1 00:00:03,590 --> 00:00:08,300 Hello, bonjour, 2 00:00:08,880 --> 00:00:14,530 welcome to the tenth tutorial of Statistical Mechanics: Algorithms and Computations 3 00:00:14,530 --> 00:00:18,530 from the Physics Department of Ecole Normale Superieure. 4 00:00:18,530 --> 00:00:22,530 We have now come to the end of a long journey 5 00:00:23,900 --> 00:00:27,600 and after almost three months of travels 6 00:00:27,600 --> 00:00:29,390 (and of struggles) 7 00:00:29,390 --> 00:00:34,220 time has come to revisit the landmark sites 8 00:00:34,220 --> 00:00:40,080 that we have seen during the last two months and a half. 9 00:00:42,730 --> 00:00:47,000 In week 1, our mascot was this pebble. 10 00:00:47,740 --> 00:00:51,300 We used it on the Monte Carlo beach 11 00:00:51,300 --> 00:00:55,300 and on the heliport of Monaco, 12 00:00:55,300 --> 00:01:00,630 as well as in the homogeneous and inhomogeneous 13 00:01:00,630 --> 00:01:02,830 3x3 pebble game. 14 00:01:04,190 --> 00:01:09,830 It illustrated such essential concepts as detailed balance, 15 00:01:10,400 --> 00:01:12,100 aperiodicity, 16 00:01:13,390 --> 00:01:18,620 and irreducibility, as well as the Metropolis algorithm. 17 00:01:20,010 --> 00:01:28,070 In the homework session, it came up in the 1/2 rule, that we used many times since. 18 00:01:28,740 --> 00:01:33,580 So, please remember the pebble from week 1. 19 00:01:38,940 --> 00:01:43,110 In week 2 we moved from pebbles to disks. 20 00:01:43,640 --> 00:01:48,580 This gave us the opportunity to discuss about molecular dynamics 21 00:01:48,580 --> 00:01:50,400 and Monte Carlo methods. 22 00:01:50,400 --> 00:01:55,040 We were plunged in deep discussions and questions 23 00:01:55,040 --> 00:01:59,920 in physics: the equivalence between Newton dynamics 24 00:01:59,920 --> 00:02:02,930 and the Boltzmann statistical approach. 25 00:02:02,930 --> 00:02:08,790 This is one of the big mysteries in physics, the so-called ergodic hypothesis, 26 00:02:08,790 --> 00:02:13,750 that you actually proved to be valid for hard disks in homework 2 27 00:02:13,750 --> 00:02:20,210 using Monte Carlo methods (direct sampling and Markov chain) and molecular dynamics. 28 00:02:25,780 --> 00:02:31,140 During the third week, we went one step further and turned our attention 29 00:02:31,140 --> 00:02:36,270 to phase transitions, which are one of the main subjects of statistical mechanics. 30 00:02:37,380 --> 00:02:44,220 With the help of this object, the clothes-pin, you have studied the case of one-dimensional hard spheres, 31 00:02:45,060 --> 00:02:51,130 and you realized that by conceiving a direct rejection-free algorithm 32 00:02:51,130 --> 00:02:58,700 it was also possible to compute the partition function and to show that the system presents no phase transition. 33 00:02:59,900 --> 00:03:08,380 Following the same way, you have also understood the origin of the famous Asakura-Oosawa depletion interaction. 34 00:03:09,510 --> 00:03:15,680 In the homework, you turned your attention to the case of two-dimensional hard disks, 35 00:03:15,680 --> 00:03:19,830 and you understood how this very same interaction 36 00:03:19,830 --> 00:03:24,990 was responsible for the existence of phase transition in this system, 37 00:03:25,190 --> 00:03:30,250 as famously shown by Alder and Wainwright in 1964. 38 00:03:35,130 --> 00:03:40,570 Week 4 was dedicated to the relation between integration and sampling. 39 00:03:40,570 --> 00:03:44,820 We moved from the sampling of points on the surface of a sphere, 40 00:03:44,820 --> 00:03:49,740 like this basket-ball here, to the Maxwell distribution of velocities in a gas 41 00:03:49,740 --> 00:03:52,530 and the Boltzmann distribution of energies, 42 00:03:52,530 --> 00:03:58,350 and also to essential concepts like discrete sampling and sampling transformations. 43 00:03:58,350 --> 00:04:04,810 In the homework we performed integration and sampling in no less than 200 dimensions. 44 00:04:09,840 --> 00:04:16,720 During week 5, you turned your attention to the case of quantum particles, 45 00:04:17,720 --> 00:04:24,000 you learned to deal with their statistics using the tool of the density matrix. 46 00:04:25,520 --> 00:04:31,190 Do you remember this slinky? It embodies the path integral 47 00:04:31,190 --> 00:04:36,320 that you could construct using three simple properties of the density matrix: 48 00:04:36,920 --> 00:04:43,890 the convolution property, the Trotter decomposition and the solution for the free particle. 49 00:04:45,300 --> 00:04:49,590 With these tools in your hands, you learned how to construct 50 00:04:49,590 --> 00:04:52,330 - using a sampling approach - 51 00:04:52,330 --> 00:04:57,660 quantum Monte Carlo algorithms, to evaluate numerically the density matrix. 52 00:04:59,840 --> 00:05:07,020 During the tutorial, you also learned how to perform the imaginary time rotation 53 00:05:07,020 --> 00:05:13,230 and to study the dynamics of one particle in one dimension. 54 00:05:18,720 --> 00:05:22,420 Week 6 was also dedicated to quantum mechanics: 55 00:05:22,420 --> 00:05:25,250 we introduced the Levy quantum path, 56 00:05:25,250 --> 00:05:29,250 a direct sampling Monte Carlo method for path integral. 57 00:05:29,250 --> 00:05:35,450 You used this method in the homework, both in its free and harmonic versions. 58 00:05:36,220 --> 00:05:39,450 In the tutorial we started to discuss about bosons 59 00:05:39,640 --> 00:05:43,630 and Bose-Einstein condensation, which takes place in 60 00:05:43,630 --> 00:05:46,040 experimental cells like this one. 61 00:05:51,280 --> 00:05:58,330 Week 7 was all about Bose-Einstein condensation, as it takes place in Alberto's experimental cell. 62 00:05:58,810 --> 00:06:04,680 It can be looked at and manipulated by lasers, so I hope you all wore your glasses. 63 00:06:06,540 --> 00:06:12,920 But seriously, it was about a true quantum phenomenon: the indistinguishability of bosons 64 00:06:12,920 --> 00:06:18,050 and about our glorious 40-lines program: harmonic_bosons.py. 65 00:06:18,810 --> 00:06:25,320 In the homework, we dealt with bosons in traps shaped like cigars, and even pancakes. 66 00:06:25,320 --> 00:06:28,760 This sounds like fun, and it was a lot of fun! 67 00:06:28,760 --> 00:06:32,760 But it was also a serious non-trivial quantum mechanics problem. 68 00:06:38,150 --> 00:06:42,440 In week 8, we returned to classical physics 69 00:06:42,440 --> 00:06:46,840 and to the Ising model of spins on a lattice 70 00:06:47,330 --> 00:06:51,750 with a rudimentary nearest neighbors interaction. 71 00:06:53,570 --> 00:06:59,430 From exact enumeration of plus and minus spins, 72 00:06:59,430 --> 00:07:06,990 using the Gray code, we moved on to a discussion of local Monte Carlo methods 73 00:07:06,990 --> 00:07:12,120 using the Metropolis and the heat-bath algorithm 74 00:07:12,120 --> 00:07:16,120 and to the fabulous cluster methods. 75 00:07:16,620 --> 00:07:21,040 And do you remember the story of coupling? 76 00:07:25,870 --> 00:07:31,820 During week 9 we went on discussing our beloved model of statistical mechanics 77 00:07:32,850 --> 00:07:38,810 In many of these models, events occur at random times 78 00:07:39,390 --> 00:07:45,940 and during the lecture we learned how to sample the times between these events 79 00:07:45,940 --> 00:07:49,940 without having to wait for them to occur. 80 00:07:49,940 --> 00:07:54,400 And in this way we constructed faster-than-the-clock algorithms. 81 00:07:56,210 --> 00:08:03,040 During the tutorial, you were introduced to a basic but very interesting 82 00:08:03,040 --> 00:08:11,690 strategy, the simulated annealing, that allows to deal with complex optimization problems. 83 00:08:12,620 --> 00:08:19,280 For instance we have studied the case of the 13-spheres problem 84 00:08:19,280 --> 00:08:26,750 and we were able to devise an algorithm which finds the best solution to this problem in a few seconds. 85 00:08:32,370 --> 00:08:35,440 So, finally, week 10 86 00:08:35,440 --> 00:08:39,700 was the alpha and the omega of Monte Carlo. 87 00:08:40,590 --> 00:08:43,900 The alpha was where all started, 88 00:08:43,900 --> 00:08:49,070 in 1777 with Buffon's needles 89 00:08:50,990 --> 00:08:55,710 thrown randomly onto a parquet, 90 00:08:56,510 --> 00:08:59,710 ant the omega was where all ended 91 00:08:59,710 --> 00:09:03,710 with the famous Levy stable distributions 92 00:09:03,710 --> 00:09:06,720 for observables with infinite variance. 93 00:09:08,010 --> 00:09:14,270 So, this was it, Statistical Mechanics: Algorithms and Computations. 94 00:09:14,270 --> 00:09:18,270 There were many subjects that we left out, 95 00:09:18,650 --> 00:09:22,670 but you will agree that there was a lot of stuff inside. 96 00:09:23,300 --> 00:09:29,550 So now it is time to present the members of the team who where not in front of the camera during all this time 97 00:09:29,550 --> 00:09:32,040 So let's start with Maxim Berman.. 98 00:09:32,040 --> 00:09:36,800 Hi everybody, I had a lot of fun producing the best animations possible for the course 99 00:09:36,800 --> 00:09:43,180 including the histograms, the cascades of configurations and the pebbles thrown on the Monte Carlo beach. 100 00:09:43,870 --> 00:09:47,180 Then there is Tommaso Comparin.. 101 00:09:47,600 --> 00:09:51,180 Ciao, it's me who looked after many of the programs 102 00:09:51,370 --> 00:09:55,360 and also typed the subtitles for the course, including the ones you are reading now. 103 00:09:55,360 --> 00:09:57,380 I hope this turned out useful for you! 104 00:09:58,130 --> 00:10:00,530 Then there is Emilie Noblet.. 105 00:10:02,540 --> 00:10:06,400 So hello everybody, I was the camera woman on this project, 106 00:10:06,570 --> 00:10:10,400 entirely shot in the green-screen technology. 107 00:10:11,050 --> 00:10:16,030 In addition there was more camera by Frederic Borjat, 108 00:10:16,030 --> 00:10:19,050 and our editor Baptiste Ribrault, 109 00:10:19,050 --> 00:10:22,290 who put all these pieces so nicely together. 110 00:10:23,140 --> 00:10:26,290 So thanks to all of you, all over the world 111 00:10:26,290 --> 00:10:29,010 for your interest and your active participation: 112 00:10:29,010 --> 00:10:32,040 *you* made this course happen. 113 00:10:33,080 --> 00:10:35,480 And now, let's have a party!