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Of course we have a nice netlogo model to illustrate our slot machine. This was kindly written for us by our TA,
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John Balwit. If I click on reset, it shows me the slot machine, and, we have our three windows, with our fruits,
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and I can set the number of pulls that I want---I'm going to set it to one, to start out with, and click on
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"Pull Lever N Times". The lever is pulled once, we get a new microstate, pull it again---oh, we get three
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of a kind there, very lucky! Ok, so, we can set the macrostate we're interested in. I'm going to set it
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as three of the same kind, ok... and, I can ask "how many times will I be likely to see that in, say, 1000
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pulls". Well, let's figure that out, and I'm going to do that by writing a note. Ok, so my note is going to be, say,
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"Probability of seeing macrostate 'three of the same'"---well, we said that there were five microstates that were included
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in that macrostate, that is, cherry cherry cherry, lemon lemon lemon, et cetera et cetera, and there were 125
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possible microstates, so that probability is going to be equal to 5 divided by 125---put that all on one line---
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five divided by 125, which is equal to .04. So that's the probability, if we pull the lever once, that we'll
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see three of the same. Four percent. Therefore, the expected number of times we'll see our
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macrostate in 1,000 pulls, that's going to be equal to the probability for one pull times the number of pulls,
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which is equal to 40. So let's see how close we are---that's expected, of course there's some randomness
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here. So I'm going to reset, and pull the lever 1,000 times. And, we can speed it up a little
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bit, perhaps... and we're seeing---it slows down a little bit, when it hits a jackpot, so we can
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see the jackpot---but you can see that, here's our number of times its seeing the goal macrostate---
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three of the same kind---and here's the number of times it's seeing the non-macrostate---
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this is like the "win" macrostate and the "loose" macrostate---and so
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we're experimentally verifying our theory, which says that we'll see it about 40 times---well, 46.
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Of course, if we run it again and again and again and averaged them all
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hopefully they would come out close to 40. So you'll be able to use this
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to do some of the homework problems, which involves some of other kind of macrostates,
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which we'll look at a little bit later. Now let's bring all this back to our discussion of
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entropy and statistical mechanics. So, remember our netlogo model, the two gas model,
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where we had two rooms, and one room contained the slow particles, and the other room contained the
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fast particles, and we opened up a gap, and they started to mix, and they started to mix,
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so we got this at the start, and this and the finish, and we said that the entropy here was lower
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than the entropy here, that is, by the second law of thermodynamics, entropy
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increased. Well, in our new language, of microstates and macrostates, we can say that
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the microstate of the system is the position and velocity of every particle---that's kind of like our
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position and identity of each of the fruits of the slot machine. Here, we have one macrostate---
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all fast-moving particles on the right, and all slow particles are on the left---and over here
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we have another kind of macrostate---fast and slow particles are completely
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mixed. Well, if you think about it, our macrostate on the left hand side corresponds to fewer
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possible microstates than the macrostate on the right hand side---that is, there is more ways that different
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particles could be arranged, in terms of position and velocity, to create a completely mixed,
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fast-and-slow particle macrostate, than there are ways in which you can create this more ordered
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macrostate. So here, on this side, of course, there's many different ways in which the blue particles
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could be individually arranged and the red particles could be individually arranged in order for
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all fast particles to be on the right and all slow particles on the left, it's just that
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there's fewer such arrangements than there are arrangements in which they're
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all mixed. So there's many different places this little red particle could be or this little blue
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particle could be so that these are all mixed---and that's true of all of the different particles.
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So that is the statistical mechanics notion of higher and lower entropy, and it
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corresponds very well with our intuitive notions of "more disordered" and "more ordered"
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states. This gives us a new way to state the second law of thermodynamics. First, our
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original way we said that in an isolated system, entropy will always increase until
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it reaches a maximum value, but now, we can look at the statistical mechanics version of
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the second law, which says that in an isolated system, the system will always
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progress to a macrostate that corresponds to the maximum number of microstates.
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Well, Boltzmann's definition of entropy is conveniently engraved on his tomb
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in Vienna, so noone ever has to forget it, and his definition says, the entropy S
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of a macrostate is some number k times the natural logarithm---that's this "log",
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the "natural logarithm"---of the number W of microstates corresponding to that
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macrostate. Well, k is called Boltzmann's constant---the constant and the logarithm are just
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for putting the entropy in particular units. So, you could really look at it as
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S equals W---Boltzmann's entropy equals, or is proportional to, in some sense, the number
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of microstates corresponding to the macrostate. So entropy is a measure of a macrostate,
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and it measures how many microstates correspond to that macrostate. So the
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general idea is that, the more microstates give rise to a macrostate, the more probable that
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macrostate is. So our slot machine, the macrostate of "loose" was much more
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probable than the macrostate of "win", and we saw that many more microstates
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corresponded to the "loose" macrostate than the "win" macrostate. Intuitively, high
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entropy just means a more probable macrostate. Or, given our gas example, it's
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much more probable that, if the door is open here, the molecules will mix, than that they'll
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just stay or re-arrange this state in which all the fast ones are on the right and
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all the slow ones are on the left. It's much more probable that they'll be mixed, and be in this state,
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than in this state, so we say that this state has higher entropy than this state.
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We can now do a final restating of the second law of thermodynamics, using
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our statistical mechanics terminology, and we can say that, in an isolated system,
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the system will tend to progress to the most probable macrostate. Well, this may seem like
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a tautology, but actually, it's one of the most profound ideas in all of physics, and it gives meaning
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to the notion of "time". You'll find some optional readings suggested on our Course Materials page
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that delve much more deeply into this idea than I have time for in this course.