1 00:00:00,150 --> 00:00:01,830 So now everything is prepared. 2 00:00:01,980 --> 00:00:09,420 We have our array of the coordinates and of the magnetic moments that we have generated randomly and 3 00:00:09,450 --> 00:00:17,220 we have defined our function for a single metropolis step, which includes the information about the 4 00:00:17,220 --> 00:00:20,880 energy that we have implemented according to the exchange interaction. 5 00:00:21,690 --> 00:00:24,450 So now we can go ahead and run the actual one to kind of. 6 00:00:25,470 --> 00:00:27,510 So I said already, this is pretty simple. 7 00:00:27,510 --> 00:00:29,760 It's just a loop, basically. 8 00:00:30,330 --> 00:00:33,090 So we must specify how many steps we will do. 9 00:00:33,180 --> 00:00:37,410 So a number of steps and we can do really many of those. 10 00:00:37,410 --> 00:00:38,970 We will go up to a million other things. 11 00:00:38,970 --> 00:00:45,120 So let's start with 10K and then we loop for I in range. 12 00:00:45,780 --> 00:00:47,370 No steps. 13 00:00:49,470 --> 00:00:54,030 And then what we do is we just write this one. 14 00:00:55,740 --> 00:01:01,530 And the first output or with the zero index will be the magnetization, which is updated. 15 00:01:02,130 --> 00:01:09,450 And the second outputs with the index one will be the energy so and g change. 16 00:01:12,510 --> 00:01:14,070 And this will be like this. 17 00:01:15,270 --> 00:01:24,150 OK, so the thing is, we don't really need the information about the change of the energy because the 18 00:01:24,150 --> 00:01:32,490 way that reprogrammed it is that we calculate the energy two times anyway, so we don't need to load 19 00:01:32,490 --> 00:01:32,920 it here. 20 00:01:33,390 --> 00:01:38,020 So this means we don't actually have to have the information here about the energy. 21 00:01:38,850 --> 00:01:44,070 So it turns out that this would already be sufficient. 22 00:01:45,060 --> 00:01:51,630 And the only thing the energy information is nice forward to really track our progress. 23 00:01:51,900 --> 00:01:54,330 And so I will write here that this is optional. 24 00:01:56,250 --> 00:02:06,840 So it is optional in the sense that we just write the energy in the beginning is just an energy exchange 25 00:02:07,260 --> 00:02:07,870 off mark. 26 00:02:08,220 --> 00:02:12,660 So this is really calculating once the total energy of the whole sample. 27 00:02:13,290 --> 00:02:20,580 And then we will store your energy list, as is an empty list. 28 00:02:20,950 --> 00:02:27,240 Or we could also add the particular value here from the start, but we could also make it empty for 29 00:02:27,240 --> 00:02:27,760 the beginning. 30 00:02:28,290 --> 00:02:32,640 And now we say in the loop, it's also optional. 31 00:02:32,640 --> 00:02:38,940 Of course, we you update the list or we will add the values to the list. 32 00:02:39,150 --> 00:02:53,330 So we say energy is equal to energy plus energy change and then energy list dot append energy. 33 00:02:54,900 --> 00:02:56,800 But as I said, these took two months. 34 00:02:56,850 --> 00:03:01,770 Commands are optional and these two commands are optional, basically, this one is the only one we 35 00:03:01,770 --> 00:03:02,070 need. 36 00:03:03,270 --> 00:03:07,270 OK, so now let's run it and you see it's already finished. 37 00:03:07,320 --> 00:03:09,390 These are 10000 loops. 38 00:03:09,840 --> 00:03:11,670 And let's see what happens. 39 00:03:11,820 --> 00:03:16,200 And before we look at the magnetization, let us look at the energy. 40 00:03:16,650 --> 00:03:22,850 So Peel Dot plot will be range 40. 41 00:03:23,070 --> 00:03:28,410 So for for the x axis, this will be just indices or in the index for the number of to step. 42 00:03:29,340 --> 00:03:32,400 So range no steps. 43 00:03:32,880 --> 00:03:39,270 And then we write here plus one because we included the starting value here and for the y axis we plot, 44 00:03:39,270 --> 00:03:41,580 of course, the energy so energy list. 45 00:03:42,750 --> 00:03:48,390 And you see, this is now the energy and this is the number of the step. 46 00:03:49,500 --> 00:03:50,960 So I will not add labels here. 47 00:03:50,970 --> 00:03:52,320 You can do this yourself. 48 00:03:52,320 --> 00:03:53,410 It's pretty easy to do. 49 00:03:53,430 --> 00:03:58,530 We have done it so many times already and you see at the beginning, we start at an energy that's very 50 00:03:58,530 --> 00:04:02,070 close to zero because at the beginning everything was randomly. 51 00:04:02,520 --> 00:04:06,630 And then the energy decreases and decreases and decreases. 52 00:04:07,050 --> 00:04:11,100 And sometimes it stays maybe constant as here, but it never increases. 53 00:04:11,280 --> 00:04:15,630 And you see, we really go to a very, very negative energy here. 54 00:04:15,630 --> 00:04:16,800 So it looks pretty good. 55 00:04:17,910 --> 00:04:21,029 But also, you see that the energy is still decreasing. 56 00:04:21,029 --> 00:04:25,020 So it seems as if the simulation isn't finished yet. 57 00:04:25,950 --> 00:04:29,100 So let's see what the profile actually looks like. 58 00:04:29,280 --> 00:04:34,920 So I will just scroll up here and copy this code from the plot. 59 00:04:35,550 --> 00:04:39,930 And since we have updated the magnetization, we can just run it as it is. 60 00:04:41,460 --> 00:04:44,430 And here is our new magnetization. 61 00:04:45,420 --> 00:04:51,600 So I think it doesn't look as confusing as before, but it doesn't look like a ferromagnetic yet. 62 00:04:52,050 --> 00:04:54,660 So probably it's just not finished yet. 63 00:04:55,830 --> 00:05:03,720 So I would say, let's go to 100000 steps, and if we would run rather just right away, we would continue 64 00:05:03,720 --> 00:05:04,710 from this point. 65 00:05:05,220 --> 00:05:10,920 This would also be OK, but I want to start from the beginning, so I just restart and run the whole 66 00:05:11,100 --> 00:05:11,610 notebook. 67 00:05:13,890 --> 00:05:15,630 So now, of course, it takes a bit of time. 68 00:05:16,170 --> 00:05:23,190 And here you see on the energy once again, starting at zero and you see this time we approach a stage 69 00:05:23,190 --> 00:05:28,950 where the energy almost remains constant, so it still keeps on decreasing a bit. 70 00:05:28,950 --> 00:05:33,150 So maybe we are not perfectly at the minimum yet, but it looks much better this time. 71 00:05:34,590 --> 00:05:37,410 But still, it is kind of difficult. 72 00:05:37,800 --> 00:05:45,420 Still, it looks non linear, but we have here some areas where it is already quite thorough magnetic. 73 00:05:46,320 --> 00:05:53,760 So maybe the problem is just that this system size is still too large and we could not just increase 74 00:05:53,760 --> 00:05:59,460 the number of steps, but maybe it's easier to just decrease the size of the system and go to a length 75 00:05:59,460 --> 00:06:00,390 equal to 20. 76 00:06:01,870 --> 00:06:03,850 So once again, restart and run all. 77 00:06:09,680 --> 00:06:10,210 OK. 78 00:06:10,460 --> 00:06:18,680 Still didn't converge, so you see, it sometimes is really tricky to arrive at a good solution in the 79 00:06:18,680 --> 00:06:23,840 Monte Carlo because everything is totally random and it's never guaranteed that you get a good result. 80 00:06:24,140 --> 00:06:28,880 You have to really play with the parameters and run these things many times. 81 00:06:29,450 --> 00:06:31,150 So you see, I have added here another zero. 82 00:06:31,160 --> 00:06:35,900 We are now at one million steps and this will now probably take a few seconds. 83 00:06:36,740 --> 00:06:37,360 But let's see. 84 00:06:39,670 --> 00:06:42,100 We are now running the loop. 85 00:06:43,930 --> 00:06:50,170 And it's still running, so you see, it's not anymore a question of a few seconds, but it takes a 86 00:06:50,170 --> 00:06:50,770 bit of time. 87 00:06:51,940 --> 00:06:54,580 But I think should finish pretty soon. 88 00:06:55,180 --> 00:06:57,730 And then I hope we will get a good result this time. 89 00:06:58,300 --> 00:07:02,350 And you see here, the line is pretty much straight at the end. 90 00:07:02,350 --> 00:07:05,770 So I'm pretty confident that this time it works. 91 00:07:05,770 --> 00:07:08,050 Better still didn't work out. 92 00:07:08,080 --> 00:07:08,680 Oh man. 93 00:07:08,870 --> 00:07:13,810 OK, but you see here it's it's getting somewhere very audits. 94 00:07:14,380 --> 00:07:17,500 So in neighboring moments are almost parallel here. 95 00:07:18,190 --> 00:07:21,610 So it looks pretty good already, but not perfect. 96 00:07:22,180 --> 00:07:29,530 So the thing is, you now have to either increase the number of steps even further or you have to just 97 00:07:29,530 --> 00:07:34,630 rerun the whole thing many, many times and then look every time at the energy. 98 00:07:34,990 --> 00:07:46,030 So for example, here what we can do is we can see an energy list and then minus one, which will be 99 00:07:46,030 --> 00:07:47,770 the last entry from the energy list. 100 00:07:48,190 --> 00:07:55,540 And we can now also check if we did everything correctly by writing Energy mix change. 101 00:07:56,560 --> 00:08:00,580 Not sorry, not with these brackets, but with these ones. 102 00:08:02,980 --> 00:08:05,170 Yeah, OK, at least we did everything correctly. 103 00:08:05,530 --> 00:08:12,970 So on the magnetization, the energy is updated every time in our list and we did it correctly because 104 00:08:12,970 --> 00:08:16,540 when we recalculate the energy at the end, it gives us the correct results. 105 00:08:18,760 --> 00:08:23,770 So we what we would have to do is we would have to rerun this whole simulation many, many times. 106 00:08:24,040 --> 00:08:27,640 And then the correct solution would be the one with the lowest energy. 107 00:08:28,390 --> 00:08:33,490 So I will not stop the recording and run this a few times and see if I will get a better result in minus 108 00:08:33,490 --> 00:08:34,780 seven six three. 109 00:08:36,340 --> 00:08:42,039 So I have rerun this whole thing several times, and the best result that I got was an energy, which 110 00:08:42,039 --> 00:08:43,110 is even lower. 111 00:08:43,120 --> 00:08:44,890 It's minus seven seven nine. 112 00:08:46,510 --> 00:08:53,800 From our initial thought about the energy, we can actually tell what the correct solution would be. 113 00:08:54,280 --> 00:08:57,010 It would be minus eight hundreds. 114 00:08:57,490 --> 00:09:00,160 So we are still pretty far from the correct solution. 115 00:09:00,640 --> 00:09:02,710 And you see, this is what we end up with. 116 00:09:03,160 --> 00:09:09,010 It looks pretty ferromagnetic at some regions, for example, here looks ferromagnetic even here. 117 00:09:09,280 --> 00:09:15,040 But I think the main problem is that we always end up with some strange structure like this world here 118 00:09:15,430 --> 00:09:19,420 and these worlds, and they are actually very hard to get rid of. 119 00:09:19,480 --> 00:09:26,350 So just by changing here, a single magnetic moment will never really help that much to get really to 120 00:09:26,350 --> 00:09:27,400 the energetic minimum. 121 00:09:28,240 --> 00:09:34,840 And this is the problem of our very first version of the of the Monte Carlo algorithm. 122 00:09:35,290 --> 00:09:41,230 It is that we can get stuck in some energetic minimum that is just a local minimum. 123 00:09:41,500 --> 00:09:48,430 So it describes some rather stable configuration, but not the actual global energetic minimum. 124 00:09:49,180 --> 00:09:56,860 And so in the following lecture, I will show you an update to our code that helps us to better converge 125 00:09:56,870 --> 00:10:01,300 the result and to get better results and also in the shorter amount of time. 126 00:10:02,020 --> 00:10:05,290 So let's stay tuned and let's improve our code in the next lecture.