's it. 156 00:12:49,830 --> 00:12:55,650 OK, so now we compare the energies, this is all fine, and now the only thing that we have to do is 157 00:12:56,010 --> 00:13:02,100 we have to see how this the energy change, so we write energy. 158 00:13:04,140 --> 00:13:08,730 Change is equal to energy. 159 00:13:08,730 --> 00:13:11,520 New minus energy. 160 00:13:12,030 --> 00:13:12,420 Old. 161 00:13:15,100 --> 00:13:18,560 Because here we accept the change and we update the energy. 162 00:13:18,640 --> 00:13:19,750 So we have to write this. 163 00:13:20,260 --> 00:13:23,710 And here in this case, we have to say energy 164 00:13:26,200 --> 00:13:28,150 change is equal to zero. 165 00:13:28,690 --> 00:13:35,500 And then we must return not only the magnetization, but we must also return to change of the energy. 166 00:13:38,100 --> 00:13:45,300 So the thing is we could now even increase the speed of the code even further, because you see in every 167 00:13:45,300 --> 00:13:47,560 step we calculate here the old energy. 168 00:13:47,910 --> 00:13:49,810 And then we calculate the new energy. 169 00:13:50,340 --> 00:13:55,590 And then in the next step, we should already know what the new energy is, but we calculated once again. 170 00:13:56,250 --> 00:14:02,550 So we could save this by adding here another parameter, which is basically the current energy that 171 00:14:02,970 --> 00:14:04,380 is given here. 172 00:14:04,380 --> 00:14:09,690 But I think this would now be maybe a bit too much and a bit too much optimization. 173 00:14:10,020 --> 00:14:16,020 This would increase our speed of the code by a factor of two, which is, of course, nice, and we 174 00:14:16,020 --> 00:14:19,740 could do this later on to improve the code, but it's not really necessary here. 175 00:14:20,280 --> 00:14:26,910 What was really necessary or what was really, really useful, was using these contributions only because 176 00:14:26,910 --> 00:14:35,220 this really, really increases the speed of our code, because now the energy calculation will just 177 00:14:35,220 --> 00:14:39,030 contain for four terms here. 178 00:14:39,030 --> 00:14:46,350 And then when we calculate the whole thing, it would contain so, so many terms, it would contain 179 00:14:46,710 --> 00:14:50,520 four times the number of moments in our sample. 180 00:14:50,520 --> 00:14:56,250 So this would make our code especially slow if we would consider very, very large samples. 181 00:14:56,880 --> 00:15:02,250 So I think this change was really worth it for the other change is nice, but not really necessary. 182 00:15:03,270 --> 00:15:08,070 OK, so let's see if it worked because it was quite a lot of code here. 183 00:15:08,580 --> 00:15:12,720 I think it wasn't really that difficult because it's just what written what's written down here, but 184 00:15:12,720 --> 00:15:15,580 still could happen that I made some error. 185 00:15:15,600 --> 00:15:16,320 So let's see. 186 00:15:16,830 --> 00:15:23,730 We just right step exchange magnetization and it gives him that role, of course. 187 00:15:25,410 --> 00:15:26,550 So where's the error? 188 00:15:27,770 --> 00:15:29,400 Oh, yeah, of course here. 189 00:15:30,060 --> 00:15:33,750 So you see, the error messages are pretty good in Python. 190 00:15:34,380 --> 00:15:43,470 So it says the function energy exchange contribution that we call here misses two required positional 191 00:15:43,470 --> 00:15:45,870 arguments, which are called X and Y. 192 00:15:46,500 --> 00:15:48,890 And of course, this is because I have changed it. 193 00:15:48,900 --> 00:15:54,480 First, I wanted to calculate only the total energy, but now I only want to calculate the contribution. 194 00:15:54,660 --> 00:15:57,990 So we need the other two arguments here. 195 00:15:58,830 --> 00:16:03,090 So we just write X come away and here as well. 196 00:16:04,410 --> 00:16:07,980 So let's try again and we have another error. 197 00:16:11,820 --> 00:16:18,540 Energy exchange takes one positional argument, but three were given, I think, yeah, I missed here. 198 00:16:19,020 --> 00:16:20,700 The update for the contribution? 199 00:16:21,600 --> 00:16:21,990 OK. 200 00:16:23,700 --> 00:16:24,990 And this time it worked. 201 00:16:25,290 --> 00:16:25,660 Nice. 202 00:16:25,680 --> 00:16:31,800 OK, so we have no more output of stuff exchange Mac, which are these two values, so we could now 203 00:16:31,800 --> 00:16:39,630 write zero, which would be the updated magnetization or one which would be the change of the of the 204 00:16:39,630 --> 00:16:40,110 energy. 205 00:16:40,680 --> 00:16:42,690 So you see, in this case, the change is zero. 206 00:16:42,720 --> 00:16:49,150 This is probably because we generated a change that increased the energy, so we declined it. 207 00:16:49,170 --> 00:16:51,060 So here it would give us zero. 208 00:16:51,600 --> 00:16:57,750 And if we repeat this, OK, yeah, you see, now the energy decreased by zero point six five. 209 00:16:58,180 --> 00:16:59,100 Let's do it again. 210 00:16:59,250 --> 00:17:07,140 Zero declined, except that negative energy, except that negative energy, except the negative energy 211 00:17:07,560 --> 00:17:08,040 and so on. 212 00:17:08,160 --> 00:17:10,920 So you see the energy change either zero. 213 00:17:10,920 --> 00:17:17,280 This is when it's declined or it's negative where it's accepted, but it will never be positive. 214 00:17:18,450 --> 00:17:20,640 OK, so now this was the most difficult part. 215 00:17:20,880 --> 00:17:22,770 We have implemented the metropolis step. 216 00:17:23,160 --> 00:17:27,660 Now we can run the Monte-Carlo, which is really not, not much left to do. 217 00:17:27,660 --> 00:17:34,560 We will just loop over the metropolis up and then see how will this configuration change and how it 218 00:17:34,560 --> 00:17:36,240 will turn into a fairer mechanism.