1 00:00:03,150 --> 00:00:07,550 Hello, bonjour, dia duit, hei, 2 00:00:08,580 --> 00:00:14,190 Welcome to the fourth week of Statistical Mechanics: Algorithms and Computations, 3 00:00:14,190 --> 00:00:17,940 from the Physics Department of Ecole Normale supérieure. 4 00:00:19,440 --> 00:00:24,800 This week we will advance our understanding along two directions 5 00:00:26,220 --> 00:00:33,580 from the physical point of view, we have so far thoroughly studied the first pillar 6 00:00:33,580 --> 00:00:37,580 of statistical mechanics: the equiprobability principle. 7 00:00:38,930 --> 00:00:43,090 Time has come to consider the second pillar 8 00:00:43,780 --> 00:00:51,900 which comes from the Maxwell distribution of velocities and gives the Boltzmann distribution of energy. 9 00:00:54,420 --> 00:01:01,660 This second pillar will allow us to heal the dissimmetry between 10 00:01:01,660 --> 00:01:05,290 the molecular dynamics simulations 11 00:01:05,290 --> 00:01:09,290 that is the solution of Newton's equations of motion 12 00:01:10,140 --> 00:01:13,290 and the Monte Carlo approach. 13 00:01:13,290 --> 00:01:16,890 So far, the molecular dynamics 14 00:01:16,890 --> 00:01:23,560 had velocities and positions, whereas the Monte Carlo only had the positions. 15 00:01:24,480 --> 00:01:27,550 What we learn today will allow us 16 00:01:27,550 --> 00:01:31,550 to overcome the dissimmetry 17 00:01:31,550 --> 00:01:34,890 between molecular dynamics and Monte Carlo. 18 00:01:35,300 --> 00:01:39,510 To get there, we must deepen our understanding of sampling 19 00:01:40,260 --> 00:01:44,990 and this is the second direction along which we will advance this week. 20 00:01:45,830 --> 00:01:50,590 So far, sampling meant that we threw pebbles, 21 00:01:50,590 --> 00:01:55,060 placed disks, or fixed clothes-pins 22 00:01:55,060 --> 00:01:56,550 on a washing line. 23 00:01:57,910 --> 00:02:01,230 Now, we will connect this sampling problem 24 00:02:01,230 --> 00:02:05,230 with the underlying process of integration. 25 00:02:05,230 --> 00:02:11,630 The sampling problem will accompany us also in this week's tutorial, tutorial 4. 26 00:02:12,670 --> 00:02:17,200 We will start with a very simple Saturday night problem 27 00:02:18,330 --> 00:02:24,870 on your free evening you have many options and you can give them probabilities 28 00:02:26,350 --> 00:02:29,760 you can go out with friend, see your family, 29 00:02:29,760 --> 00:02:33,760 do homework, do sports. 30 00:02:33,760 --> 00:02:37,760 All of these options have statistical weights 31 00:02:37,760 --> 00:02:45,950 and you may think at first sight that the problem is writing down these statistical weights 32 00:02:45,950 --> 00:02:51,720 but it isn't. The difficult part is doing the sampling. 33 00:02:52,870 --> 00:02:58,100 In tutorial 4, you will learn how to sample this distribution 34 00:02:58,100 --> 00:03:00,900 along different approaches, 35 00:03:02,180 --> 00:03:06,250 and how to do it if you not only have 4 or 5 options, 36 00:03:06,250 --> 00:03:10,250 but thousand, millions, or billions, 37 00:03:10,250 --> 00:03:14,250 how to sample such a discrete distribution 38 00:03:14,250 --> 00:03:18,250 efficiently, or event optimally 39 00:03:18,250 --> 00:03:22,250 and how to take the continuum limit. 40 00:03:22,250 --> 00:03:26,250 All of these are central subjects 41 00:03:26,250 --> 00:03:30,250 of daily life at work or study 42 00:03:30,250 --> 00:03:34,250 but - as you see - also for the night life. 43 00:03:35,250 --> 00:03:39,200 In this week's homework, homework session 4, 44 00:03:39,430 --> 00:03:43,530 you will be concerned again with the connection between sampling 45 00:03:43,530 --> 00:03:44,740 and integration. 46 00:03:46,080 --> 00:03:49,410 In fact, you will compute the volume of the sphere 47 00:03:50,160 --> 00:03:52,880 not in three dimensions, like here, 48 00:03:52,880 --> 00:03:56,310 but in 100 or 200 dimensions 49 00:03:56,830 --> 00:04:00,650 this is what Monte Carlo methods are really good for. 50 00:04:02,440 --> 00:04:09,680 You will learn for yourself how the sampling approach really goes at top speed in high dimensions 51 00:04:09,680 --> 00:04:14,120 and how you can evaluate an integral there. 52 00:04:15,230 --> 00:04:21,150 You see, there are lots of exciting developments in week 4 of our course, 53 00:04:21,150 --> 00:04:24,070 so let's get started. 54 00:04:29,390 --> 00:04:36,050 As we just mentioned, time has come to connect sampling to integration, 55 00:04:36,900 --> 00:04:40,050 let us look back at the children's game: 56 00:04:41,800 --> 00:04:50,030 pi/4 came out as the ratio of the volume of the circle to the volume of the square. 57 00:04:51,500 --> 00:04:58,610 More precisely, we find that the number of hits divided by the number of trials 58 00:04:58,610 --> 00:05:03,450 is approximately equal to the ratio of two integrals. 59 00:05:04,630 --> 00:05:09,500 The integral over dx and dy on the square 60 00:05:09,500 --> 00:05:13,940 of O(x,y) pi(x,y) 61 00:05:13,940 --> 00:05:19,590 divided by the integral over the square of pi(x,y). 62 00:05:19,590 --> 00:05:28,170 In this equation O(x,y) is a function, an observable 63 00:05:28,170 --> 00:05:33,650 and it is 1 inside the circle and 0 outside, 64 00:05:34,400 --> 00:05:42,000 and pi(x,y) is the uniform probability distribution on the square. 65 00:05:42,670 --> 00:05:48,960 By throwing pebbles, we in fact compute the integral in dx dy 66 00:05:48,960 --> 00:05:51,620 pi(x,y). 67 00:05:52,190 --> 00:06:00,890 But notice: on the left side the probability distribution pi(x,y) is absent 68 00:06:01,460 --> 00:06:06,990 it is the samples that are drawn from this distribution. 69 00:06:08,550 --> 00:06:16,300 Furthermore the dimension of the integral (the fact that the integral on the right is in two dimensions) 70 00:06:16,300 --> 00:06:18,890 does not appear on the left: 71 00:06:19,760 --> 00:06:22,890 there are no multiple sums along all dimensions. 72 00:06:23,560 --> 00:06:31,210 This explains why Monte Carlo sampling methods excel in high dimensions. 73 00:06:32,090 --> 00:06:38,400 For integrals, we have substitution rules: changes of variables. 74 00:06:38,400 --> 00:06:43,460 And we we'll next see how these changes of variables 75 00:06:43,460 --> 00:06:46,720 relate to the sampling approach. 76 00:06:47,300 --> 00:06:51,180 Our example will be the Gaussian integral 77 00:06:51,180 --> 00:06:56,140 integral from (-infinity) to infinity in dx 78 00:06:56,140 --> 00:07:01,670 over sqrt(2 pi) exp(- x^2 / 2). 79 00:07:02,700 --> 00:07:06,680 The value of the integral is I=1. 80 00:07:06,680 --> 00:07:14,690 And to show it, generations of students have learned to square the integral 81 00:07:16,330 --> 00:07:23,600 so we write I^2 = (integral in x) times (integral in y) 82 00:07:24,920 --> 00:07:28,570 Now let's do a first substitution 83 00:07:29,170 --> 00:07:33,630 we go from x and y to r and phi. 84 00:07:33,630 --> 00:07:37,630 x = r cos(phi), 85 00:07:37,630 --> 00:07:42,240 and y = r sin(phi) 86 00:07:44,610 --> 00:07:46,460 This gives the following integral. 87 00:07:47,310 --> 00:07:53,830 Now, we do a second substitution from r to psi 88 00:07:56,050 --> 00:08:02,680 and then finally we do a substitution from exp(-psi) 89 00:08:02,680 --> 00:08:05,400 to Upsilon, 90 00:08:06,190 --> 00:08:16,220 and we arrive at the formula I^2 = integral from 0 to 2pi dphi over 2pi 91 00:08:16,760 --> 00:08:20,850 times the integral from 0 to 1 dUpsilon. 92 00:08:22,630 --> 00:08:34,120 We can do these two integrals, and confirms that I^2=1, which means that I=1. We have computed the Gaussian integral. 93 00:08:35,680 --> 00:08:40,060 But in fact we don't care about the value of this integral 94 00:08:40,060 --> 00:08:46,600 what we do is to use our relationship between integration and sampling. 95 00:08:47,660 --> 00:08:55,640 So we can see that phi in fact is a random number between 0 and (2 pi) 96 00:08:56,730 --> 00:09:01,690 and Upsilon is a random number between 0 and 1. 97 00:09:03,410 --> 00:09:07,870 So now let's take a little crab's walk back 98 00:09:07,870 --> 00:09:10,690 from the variables 99 00:09:10,690 --> 00:09:16,150 phi and Upsilon (the samples phi and Upsilon) 100 00:09:16,150 --> 00:09:21,660 back to x and y. So the samples phi and Upsilon 101 00:09:21,660 --> 00:09:25,660 give samples phi and psi 102 00:09:26,620 --> 00:09:32,800 phi and r, and finally x and y. 103 00:09:32,800 --> 00:09:38,750 Let us discuss as what it really is, namely an algorithm 104 00:09:38,750 --> 00:09:41,210 gauss_test.py 105 00:09:42,220 --> 00:09:46,620 Its graphic version (gauss_test_movie.py) 106 00:09:46,620 --> 00:09:48,830 allows you to see 107 00:09:48,830 --> 00:09:54,420 that the flat input distributions of phi and Upsilon 108 00:09:55,210 --> 00:10:01,860 get transformed into distributions of x and y that are Gaussians. 109 00:10:02,730 --> 00:10:11,860 So we have a distribution exp(-x^2/2) and a distribution exp(-y^2/2) and the two are independent. 110 00:10:13,410 --> 00:10:16,710 So what you see here allows you to understand 111 00:10:16,710 --> 00:10:23,070 that the substitution rules (the changes of variables in integrals) 112 00:10:23,070 --> 00:10:26,090 also apply to the samples. 113 00:10:27,000 --> 00:10:34,230 Before going on, please take a moment to download, run and modify the programs we just discussed. 114 00:10:35,270 --> 00:10:39,520 On the Coursera website, you'll find the program gauss_test.py. 115 00:10:40,340 --> 00:10:46,940 Understand that phi and Upsilon are numbers, samples, 116 00:10:46,940 --> 00:10:54,190 and that these samples are transformed into numbers, samples of x and y 117 00:10:54,550 --> 00:10:57,040 taken from a Gaussian distribution. 118 00:10:58,040 --> 00:11:01,040 Take the program apart 119 00:11:02,040 --> 00:11:05,040 in the graphic version of gauss_test.py 120 00:11:06,220 --> 00:11:09,040 you see the variable Upsilon: 121 00:11:09,040 --> 00:11:15,630 it's a uniform between 0 and 1. Check that this is actually uniform. 122 00:11:16,580 --> 00:11:20,030 Then there is an intermediate variable psi 123 00:11:20,700 --> 00:11:25,640 - log(Upsilon). What is its distribution? 124 00:11:25,640 --> 00:11:33,910 Check it out, and compare it to the analytical calculation we did a few minutes ago. 125 00:11:39,550 --> 00:11:45,310 Now let us consider vectors or lists of Gaussians 126 00:11:45,310 --> 00:11:47,920 not just a single one. 127 00:11:48,960 --> 00:11:55,190 We start in two dimensions, with x and y two independent Gaussians. 128 00:11:55,830 --> 00:11:59,790 This is shown here in this program, and note 129 00:11:59,790 --> 00:12:05,250 that we have gone from our home-grown version of gauss_test.py 130 00:12:05,250 --> 00:12:10,020 to the Python version random.gauss. 131 00:12:10,320 --> 00:12:16,010 It uses basically the same algorithm, called the Box-Muller method. 132 00:12:18,200 --> 00:12:25,110 Output of gauss_2d_movie.py is shown here. 133 00:12:25,670 --> 00:12:30,930 You see a first sample, a second sample, a third sample, and so on. 134 00:12:32,070 --> 00:12:45,650 What you see is a cloud of points. Each point is an independent Gaussian in x and an independent Gaussian in y. 135 00:12:46,920 --> 00:12:50,600 And note that the distribution (the cloud) 136 00:12:50,600 --> 00:12:56,880 is isotropic, it is invariant under rotations. 137 00:12:57,600 --> 00:13:03,170 This situation is unique, there's only Gaussians that can do it. 138 00:13:03,170 --> 00:13:08,290 Remember that we already had distributions of x and y, 139 00:13:08,290 --> 00:13:15,370 in the children's game it was a square and it changed shape when it was rotated. 140 00:13:16,240 --> 00:13:24,380 To understand why this distribution is isotropic, let's look at the old program gauss_test.py 141 00:13:25,170 --> 00:13:31,960 you see that there is a variable phi which is uniform between 0 and (2pi) 142 00:13:32,730 --> 00:13:38,650 and the point x,y is oriented along this axis. 143 00:13:39,670 --> 00:13:48,740 Another of choice of phi would simply rotate the point x,y along the origin. 144 00:13:50,020 --> 00:13:57,570 You can directly check this also in out little crab walk we took a few minutes ago. 145 00:13:58,890 --> 00:14:04,300 You see we arrived from independent distributions x and y 146 00:14:04,300 --> 00:14:13,110 at a distribution integral from 0 to (2 pi) dphi over (2 pi), times an integral in r 147 00:14:13,110 --> 00:14:20,850 There is no function depending on phi, which means that the variable phi is independent. 148 00:14:21,910 --> 00:14:28,210 We will considerably deepen and generalize this for higher dimensions in a few minutes. 149 00:14:28,760 --> 00:14:39,470 Gaussians are unique in that independent distributions in x and y give an isotropic distribution in two dimensions. 150 00:14:40,380 --> 00:14:44,740 This property is general for Gaussians in any dimensions. 151 00:14:46,420 --> 00:14:51,430 The 3D case is shown here in gauss_3d.py 152 00:14:52,620 --> 00:14:59,330 each point is a vector x,y,z, and all three are independent Gaussians. 153 00:15:00,950 --> 00:15:08,350 So output of the movie version is shown here: one point, two points, three points, and so on.. 154 00:15:08,350 --> 00:15:12,350 You see that the distribution is isotropic. 155 00:15:13,730 --> 00:15:20,640 And in the movie version we can even turn around the distribution and look at it from all angles. 156 00:15:22,360 --> 00:15:24,640 Does this make more adventurous? 157 00:15:25,450 --> 00:15:29,510 Yes of course, let's give this distribution a haircut. 158 00:15:29,510 --> 00:15:33,510 Let's take each point x,y,z 159 00:15:33,510 --> 00:15:36,160 and renormalize it to one. 160 00:15:37,100 --> 00:15:43,470 So we take x,y,z independent Gaussians, and divide this vector 161 00:15:43,470 --> 00:15:50,300 by the square root of (x^2 + y^2 + z^2). 162 00:15:51,450 --> 00:15:56,060 So now we have random points on the unit sphere. 163 00:15:57,720 --> 00:16:02,310 And as the distribution initially was isotropic, 164 00:16:02,310 --> 00:16:06,310 it remains isotropic on the unit sphere. 165 00:16:07,830 --> 00:16:16,010 So we found an algorithm to put random pebbles onto the surface of the sphere. 166 00:16:16,730 --> 00:16:23,150 Understand that isotropy does not simply mean that the points are on the surface of the sphere 167 00:16:23,150 --> 00:16:26,230 they are so by construction 168 00:16:26,230 --> 00:16:30,230 because we renormalized each point x,y,z 169 00:16:30,230 --> 00:16:31,880 to have a radius = 1. 170 00:16:31,880 --> 00:16:37,330 Isotropy means that the distribution of points (the probability distribution) 171 00:16:37,330 --> 00:16:40,700 is uniform on the surface of the sphere. 172 00:16:41,480 --> 00:16:48,300 So here is the program direct_surface.py in general dimensions. 173 00:16:48,300 --> 00:16:57,500 It puts points randomly with a uniform distribution on the surface of the hypersphere in d dimensions. 174 00:16:57,500 --> 00:17:01,500 Now, let us follow mathematically what is going on. 175 00:17:02,240 --> 00:17:06,240 We have to use the concept of sample transformation. 176 00:17:06,910 --> 00:17:18,450 Remember: sample transformation means that the changes of variables in the integral apply directly to the samples. 177 00:17:18,450 --> 00:17:27,490 So the integral that we are concerned with is the integral dx_0, dx_1, .., dx_(d-1) 178 00:17:27,490 --> 00:17:36,390 exp(-x_0^2) exp(-x_1^2) .. exp(-x_(d-1)^2) 179 00:17:38,030 --> 00:17:41,430 Now let's do a change of variables 180 00:17:41,430 --> 00:17:46,710 from x_0, .., x_(d-1) 181 00:17:46,710 --> 00:17:50,710 to r, the radius variable 182 00:17:50,710 --> 00:17:53,380 and an angular variable Omega, 183 00:17:53,380 --> 00:17:55,360 the solid angle. 184 00:17:57,080 --> 00:18:05,620 Integrated over the whole space, dOmega gives (2 pi) in two dimensions 185 00:18:05,620 --> 00:18:08,410 and (4 pi) in three dimensions. 186 00:18:10,370 --> 00:18:13,000 So now see that the integrand 187 00:18:14,140 --> 00:18:20,720 depends only on r, it is equal to exp(-r^2 /2) 188 00:18:21,840 --> 00:18:28,190 and the differential dx_0 dx_1 .. dx_(d-1) 189 00:18:28,190 --> 00:18:32,770 yields r^(d-1) dOmega dr. 190 00:18:34,430 --> 00:18:41,590 So, for our d Gaussians, we end up with a distribution 191 00:18:41,590 --> 00:18:46,970 integral dr r^(d-1) 192 00:18:47,780 --> 00:18:52,360 exp(-r^2 / 2) integral over dOmega. 193 00:18:53,610 --> 00:18:57,220 It is two independent integrals 194 00:18:58,170 --> 00:19:03,400 one in r, one in Omega. There's no function depending on Omega, 195 00:19:03,400 --> 00:19:11,930 which shows that we are isotropic in angles on the surface of the sphere in d dimensions. 196 00:19:13,200 --> 00:19:15,930 Finally, let's do a trick 197 00:19:15,930 --> 00:19:23,400 let's renormalize the radius r to be on a surface: r=1 198 00:19:23,400 --> 00:19:35,040 and then blow it up again to have a distribution pi(r) = r^(d-1) between 0 and 1 199 00:19:35,040 --> 00:19:42,150 rather than r^(d-1) exp(-r^2/2) from 0 to infinity. 200 00:19:43,850 --> 00:19:50,770 What you obtain is written in this program: direct_sphere_3d.py 201 00:19:50,770 --> 00:19:56,660 and it is a random point inside the three dimensional unit sphere. 202 00:19:57,290 --> 00:20:01,140 So output is shown here: one point, two points, three points, .. 203 00:20:02,570 --> 00:20:09,140 and we can turn it around and we have a random point inside the surface of the three dimensional sphere. 204 00:20:10,200 --> 00:20:18,440 There's also a general program that allows you to sample a random point inside the d-dimensional hypersphere. 205 00:20:18,440 --> 00:20:21,760 Before continuing, please take a moment 206 00:20:21,760 --> 00:20:26,310 to download, run and modify the programs we discussed in this section 207 00:20:26,310 --> 00:20:30,640 that means: take them apart and put them back together. 208 00:20:31,590 --> 00:20:40,570 On the Coursera website, you'll find the programs direct_sphere_3d.py and direct_surface_3d.py 209 00:20:41,710 --> 00:20:46,960 These short and sweet programs are marvelous 210 00:20:46,960 --> 00:20:51,580 and they should have their place in the national museums all over the world. 211 00:20:57,200 --> 00:21:03,030 Molecular dynamics concerns positions and velocities, 212 00:21:03,030 --> 00:21:08,640 whereas the Monte Carlo method considers only the positions. 213 00:21:10,460 --> 00:21:21,880 Why the velocities disappear from our Monte Carlo program and how we can make them come back deserves a most thorough answer. 214 00:21:24,060 --> 00:21:32,930 To incorporate velocities into statistical mechanics, we again use the equiprobability principle. 215 00:21:35,170 --> 00:21:37,770 For hard-disks in a box, 216 00:21:38,970 --> 00:21:48,080 during an event-driven molecular dynamics simulation, the energy is equal to the kinetic energy 217 00:21:48,080 --> 00:21:50,440 and it is fixed. 218 00:21:51,430 --> 00:21:58,490 Kinetic energy is equal to 1/2 m sum over the velocities square 219 00:21:58,490 --> 00:22:04,540 v_0^2 + .. + v_(N-1)^2 220 00:22:05,400 --> 00:22:11,220 In this equation, each velocity has two components 221 00:22:11,220 --> 00:22:16,590 v_0 = v_0x, v_0y 222 00:22:17,280 --> 00:22:29,530 So that the square v_0^2 is equal to v_0x ^2 + v_0y ^2 223 00:22:32,130 --> 00:22:38,060 In this equation you see that the sum over the squares is fixed 224 00:22:38,710 --> 00:22:43,730 just like on a circle, or on a surface of a sphere. 225 00:22:43,730 --> 00:22:49,660 Only in higher dimensions, any legal set of velocities 226 00:22:49,660 --> 00:22:55,930 is a point on the 2N-dimensional hypersphere 227 00:22:55,930 --> 00:23:03,350 of radius sqrt(2 E_kin / m) 228 00:23:03,350 --> 00:23:10,400 for 4 particles in a box, this forms a vector 229 00:23:10,400 --> 00:23:17,140 of 8 components (an 8-dimensional vector) on the surface of the 8-dimensional hypersphere. 230 00:23:19,060 --> 00:23:36,450 The equiprobability principle applied to these velocities means that the statistical weight pi(v_0, v_1, .., v_(N-1)) must be a constant if 231 00:23:36,450 --> 00:23:45,230 the vector (v_0.. v_(N-1)) is on the surface of these hypersphere, and 0 otherwise. 232 00:23:47,150 --> 00:23:51,410 This means that the set of velocities 233 00:23:51,410 --> 00:23:57,400 is a random vector on the surface of the 2N-dimensional hypersphere. 234 00:23:58,170 --> 00:24:07,070 Fortunately, we just became expert in the subject of sampling a random point on the surface of a hypersphere, 235 00:24:07,720 --> 00:24:12,850 using 2N independent Gaussian random numbers. 236 00:24:14,160 --> 00:24:22,140 Remember that the algorithm direct_surface.py involves a rescaling of velocities. 237 00:24:23,110 --> 00:24:33,550 but that this rescaling became unnecessary in high dimension if the variance of the Gaussian was chosen correctly. 238 00:24:34,700 --> 00:24:41,670 In our case, we have to choose each component v_x or v_y 239 00:24:41,670 --> 00:24:52,090 according to a probability pi(v_x) proportional to exp(-v_x^2 / (2 sigma^2)) 240 00:24:53,490 --> 00:25:03,390 where sigma = sqrt((2/m) (Ekin / 2 N)) 241 00:25:03,390 --> 00:25:16,700 or more generally in d dimensions rather than two dimensions: sqrt((2/m) (Ekin / dN) ) 242 00:25:17,410 --> 00:25:28,020 Look here, at the distribution of the radius r in the algorithm direct_surface.py before rescaling 243 00:25:29,180 --> 00:25:36,710 and see that it becomes sharper and sharper with increasing particle number right at the correct value 244 00:25:37,710 --> 00:25:45,460 this means that each velocity component is a Gaussian random number. 245 00:25:46,690 --> 00:25:56,040 This is the Maxwell distribution. It is really famous, yet we have derived it by ourselves. 246 00:25:58,570 --> 00:26:08,550 The total kinetic energy divided by d N is the mean energy per degree of freedom 247 00:26:09,360 --> 00:26:13,500 and it is equal to one half kB times T 248 00:26:13,500 --> 00:26:18,410 where T is the temperature in Kelvin 249 00:26:18,410 --> 00:26:22,410 and kB is the famous Boltzmann constant. 250 00:26:22,410 --> 00:26:32,680 We find that the variance sigma^2 of the Gaussians describing the velocity distribution in our system 251 00:26:32,680 --> 00:26:39,470 is given by sigma^2 = kB T / m 252 00:26:39,470 --> 00:26:44,530 And we finally arrived at the probability distribution 253 00:26:44,530 --> 00:26:55,010 of each velocity component of each particle, pi(v) proportional to exp(-v^2) 254 00:26:56,360 --> 00:27:04,890 Now remember that the velocity of each particle was given as v = vx,vy 255 00:27:06,050 --> 00:27:13,360 so the velocity distribution of the absolute value of the velocity 256 00:27:13,360 --> 00:27:24,010 is given by pi(v) proportional to v exp(-v^2) 257 00:27:24,740 --> 00:27:33,420 because the volume element dvx dvy can be written as 2 pi v dv 258 00:27:35,330 --> 00:27:43,360 In three dimensions, we do the same thing with vx,vy,vz 259 00:27:44,570 --> 00:27:51,400 the probability distribution for each component is unchanged: it is a Gaussian 260 00:27:51,880 --> 00:28:03,640 but the probability distribution of the absolute value of the velocity is now proportional to v^2 exp(-v^2), as shown here 261 00:28:05,050 --> 00:28:09,860 We can compare the Maxwell distribution that we just derived 262 00:28:09,860 --> 00:28:17,960 with the simulation results obtained by event_disk_box.py in the second week 263 00:28:17,960 --> 00:28:27,550 and check that the histogram of each velocity component of an individual particle is a Gaussian 264 00:28:27,550 --> 00:28:31,550 The probability distribution is like exp(-v^2) 265 00:28:33,040 --> 00:28:43,300 Furthermore, we can check that the probability distribution of the absolute value of the velocity of each particle 266 00:28:43,300 --> 00:28:49,490 is also given by a Gaussian: v exp(-v^2) 267 00:28:49,490 --> 00:28:57,740 So we find that in order to introduce velocities into the Monte Carlo scheme 268 00:28:57,740 --> 00:29:07,700 we have to sample independently from the particle positions the velocities from a Gaussian with some rescaling 269 00:29:08,200 --> 00:29:19,550 We thus established the complete symmetry between the molecular dynamics approach and the Monte Carlo sampling approach. 270 00:29:24,290 --> 00:29:28,550 So finally let us look again at the Maxwell distribution 271 00:29:28,550 --> 00:29:34,940 which has the form exp(- energy / kB T) 272 00:29:34,940 --> 00:29:39,220 this is for one particle and one component 273 00:29:39,220 --> 00:29:47,530 but it also applies to a subsystem of particles inside a larger simulation box, 274 00:29:47,530 --> 00:29:55,470 for example cut-off by boxes that allow exchange of energy and momentum. 275 00:29:56,710 --> 00:30:11,710 Such a subsystem has energy E that may vary and the probability distribution pi(E) is proportional to exp(-E / kB T) 276 00:30:13,030 --> 00:30:23,880 We have arrived at the Boltzmann distribution, the second pillar of statistical mechanics, and we have done it all by ourselves. 277 00:30:28,430 --> 00:30:36,300 In conclusion we have studied in this fourth lecture of Statistical Mechanics: Algorithms and Computations 278 00:30:36,300 --> 00:30:41,830 the intricate relationship between sampling and integration 279 00:30:43,160 --> 00:30:50,670 the change of variables in integration was mirrored in the process of sample transformation 280 00:30:52,440 --> 00:31:01,350 The most important such change of variable or the most important sample transformation 281 00:31:01,350 --> 00:31:09,060 appeared in multi-dimensions from Gaussians to an isotropic distribution 282 00:31:09,060 --> 00:31:17,850 This magnificent relationship between Gaussians and isotropic distributions 283 00:31:17,850 --> 00:31:22,560 was very well understood by the founders of statistical mechanics 284 00:31:22,560 --> 00:31:29,660 and it produced the Maxwell distribution, and by generalization the Boltzmann distribution 285 00:31:29,660 --> 00:31:36,760 It is now time for you to study the simple programs that come with this lecture 286 00:31:36,760 --> 00:31:42,730 the math is simple and the concepts are clear 287 00:31:42,730 --> 00:31:46,730 yet the implications are far-reaching. 288 00:31:48,500 --> 00:31:56,420 What we discussed here will be considerably deepened in this week's tutorial and homework sessions. 289 00:31:58,390 --> 00:32:08,080 Next week we'll take out first steps into the field of quantum physics and quantum statistical mechanics. 290 00:32:09,430 --> 00:32:15,590 In the meantime, have fun with tutorial 4 and homework session 4. 291 00:32:16,260 --> 00:32:22,300 And see you again next week at Statistical Mechanics: Algorithms and Computations.