WEBVTT 00:00.000 --> 00:12.800 So, today we're going to talk about microstates, multiplicity and macrostates. 00:12.800 --> 00:22.880 Now, we've talked about reversible or irreversible before. 00:22.880 --> 00:31.200 And I'd like you to have a look at this video. 00:31.200 --> 00:36.120 Is this a possible scenario? 00:36.120 --> 00:42.120 Or what is it that seems impossible to us? 00:42.120 --> 00:52.600 In this case, we've already done the simulations in molecular dynamics of these particles and 00:52.720 --> 00:56.760 we know that they are following Newton's law of motion. 00:56.760 --> 01:02.760 We know that these are reversible dynamics. 01:02.760 --> 01:08.600 So this is, in fact, a possible evolution. 01:08.600 --> 01:13.000 These are following the Newton's laws of motions. 01:13.000 --> 01:22.000 They just started in a very particular configuration that works backwards into a very special state. 01:23.600 --> 01:30.600 On the other hand, we know perfectly well that something like this never happens. 01:30.600 --> 01:38.600 So, for one body, it is possible to have reversible motion. 01:38.600 --> 01:47.600 It is possible for my son to be shot out of the water and back up to Pantan, although it's very unlikely. 01:48.600 --> 01:57.600 However, for multi-body systems, like all the water particles, and for the particles in this simulation, 01:57.600 --> 02:07.600 we know that the reversible laws of motion make it possible for all beads to move to one side. 02:07.600 --> 02:15.600 But we know that the irreversible collective dynamics tells us that it will never happen. 02:16.600 --> 02:21.600 So, we have a notion of the macroscopic definition of equilibrium. 02:21.600 --> 02:31.600 In the case of this simulation here, it's homogeneous particle distributions. 02:31.600 --> 02:36.600 But what is a good microscopic definition of equilibrium? 02:36.600 --> 02:39.600 That's what we're going to look at today. 02:40.600 --> 02:47.600 Now, I'm going to introduce a new concept, microstates and macrostates. 02:47.600 --> 02:51.600 So, let's take one example. 02:51.600 --> 02:59.600 Let's toss three coins that can give us heads or tails. 02:59.600 --> 03:06.600 And we'll ask the question, what is the most likely outcome of tossing these three coins? 03:07.600 --> 03:15.600 Well, these exercises you've been doing since early on in your school years. 03:15.600 --> 03:21.600 And you know that, so, but we're going to put some new words onto it. 03:21.600 --> 03:26.600 I'm going to call a microstate the... 03:27.600 --> 03:31.600 Now, there's my pointer. 03:31.600 --> 03:37.600 A microstate is the state of all the coins. 03:37.600 --> 03:45.600 So, each coin has a state Si, it should be a subscript, S and subscript i. 03:45.600 --> 03:51.600 So, if it's head, Si is one. If it's tails, Si is zero. 03:52.600 --> 03:57.600 Now, we know that if we toss good coins, 03:57.600 --> 04:04.600 all the... there's the same probability of getting heads and tails. 04:04.600 --> 04:09.600 This means that all the microstates are equally likely. 04:09.600 --> 04:14.600 Now, we're going to call a macrostate, in this case, 04:14.600 --> 04:20.600 the sum of all... the sum over all the microstates. 04:20.600 --> 04:25.600 So, this can mean the possible macrostates we can have, 04:25.600 --> 04:32.600 zero, one, two or three, meaning all tails, all heads, 04:32.600 --> 04:38.600 or one heads, the rest tails, two heads, the rest tails. 04:38.600 --> 04:42.600 So, which is the most likely macrostate? 04:42.600 --> 04:49.600 Well, we can start by just listing all the microstates. 04:49.600 --> 04:53.600 So, macrostates sum equals zero. 04:53.600 --> 04:59.600 Well, there's only one. All of them are tails. 04:59.600 --> 05:02.600 Macrostates n equals one. 05:02.600 --> 05:06.600 Well, there are three different possibilities for each of these coins. 05:06.600 --> 05:13.600 They can be either heads or tails, and there are three possibilities. 05:13.600 --> 05:18.600 The same for macrostate two, two heads. 05:18.600 --> 05:20.600 There are three different possibilities. 05:20.600 --> 05:24.600 For three heads, there is just one single possibility. 05:24.600 --> 05:28.600 All of them fall down on heads. 05:28.600 --> 05:34.600 So, when we are... which is the most likely macrostate? 05:34.600 --> 05:40.600 Well, we see that there are eight different macrostates... 05:40.600 --> 05:43.600 there are eight different microstates. 05:43.600 --> 05:47.600 So, there are two to the third possible microstates, 05:47.600 --> 05:54.600 and the probabilities of each of the macrostates 05:54.600 --> 06:04.600 is the number of different microstates giving this macrostate. 06:04.600 --> 06:08.600 So, for zero, the probability is one eighth. 06:08.600 --> 06:11.600 For n equals one, it's four eighths. 06:11.600 --> 06:13.600 n equals two, it's four eighths. 06:13.600 --> 06:16.600 n equals three, it's one eighth. 06:17.600 --> 06:19.600 So, you've done this several times, 06:19.600 --> 06:23.600 but now we've just put the new names micro and macrostates onto it. 06:25.600 --> 06:30.600 Now, this new concept, microstate and macrostate, 06:30.600 --> 06:37.600 we're going to use it for the simulations that we've done before. 06:37.600 --> 06:42.600 Here, we've got the two-sided box, 06:42.600 --> 06:46.600 where, with the algorithmic model, we are switching from... 06:46.600 --> 06:49.600 we're saying it can go to the left or the right, 06:49.600 --> 06:54.600 and over time, we know that we get the equal distribution. 06:54.600 --> 07:00.600 So, we start with all the particles on one side, 07:00.600 --> 07:07.600 and then we end up with a fluctuation around having the number of particles 07:08.600 --> 07:12.600 equally distributed on both sides. 07:12.600 --> 07:17.600 So, now we're talking about microstates. 07:17.600 --> 07:22.600 Well, we've done this with different models for molecular dynamics, 07:22.600 --> 07:25.600 the ideal gas. 07:25.600 --> 07:32.600 Then, a microstate is a particular set of positions and velocities. 07:32.600 --> 07:35.600 The problem with this is that these are not countable, 07:35.600 --> 07:38.600 so we're not going to use them now. 07:38.600 --> 07:44.600 In the case of a random walk, we can count them if they are in a grid. 07:44.600 --> 07:48.600 So, then the microstates is the set of all possible... 07:48.600 --> 07:54.600 is the set of positions that are being used. 07:54.600 --> 07:58.600 For the algorithmic model, 07:58.600 --> 08:03.600 then the single particle states are like in the coin problem. 08:03.600 --> 08:07.600 Then, S i equals 1 is, for example, having... 08:07.600 --> 08:12.600 is for a particle i being on the left side. 08:12.600 --> 08:17.600 S i equals 0, pi, the particle is on the right side. 08:19.600 --> 08:27.600 So, the microstate in this time is the number of particles on the left-hand side. 08:27.600 --> 08:32.600 It's the sum over all the microstates, 08:32.600 --> 08:35.600 just as in the example of the coins. 08:35.600 --> 08:41.600 And the possible microstates is from 0 to the total number of particles. 08:41.600 --> 08:46.600 So, let's try to count it. 08:46.600 --> 08:49.600 We'll just make an example like with the coins. 08:49.600 --> 08:51.600 Here we've got five particles. 08:51.600 --> 08:55.600 I've given them different colours because they are... 08:55.600 --> 08:57.600 we can discern them. 08:57.600 --> 09:01.600 They are like the coins. 09:01.600 --> 09:04.600 They are distinguishable, we say. 09:04.600 --> 09:07.600 And this box has a left and a right side. 09:07.600 --> 09:11.600 We've got five particles. 09:11.600 --> 09:14.600 We can distinguish them, so we can label them, 09:14.600 --> 09:17.600 particle 1 to 5. 09:17.600 --> 09:21.600 And we've got the particle states, 09:21.600 --> 09:26.600 and the macrostates is the sum over the microstates. 09:27.600 --> 09:33.600 Now, let's list the possible microstates of macrostate n equals 1. 09:33.600 --> 09:43.600 So, there are five different ways we can have macrostate n equals 1, 09:43.600 --> 09:45.600 which is given here. 09:45.600 --> 09:48.600 These are the five particles, 09:48.600 --> 09:54.600 and they can be represented of green 0, blue 0, red 0, yellow 0, 09:54.600 --> 09:58.600 and violet 1, and so on. 09:58.600 --> 10:07.600 So, there are five microstates for macrostate n equals 1, 10:07.600 --> 10:13.600 and we will now use the symbol omega of small n, big N, 10:13.600 --> 10:16.600 for the multiplicity, 10:16.600 --> 10:21.600 the number of multiplicity of microstate n equals 1, 10:22.600 --> 10:24.600 is 5. 10:24.600 --> 10:31.600 So, for where small n is the macrostate, 10:31.600 --> 10:36.600 n is the number of particles, 10:36.600 --> 10:41.600 and in this case, omega of 1, 5 equals 5. 10:41.600 --> 10:44.600 The multiplicity is 5. 10:45.600 --> 10:52.600 So, let's look at the multiplicity of all the different macrostates. 10:52.600 --> 10:56.600 So, we've got macrostate n, 10:56.600 --> 11:00.600 and on the other side there is big N minus n particles. 11:00.600 --> 11:05.600 So, for n equals 0, there's just one possibility, 11:05.600 --> 11:07.600 multiplicity is 1. 11:07.600 --> 11:10.600 For n equals 1, that's what we just counted, 11:10.600 --> 11:13.600 we've got multiplicity of 5. 11:13.600 --> 11:16.600 n equals 2, the multiplicity is 10. 11:16.600 --> 11:20.600 n equals 3, the multiplicity is also 10. 11:20.600 --> 11:24.600 n equals 4, the multiplicity is 5, 11:24.600 --> 11:27.600 and here the multiplicity is 1. 11:30.600 --> 11:36.600 Now, from this we can, 11:36.600 --> 11:41.600 and I will make a small video later showing this more clearly, 11:41.600 --> 11:45.600 we can have a general formula for the multiplicity. 11:47.600 --> 11:53.600 So, the multiplicity defined as we said here for a macrostate n 11:53.600 --> 11:56.600 with number of particles N, big N, 11:56.600 --> 12:00.600 is the large N faculty 12:00.600 --> 12:04.600 divided by large N minus small n faculty 12:04.600 --> 12:08.600 times small n faculty. 12:09.600 --> 12:14.600 So, if we take N25, 12:14.600 --> 12:18.600 I see that I've made an error here, 12:18.600 --> 12:21.600 it should say 25, 12:21.600 --> 12:24.600 here it should say 3545, 12:24.600 --> 12:26.600 so I just messed it up a little bit. 12:26.600 --> 12:29.600 So, here it should say 25, 12:29.600 --> 12:33.600 N25 we see is 10 when we, 12:33.600 --> 12:35.600 whoops, 12:36.600 --> 12:42.600 and this we get from the general formula. 12:44.600 --> 12:47.600 So, the number of possible microstates, 12:50.600 --> 12:55.600 the total number of possible microstates is 2 to the 5, 12:55.600 --> 12:57.600 it's 32. 12:57.600 --> 13:01.600 So, we can actually then calculate the probabilities 13:01.600 --> 13:03.600 like we did with the coins. 13:03.600 --> 13:06.600 The probability of each microstate, 13:06.600 --> 13:08.600 macrostate, 13:08.600 --> 13:13.600 is the multiplicity 13:13.600 --> 13:16.600 divided by the total number of microstates. 13:16.600 --> 13:23.600 So, now we have a formula for the probability of a macrostate. 13:26.600 --> 13:29.600 So, to sum up the multiplicity of a system 13:30.600 --> 13:33.600 with large N particles in a macrostate 13:33.600 --> 13:38.600 with N particles on the left-hand side 13:38.600 --> 13:44.600 is given by this formula of multiplicity. 13:45.600 --> 13:51.600 The total number of microstates is 2 to the N, 13:52.600 --> 13:55.600 and the probability of a macrostate 13:55.600 --> 13:58.600 with N, small n on the left side, 14:00.600 --> 14:03.600 is given by this formula. 14:03.600 --> 14:08.600 So, the conclusion about that of our initial question, 14:08.600 --> 14:10.600 what is the equilibrium? 14:10.600 --> 14:16.600 So, macroscopically we said it's a homogeneous distribution. 14:16.600 --> 14:21.600 Now we've actually got a microscopic rule for this. 14:21.600 --> 14:24.600 It's the most likely macrostate. 14:24.600 --> 14:31.600 So, most likely means the macrostate with the highest multiplicity 14:31.600 --> 14:34.600 and therefore the highest probability.