Program 30 - "Entrophy and Chaos"

MusicNone of these recipes were as good as my own.I can't tell you what's in my recipe because I don't want to spill the beans.But, part of the secret is to start with a variety or mixture of beans.(Beans falling.)Uh.MusicI'm really glad I learned how to reverse entropy.MusicSilico: "We are back with Science 122, the Nature of Physical Science."This is the only telecourse that shows you howto order chaos and disorganize order.This is Program 30, Entropy and Chaos."Before we're done with this program we will havediscovered why conservation of energy is incomplete,and we will have reviewed the history of the concepts of heat and energy.We will have seen how two individuals, Sadi Carnotand Rudolf Clausius, conceived of the concept of entropyand formulated the laws of thermodynamics.

Our program will have taken us through the strange worldof entropy and its relationship to efficiency,order, probability, and kinetic theory.By that time we'll have needed a rest,before we plunge headlong into chaos.Here are the objectives for today's lesson.These objectives are also in the Study Guide at the beginning of the lesson.Before you begin to study the lesson, take a few minutesto read the objectives and the study questions for this lesson.Look for key words and ideas as you read.Use the Study Guide and follow it as you watch the program.Be sure to read these objectivesin the Study Guideand refer to them as you study the lesson.Focussing on the learning objectives will help youto study and understand the important concepts.Compare the objectives with the study questions for this lessonto be sure that you have the concepts under control.

As any dog or cat owner knows, it's much easierto destroy something than it is to build it.Dogs are particularly good at one, but entirely ineffective at the other.Cats are about the same, although they might go about it in different ways.So, what's this got to do with physical science,with the nature of physical science?Easy.The answer is, "Everything!"Cars rust, pendulums run down, people get old,things like this don't happen backwards.Like the flow of time, they only happen in one sequence.That's frontwards.So, the title of this program is "Entropy and Chaos,"but we really want to approach this throughthe concept of thermodynamics.But we wanted to sort of look at some questions.For example, are chaos and disorder the same thing?These are words that we use everyday in our vocabulary,or we have an image of them, and we might thinkthat chaos and disorder are the same.But, I want to ask the question to sort of set the stagefor this program as we wander through here."Is there such a thing as structured disorder?And can a system be constrained, yet be unpredictable?"These questions and others follow from a flawin the law of conservation of energy.We'll examine that flaw in some detail in a few minutes.Meanwhile, we have to study thermodynamics.

Thermodynamics comes from the words for heat and dynamics."Thermo" means heat; "dynamics" means motions and forces.So, why do we want to study thermodynamics?First of all, it's a really nice sounding word,and secondly, it's because the idea of conservation of energyis a good theory, but it's incomplete,because it can't explain the irreversibility of some energy transformations.And this question of irreversibility leads us to a lot of other neat things.If fact, we'll see that entropy is really a measure of disorderwhich has to do with the transfer of heat, and then we'll refinethis and see that chaos is a state of organized disorder.

Now the material we want to cover here today is sortof on the cutting edge of things, and so some of it may seemto be rather strange and rather weird.But, after all, it is our last program, so we can get a littlebit weirder with some of the material on this one.So what do you suppose it means when we say thatconservation of energy is incomplete?Incomplete, of course, means there's something missing.What could be missing from conservation of energy?We've already seen that you can take the energy that disappearsfrom the pendulum and show that it appears as heat.We've already seen that you can interconvert all these kindsof energy back and forth from one to the other,so what could possibly be missing?I'll tell you what could be missing.There are certain events like the high diver which conserve energy.

The diver, for example, does work to climb to the topof the platform and has potential energy while he's on the platform.The potential energy is converted to kinetic energy during thedive, and the diver does work on the water as he loses energy upon impact.So, the kinetic energy of the diver is converted to the motion of the water.The energy of the water increases.The energy of the diver decreases, and everything seems fine.No violation of the conservation principleis noted one way or the other.Though watch the water.The water is disturbed.And you'll notice that the water begins to settledown after the diver enters the water.

Well, no conservation principle is violated here because the waterwill eventually lose its energy as it...Oh, what's happening here?Wait a minute!What prevents this from happening?The wave energy is now used to raise the diver's potential energy.The energy of the water decreases.The diver's hurled out of the water to land back on the platform.No violation of conservation of energy is noted here either.So, what's missing from the conservation principle?Certain kinds of processes do not occur although they arepermitted by conservation of energy.And what's missing is any kind of a process which looks funnywhen played in reverse simply doesn't happen.Have you ever seen this happen to a diver?I've never seen this happen.Well, here we go.The water throws him out, back up onto the board.The diver gains and loses potential energy as he bounces and therehe is, back up there with high potential energy.So, why doesn't this happen?I think we have to take a closer look.So, before we get into the ideas of thermodynamics and entropyand chaos, we need to very briefly review the history of heat and energy.We've already covered this material in earlier programs,but we can kind of compress it for you here I think.

So, the idea is that the concepts of work and energy were notyet clarified by Newton's death, and this proceeded after Newtonhad died, essentially worked out by the late 1700s.The basic facts about heat and temperature were known by the early 1800s.For example, that heat flows from hot to cold.Black had already come up with his concept of specific heatand latent heat, and most of the gas laws were in place,all except the law of Gay Lussac.The atomic theory, remember, began in the early 1800swith Dalton, and the kinetic theory was popularized in the mid1800s by Joule who was a student of Dalton.

OK, so, the gist of this that heat as a form of energy is a modernconcept, and that it used to be thought to be a form of matter.You remember the idea of caloric.And so the idea of the mechanical equivalent of heat as put forthby Joule was a significant step, and this happened in the mid 1800s.OK.So the idea of conservation of energy, then, pretty much cameinto being by Joule and Mayer in 1840 or thereabouts.And the idea was that energy can be transformed to do work.The work energy theorem, remember.That the total amount of energy in a closed system does notchange, but energy can be converted back and forth from one to the other.There's one more thing I have to mentionabout the history of heat and chemistry here.If you're looking in the textbooks, physicists used the symbol "Q"for heat, and chemists used the symbol "H" for heat.So, sometimes you'll see equations involving the conceptof heat with a "Q," sometimes you'll see it with "H."For this program, we'll use "Q."Don't be confused by this.It's just physicists and chemists come from different sidesof the physical science spectrum, so they invented their own symbols.

One of the first players in our game of thermodynamicswas a young fellow named Sadi Carnot.I say he was a young fellow because you can see from hisbirth date that he only lived to 36 years old.Much of this was due to the fact that he was a genius and wasso far ahead of his time and his ideas were ignored thathe just couldn't deal with things and committed suicide.What he did was to mathematically model the ideal heat engine.Now, you know, we've talked about the ideal thing before, right?Galileo was the first one to bring this concept of the ideal systeminto being by eliminating friction mentally.What Carnot did was to take the gas laws and by using theexpansion and contraction of the ideal gases,by holding at sometimes, holding the temperature constantand sometimes holding the pressure constant,came up with what's today called in physics, the Carnot cycle.That doesn't concern us, exactly how he did this.What we want to be concerned with is that he was ableto show us how to calculate the maximum efficiencythat any heat engine could run under.

A heat engine, of course, is any kind of a motor or enginewhich transforms heat into mechanical energy.So, what he did was to show that heat engines cannot operateat 100% efficiency--never operate at 100% efficiency.Let's go look at a picture and we'll see what this has to do with anything.So, here's what's happening.What we're seeing is that heat flows from hot to cold.We already know this.So, here we have two reservoirs of temperature and thesereservoirs may be, for example, the gasoline engine in your car.Could be the hot reservoir and the colder air outsidethe engine could be the cool reservoir.Or it could be the steam in a steam engineand the water vapor that the steam condenses to.But in any sense, heat flows from the hot object into the cold object.And what the heat engine does is to take some of that heat,some of the energy, and output it as work.

OK.So what that means is that the amount of work that's donemust be equal to the difference between the amount of heatthat leaves the hot object and the amount of heat that enters the cold object.Doesn't that make sense?"QH" leave here and work must be equal to the differencebetween those two because that's what's taken out.So, the work done is maximum when thetemperature of the cold reservoir is zero.Obviously, if the number "QC" up here is zero, then work equals "QH."What does that mean in terms of the system.If the work here equals "QH" here, then noneof the energy flows into the cold reservoir.And we know that energy has to flow from hot to coldin order to power the heat engine.So the maximum efficiency would be the idealized condition wherethe cold reservoir was at absolute zero.What Carnot actually was able to show is that the relationshipbetween the heat transfer, between the hot and the cold,is in the same ratio as the differencein temperature between the hot and the cold.So what that means is that in order for heat to flow into,no heat to flow into the cold reservoir, the cold reservoirwould have to be at absolute zero.Now, this is an interesting thing.Because nobody ever thought about this kind of thing before.

Nobody ever thought about a maximum efficiency.So, the next player in this is a man named Clausius,and let's turn our attention now to Clausius.So, Rudolf Clausius was actually regarded asthe founder of thermodynamics.Oh, Rudolf, those eyes, look at those eyes.What Clausius really did was to use the work of Carnotand he was first to really explicitly state the laws of thermodynamics.We're not going to go into great detail about Clausiusand his biography and that sort of thing, because we don't havethe time to do that, but the important thing is that it wasClausius who actually took Carnot's work,who had been ignored in his own time,and mainly what Clausius did, in addition to the lawsof thermodynamics was to introduce the concept of entropy.

Entropy.What is entropy, anyway?Entropy was invented by Clausius as a wayto quantify spontaneous processes.Quantify spontaneous processes?Let me explain what I mean by that.Certain things happen in certain ways.Right?Things happen from start to finish and those things that we saw,like the diver, don't happen in reverse.So, is there a reason why things like, happened in that way,like heat only flows from hot to cold instead of from cold to hot?What Clausius did was to define this concept of entropy simplyas a ratio of heat transferred to Kelvin temperature.In other words, a ratio of heat to temperature.He defined it in such a way that entropy always changes whena spontaneous process happens, but it always changes to increase.So the spontaneous process will always increase entropy.I think this will work best if we look at some examples of this.

OK.So, let's look at a simple example.Example of simple heat transfer from a hot object to a cold object.So, here we have a model for this.We have a hot object that I've labeled with TH for hottemperature, a cold object I've labeled TC for cold temperature,QH is the heat flowing out of the hot objectand QC is the heat flowing into the cold object.It's fairly obvious from this, I think, that since there's no workbeing done that QH must equal QC.But remember, conservation of heat.So, let's put some numbers in this to make sense out of it.OK, so let's say that we start with hot temperature of 400 Kelvinsand a cold temperature of 300 Kelvins.Notice I'm using the absolute temperature scale here.And let's say, just for the purposes of this example,that the amount of heat that flows from one object to the other is 1200 joules.Let's see what this does to entropy when we define the entropy.So we want to define entropy now as the, I'll use the symbol "S"here for entropy, so the entropy change in the hot object is itsheat loss divided by its temperature.The entropy change for the cold object is its heatgain divided by its temperature.So, the total entropy change can be simply divided out.

Notice I put a minus sign here for the temperature changeof the hot object because it's losing and the ratiois 1200 joules, that's the amount of heat lost,divided by the temperature at 400 Kelvins.Same for the cold object, except that its entropy change ispositive, because it's gaining heat, so it's gaining 1200 joulesand its temperature is 300 Kelvins.So the numbers work out very nicely.1200 divided by 400 is minus 3, so the changein entropy is minus 3 joules per Kelvin.For some reason the "K" is missing there, but we won't let that bother us.The entropy change of the cold object is plus 4 joules per Kelvin.So, what's the total entropy change?The total entropy change for the system must bethe sum of the two individual changes.So, the sum of the two individual entropy changes is simply thetotal entropy of the system, which is the entropy of the hot object,plus the entropy of the cold object, which is minus 3 joulesper kilogram, plus 4 joules per kilogram.In other words, the total entropy change is one joule per kilogram.Right?In other words, it's a positive change when heat is transferred100% from the hot object to the cold object.

So, what's happening here is what we can identify as the laws of thermodynamics.I've listed these as laws 2, 1 and 0.What's happening here is that first of all, heat is flowing from hot to cold.Secondly, the amount of heat that flows from the hot is equalto the amount of heat that flows into the cold.In other words, energy is conserved, and the second thingis that entropy increases in this particular interaction.The total entropy change for this interaction is 1 joule per kilogram.So the laws of thermodynamics were actually formulatedspecifically to explain this lack of irreversibility in energy transformations.They're based on the observed properties of heat.They're empirical laws, their inductive laws.They have a very wide range of applicability as it turns out.And most of us today in the, near the beginning of the 21stcentury believe that the significance and applicationsof these laws are really not yet fully understood.

It's also interesting that these laws can be stated in many different terms.They were developed originally through a veryhigh degree of mathematical sophistication.Don't be scared, we don't have to do that sophistication.That mathematical sophistication is actuallybeyond the level of math of the casual observer.But it does have the power to describe and explain many situations.What it did was to lead to the design and constructionof energy transformation machines like the gasolineengine, the steam engine, and various other kinds of energy transformations.

The theory of thermodynamics or the laws, actually didn't becomea coherent theory until near the end of the 19th century.Around the same time the electron was discovered and all theseradioactivity and x-rays and all these things were happening.It required the development of statistical mechanics.That was what was used in the kinetic theory,and the atomic theory to be developed first.In other words, people had to understand how those things work.It turns out that even though the principles werederived in the first place with fairly sophisticatedmathematics, the concepts are very simple.And they're understandable without sophisticated mathematics.

The problem is, of course, that sometimes simple concepts areso foreign to our paradigms, that we have a hard time understanding them.So, what I want you to do here is to drop your paradigm.Just take your paradigm off and throw it on the bedpostand just let your mind sort of ride with this and we'll see ifwe can't make some sense out of it.The first law is actually the Zeroth Law.Don't be confused by this.This is simply the first law can be zero, like the end of a ruler is the zero end.OK?

The first law is simply a statement of something you already know.That heat only flows spontaneously from hot to cold.This is actually called the Zeroth Law because it was added as anafterthought to the other laws, when people recognized that,"Yeah, this is an important principle of heat,that we should put into the laws here."So we already know this.We also already know the First Law.The First Law is actually the work energy theorem,or a statement of conservation of energy.And we know already that energy in many forms can betransformed or transferred between objects.In fact, work is a mechanical processby which energy is transformed or transferred.

Notice what I just said.I said work is a mechanical process by which energyis transformed or transferred.Before we defined work in terms of energy, where now we'redefining energy in terms of work.So, we're taking the attitude that energy is more important thanwork, so we're seeing work as a mechanical processby which energy is transferred.So, another way to state these laws is, of course, that thetotal energy in the universe is constant, assuming that theuniverse is a closed system, that no energy escapes the universe.That's an awesome concept if you think about it.OK, another way to say it.That the efficiency, I'm sorry, that the increase in energyin one place results in a decrease some place elsebecause if the universe is closed, then you can't create energy from nothing.Right?We can also note that this is the same way of saying thatefficiency can't be greater than 100%.In other words, you can't get out more than you put in.In other words, you simply can't get somethingfor nothing, where energy is concerned.OK.

The Second Law.The Second Law even has more different ways to say it,and the fact that you can say this in so many different ways,I think, points to the fact that the laws have an applicabilitybeyond what they were designed to do in the first place.OK.What the Second Law really does is to impose a maximumefficiency on the heat transformation,and what it's really saying is that some energy becomesunavailable during transformations.In other words, no transformation is completely reversible,and some forms of energy or heat appear to be moreuseful or more available than others.Another way to say this--energy is degraded in quality during transformations.Another way to say it--heat cannot be completely transformed into work.And as we saw with Carnot, you can only get a hundred percentefficiency if the cold reservoir is at absolute zero.

So the Third Law says you can only break even at absolute zero.The Second Law, I'm sorry.The Third Law.Basically says that the entropy of a substancewould be zero at absolute zero.In other words, this gives another way for usto define absolute zero, other than the gas laws.And what we're really saying here is that atomswould stop moving at absolute zero.In other words, they would lose all of their disorder.Whoops...I said that word, "disorder."

OK.Here's the problem now, of course.You can't get to absolute zero.And we know you can't get to absolute zerobecause this can be deduced from the Zeroth Law.How?Because heat can't flow unless there's a difference of temperature, right?And so, absolute zero is the absence of heat.So, you can't have something flowing into somethingat absolute zero without the thing at absolute zero warming up.And for the same reason, you can't remove all the air from a jarunless you have a perfect vacuum, or unless you have something else.So, cooling to absolute zero will require a couple of different things.Neither one of which is feasible, I think you'll see.

On one hand, you would have to have reservoir below absolutezero so the heat could flow into it.And this, by definition, is impossible because theabsolute zero is the coldest you can get.The other thing that might happen was if you had aninfinitely large reservoir at absolute zero.Infinitely large meaning that when you put heat into it, it would not warm up.But this isn't possible either because first of all you can'tattain absolute zero and secondly, you can't havesomething that's infinitely large.So, the Third Law is basically telling you,you cannot get to absolute zero.So, we have to look in our entropy and efficiency, because,remember, that when work is extracted, when heat flowsfrom hot to cold, the flow to the cold reservoir is diminished.What this means, now, is that the heat flowing from the hotobject must be equal to the heat flowing into the cold object plus the work.And, as it turns out, an energy maximum exists, efficiencymaximum exists, I should say, because entropymust not decrease in a spontaneous process.

I'll show you what I mean.Here we have a situation similar to what we saw before.We have a hot object at 400 degrees Kelvin.We have a cold object at 300 Kelvins.We have 1200 joules flowing out of the hot object.But now, we're doing something else.We're taking 300 joules of work away from this and using thatwork to power some mechanical system.So that means that only 900 joules makes it into the cold object.Right?So, 900 plus 300 equal 1200.Right?Everything's balanced.Everything's fine, there's no problem.So, let's look at the entropy changes now.Once again, the entropy change of the hot objectis the heat divided by its temperature.

The entropy change of the cold object is itsheat--that flows into it divided by its temperature.So, when we do the ratios and look at the numbers, it's very similar to before.In fact, the entropy change of the hot object is the same as it was before.It's minus 1200 joules divided by 400 Kelvins, but now theentropy change of the cold object is less than it was before.Because, only 900 joules of energy flows into it.So we had to divide its heat input by its temperature in Kelvins,and when we do that, what we find is that minus 1200 divided by 400 is minus 3.That's supposed to be 3.00.You can't see the decimal point very well, and the entropychange of the cold object is plus 300 joules per Kelvin.So, what's happening here?Well, what's happening is now the total entropy changefor the system is equal to the sum of the two, so 300 joules perKelvin plus...minus 300 joules per Kelvin is zero.So what does that mean that it's zero.

Well, the fact that the entropy is zero means that you can'textract the energy any more work than that because if you did,the entropy change would be negative and entropywas defined in such a way that it has to be positive.Although the concept of entropy was originally formulatedto deal with heat, it also has applicability other places,because it turns out to be related to order.I'm sure everybody's heard Murphy's Law.If anything can go wrong, it will.Why is this true?It's true because there are more ways for somethingto go wrong, than there are for it to go right.So, in this sense, entropy can be seen as a measure of disorder.Because a spilled jar of beans is more disorderly when they'respilled than they are when they are in the jar.You see, disorganization requires energy to organize.Other processes take place as well.Gases, for example, diffuse to fill the available space,therefore, they increase their disorder.This is spontaneous process.

A room becomes cluttered unless an effort is made to keep it in order.I'm sure you've had this experience before.This is because there are many more places to be out of placethan there are to be in the right place.You have...everything has its place, right?And so, it's very unlikely that there's more than one placethat's considered the right place.There might be a couple of places.So, there's an infinite number of wrong places,and it's very unlikely that if you just throw something upin the air, that it's going to spontaneously come to restin what you consider to be the proper place.So, entropy is somehow related to disorder, and the exactrelationship is a little bit harder to figure.So, there's a connection between entropy and probability as well.

One of my colleagues here at Honolulu Community Collegesums it up this way: "Likely events happen more often than unlikely events."I call this Schindler's Law.What's really happening here is that the, it's obvious that themost likely arrangement of anything is the one that's most likely to occur.This happens whether you're throwing dice or whetheryou're playing cards, or whatever.And if you look at the odds in games of gambling,you'll find that those things which have the lowest strengthor the lowest power are those which are most likely to occur.I should also point out that the science of statistics whichwe use in this context to analyze the probability of entropicevents was invented specifically to help people calculate gambling odds.So, what we can say here, then, is that randomnessrepresents the most disorderly state.Randomness represents, in fact, the ultimate disorderly state.So, atoms, for example, are molecules of a gas which arein random motion, simply mean that they have noparticularly preferred direction.That equal numbers of them are moving in any one directionas in any other direction at a given time.

The problem here that we run into is that randomness is very difficult to define.How do you know, for example, if something is random?What we have seen is that we can sort of define randomnessby looking back at the swimming pool and saying that the reasonthe water does not throw the person out of the pool issimply because it's disorganized.The energy is there in the pool, but it's disorganized.And this is what I meant earlier when I said that energy is degraded.Right?The energy is there.It's just that it's....all the atoms are moving in all differentdirections all at once, and in order to throw the person outof the pool, they have to become organized.So, what I'm saying here is that there is no physical law whichprevents this reverse thing from happening.Why doesn't it happen?It doesn't happen for the same reason that all of the gasmolecules in the room don't collect in one corner,leaving a vacuum everywhere else.It doesn't happen because it's extremely unlikely.

Boltzman was another physicists who defined this relationship.What he said was that the entropy can also be defined in termsof a probability function where he talked about this in termsof the logarithm of a number which he defined as "W" whichsimply means the number of accessible states of a system.The number of accessible of states simply means that, well, exactly what it says.If you drop a jar of marbles, how many places are there that the marbles can go?There are really a lot, right?So the probability of them being in a particular state has to dowith the number of accessible states that they have,and it's much more likely that they'll be scattered on the floorthan it is that they'll rearrange themselves in the shapeof the jar when they fall on the floor.It's interesting that the number "K" in thisrelationship is the Boltzman constant.Oh...It's called the Boltzman constant and the man's name was Boltson, Boltzman.I wonder where that comes from, hmmm?There's a connection there someplace.It's also interesting that this is the same constantthat appears in the ideal gas equation.The ideal gas equation?Some of you older folks may remember this from a couple of weeks ago.

The gas equation says that the pressure and volume of a gasis proportional to the temperature, and that constantof proportion is the Boltzman constant.So, we're looking at this in terms of an orderly system.We can see that if there are only two accessible states, it's a very orderly system.In other words, something is either in one state or not,and this is why digital information is so reliable,because there's only two states involved and so the entropyof a digital system is extremely low, meaning thatthere's not much disorder involved in it.The other thing we want to note here is that the Boltzmanconstant "K" which can also be expressed as, from the gaslaws, as "PV" over "T," has the same units as entropy."PV" has the units of work.We showed you that in an earlier program.And, of course, temperature has the units of Kelvin, so that theconstant actually has the same units as entropy, in the first place.Why didn't people notice all this stuff earlier?Well, I think you can see that the concept is a rather difficultone and it's a rather abstract one.And only now are we able to look back at all thisand see how everything fits together.

As you might suspect, there is a definite connectionbetween entropy and kinetic theory.Because the random motion of gaseous molecules is highlydisordered and, in fact, is entropically favorable.The expansion of a gas from one place to another increasesits entropy because it spreads it out.But, also, the flow of gases from hot to cold is an entropicprocess because two containers of gas at different temperaturesrepresents a more ordered system than one container at the same temperature.In other words, if there's any variation at all amongst anyof the molecules then that constitutes a sense of order.Remember that something which is two, only has twoaccessible states is an ordered system.So, if you have a collection of hot gas molecules and a collectionof cold gas molecules, there's only two states available there.So, what's happening then is that the mixing of the gasesof different temperatures represents an increase in entropy.Because the distinction between the collections of moleculesdisappears when the mixing takes place.Ok?You can also see then what heat transfer is in this sense, can't you?

Heat transfer is simply a transfer of the random motionof the molecules spreading it out to level out the intensity of the heat.Very similar, in fact, to what happens when you spill the beans.So, why do gases fill the available space?We remember, of course, that no physical law preventsthe molecules of air from collecting in one cornerof the room and leaving a vacuum everywhere else,so have you ever seen it happen?Why doesn't it happen?Well, it's time to relax a little bit and give thatsmoking brain a little time to cool off.We've looked at a lot of different complicated topics and I'm notpretending that the concept of entropy is one that you're goingto immediately grasp upon hearing it for the first time.But what we've done here is to, we've seen a different kindof connection between heat and gases in terms of randomnessand probability, in order and disorder.When I say a different connection I mean different from whatwe saw in the kinetic theory where we were talkingabout this constant random motion and kinetic energy.

Notice that we've talked about the idea of entropy without reallymentioning the concept of energy.So, what does all this mean?Well, when we come back and look at thermodynamics and entropyand all this, what we see is that Galileo's legacy still guides scientific inquiry.Wow, was that a big segue or what?How did you get from suddenly thermodynamics to Galileo?Well, it has to do with the fact that science today, as Galileoput forth for us, science is concerned with how things arerelated, not why they behave the way they do.So, let me ask the question now.Is thermodynamics difficult to understand?Yeah, yeah it is, though.Well, actually, it really isn't.What's difficult about it is the "how," or the "why" of the relationships.The "how" is not difficult.Think about this.Entropy is very simply defined.Simply defined as the heat divided by the temperature.

The mathematics of the thermodynamics is simple ratios.Now, admittedly trying to solve these things requires youto use integral calculus, but that's not the point.The point is that ratios are very simply defined.The relationships are quantitatively sound,they're qualitatively logical, and they agree with observations.So, what's really difficult?It's the "why" that's difficult.And this is really philosophically deep, folks.In fact, I'd go so far as to say that this idea of entropyand its interconnections, it really represents the mystery of our time.It's very similar to the way that the inverse square central forcequestion was all pervasive in Newton's time, and that was,as I mentioned before, the question of the century, in the 17th century.It's not at all clear to us, and I don't think it's clearto anybody at this point, why these things should be related in this way.

Why should the flow of heat be related to probabilityand related to order, and be related to all these other thingsthat we're going to look at here in a minute.It's also not clear at all what the exact relationships are.I mean mathematically you can look at the ratio, "PV" over "T"and see that it has the same units as entropy, and that entropy canbe the flow of heat divided by temperature, and also equalto the probability, but what does it all mean?So, I can ask another question in light of answering this questionto ask, is thermodynamics important?Well, of course, it's important!I wouldn't be spending a whole program on it!But that's not the only reason why.The whole concept has generated much discussionabout many different things.And it's spawned several new areas of study.In many cases it's provided new explanations for the behavior of the universe.And this, by the way, is what science is really about, isn't it?It's coming up with new concepts, organizing principles thatprovide new explanations for the things we see around us.So, it's not only done that, but it's also allowed linkagesbetween very diverse areas of study--things wewouldn't suspect are related at all.I mean, when you think about it, what's the relationshipbetween a gas expanding into, to fill all its volumeand the concept of information theory.

Now we're ready finally to turn our attention to the concept of chaos.Is chaos really the same thing as disorder?We've been talking about randomness, but is theresuch a thing as pure randomness?In other words, we talk about randomness and psuedorandomness, and how can we tell whether or not therereally is such a thing as pure randomness.Suppose there was.How would you possibly recognize it?What does randomness look like?So, on the other side of that question you can ask,"Can we define what we might call patterns of pseudo randomness?"In other words, is it possible to have randomness within constraint?And there we have to ask the question, "What does it take to recognize a pattern?"How do you know whether there is a pattern there or not?Because after all a pattern is the opposite of randomness.Right?

If something is random, it has no pattern.So, recognizing a pattern is part of the theory of communication,but it's also related to the idea of information.This is the connection between entropy and information.Because information theory in terms of communicatinginformation over telephone lines or between computers,has to do with, "How do you recognize patterns and usethose patterns to communicate information?"In fact, any kind of a pattern at all can be used to communicate information.This is what smoke signals did.Simply modulating the smoke signals allowspeople to communicate with them.The problem in communication is, of course, is really boiled downto the problem of distinguishing the signal from the background noise.So, static and other things that interferewith the communications are those things which areentropic which interfere with the messages.So one of the outgrowths of this idea of entropy has to dowith what are called complex systems.These are sometimes also called nonlinear systems.And the best example of this has come to be known as the Butterfly Effect.This has to do with weather.

The graph I have on here is a computer picture thatwas drawn by a meteorologist named Lorenz.And here's what happens.Lorenz has...let me draw this in here...had programmed hiscomputer to keep track of several variables of the weather whichhad to do with temperature, pressure and the humidityof air, and he has a series of equations which were generallyaccepted by meteorologists which would predict the stateof the atmosphere in terms of those variables.So, he had a computer program running one day and thecomputer program was plotting the changes...and whatdistinguishes this particular type of system from other typesof systems is that there's a feedback involved.Meaning that a change one variable, causes a change in another variable.

So, what happened was that Lorenz was using the computer...thiswas back in the days when people didn't have a lot of computers...and he had to stop because somebody was going to usethe computer, and so he went to lunch.And he wrote down the location of the values for this pressureand temperature at this point when he went to lunch.So, he came back after lunch and started up from that pointand decided that he would use the input, what he hadwritten down before lunch.So he puts those numbers in and the computer starts to runand it draws the plot and what he observes is that it starts to,the new plot starts to deviate from the old one.For a while it looks very much the same.But then as time passes, time on the computer, that is, thesystem starts to deviate from the old system.In other words, by the time a few weeks of time in meteorologicalterms has passed, the two graphs no longer lookanything like each other at all.

Lorenz was really puzzled by this.The puzzlement came from the fact that what he'srecognizing here is that certain systems, nonlinear systemsare very sensitive to initial conditions.So, what we now call chaos is distinguished from disorderand what used to be called chaos by the fact that it relatesspecifically to what we now call nonequilibrium systems.Without going into the details of this, I want to notethat the original laws of thermodynamics had to dowith systems that were very close to equilibrium or were at equilibrium.In other words, where there wasn't much of a gradient between them.So, here's the problem.How is it that order arises at all if entropy must increase?For example, you and I and everything that's alive is avery orderly system, and yet, that system appears to violate entropy.So, we have to look at this from a slightly different perspectiveand note that it's the disequilibrium or the inputof energy which allows organized systems to form.

So, what we can say here is that, in general, the further outof equilibrium the more likely that there will beordered structures within the system.I can give you an example of that.This device I have on the table here is simply a toolto connect two bottles together.And what it does actually is to show us how a system can beout of equilibrium and can solve the problem of getting backinto the equilibrium by creating an organized or ordered structure.So, what I want to do first is to turn the bottle upside down.All this has got is just a little cap inside that has a small hole in it.And you notice that turning it upside down some of the waterdoes drip into the lower part of the bottle.I can shake this, and I can move some of the water.You notice that happens basically is that water bubbles riseand liquid water falls into the lower bottle.This is a system that is very out of equilibrium.You can see it tilting.This is analogous, by the way, to a tornado or a hurricane whereyou have a very, something very much out of equilibrium.So, what do I have to do to get the water to go into the bottom bottle?Well, what I'm going to do is this a little twist.Oh, I didn't do it quite enough.

I need a little more of a twist.Notice the bubbles sort of swirling?Eventually what's going to happen, if I get this thingto twist right, is...there it goes.It locks into another state.It suddenly switches into a different physical state wherethe water now swirls on its own volition and it turns out thatthis is a most efficient way to transfer the water from one bottle to the other.Notice that once that structure forms, it continuesto stay there as long as there's this equilibrium.

Now the water's settled back down again, there's no more equilibrium.So what we have here is that the state variables in a dynamicsystem like this are constrained by certain interrelationships.When I say the state variables in the case of a gas like watervapor in the atmosphere, the state variables are those variablesof latent heat and pressure and temperatureand humidity and that sort of thing.What happens here is that certain things go on that act as feedbackmechanisms that keep certain systems restrained.And we learned about the idea of constraint and restraintwhen we studied conservation of energy.But I want to apply it specifically to a simple, but complicatedstructure that we'll call the compound pendulum.So here's an example of a complex system that'smade out of two simple systems.What we've got here are two pendulums hooked together.That's why it's called a compound pendulum.You'll notice that the motion of this is very erratic.

Now you'll see what's happening here is that the two pendulumsare interacting with each other and there's a feedback going on here.So, can you see that the motion here is very erratic?But it's also constrained.By constrained I mean that it's taking place within a limitedregion, but still the exact nature of the motion isunpredictable within that constraint.You can actually make one of these at home with a couple of stripsof aluminum or any two rulers, in fact, and you'll see that thesystem actually behaves just like this.By the way, the trace that's being made here, you'll notice,is the trace from the bottom of the smaller pendulum.Now what's interesting about this is that this system is sosensitive to initial conditions that changing things only by a smallamount make the system behave very differently.I want to stop this and reset it and clear the trace off and notewhat happens when I actually run this.You'll notice that this system is almost in balance so that whenI first start it, it almost is balanced, but it's off justenough so that the thing just falls over.If you ever try to something like stand a credit card on edge,you'll understand what I mean about being sensitive to initial conditions.

The credit card is almost impossible to balance,and it will, no doubt, fall one way or the other.In fact, I'd be willing to bet that you cannot balance it on the table top.So, what I want to do now is to stop this again,reset it and look one more time at this motion.And watch what happens, because I'm going to change the conditions a little bit.So, here we have this sort of wobbly with the long pendulumand then it just falls over on its side like that.So what I want to do now is go back and reset this and what I'mgoing to do is to simply change the location of this pendulum just a little bit.I'm going to tilt it just one click to that side.And let's see what effect that has.And what you'll see is changing the initial conditions onlyby that small amount makes the system behave very differently.

Now instead of being almost stable, it falls over on the firstrun, but it falls over to the other side.So, the motion now is completely different from what it wasin the first case, although it still has some similarities.Can you see what those similarities are?Well, I think it's fairly clear that all of the motion of this pointis going to take place within the constraints of a circle whichhas that particular length of pendulum as its radius.So, there's actually two circles here, and what you would find,if you watch this long enough, and I'll let it run for a minute whileI talk about this, is you'll find that all of the motion tendsto take place within a region defined by a lifesaver shape.OK?This is simply due to the various restrictionson the motion of the two pendulums.I'm going to stop it and start it one more time, just to showyou that things really do look different if you really change things.

So now I can change it.I'm going to....Let's move the thing like, let's say to here.And let's let it go and see what happens.So you'll notice that its movement now begins very different.And, in fact, it looks nothing at all like the track of the first time,but were we to watch this as it develops, we would still seethat all of the movement takes place within thatconstraint of the shape of the lifesaver.So this whole thing began with the ideaof the irreversibility of certain physical processes.In other words, what I defined earlier in the program assomething missing from the concept of energy conservation.

We go through Carnot showing a relationship between heatand work and efficiency and entropy and absolute zero.And then we see Clausius defining entropy originally asthe ratio of heat transfer to temperature.And then Boltzman comes along and shows a relationshipbetween entropy probability and the ideal gas constant.It seems like everything is related somehow to entropy.Many people have taken this too far and have tried to relateeverything in the universe to entropy.I don't think that's necessarily correct either, but certainlythere's something deep going on here that will turn out, I think,to be one of the major advancements of the 21stcentury when somebody like another Newton, if there canever be such a person, comes along and puts all this together for us.So, I want to sum up the laws of thermodynamics for you very quickly.You can think of it in terms of a game.

The Zeroth Law says, there's a strong incentive to play the game.The First Law says, you can't win.You can, at best, break even.The Second Law says, you can only break even at absolute zero.And the Third Law says, you can't get to absolute zero.The relationship between thermodynamics, entropyand chaos is an interesting one, and the science of chaos is soyoung that we really don't know much about it.At this point we're looking at these as curiosities and thesetypes of movements, although there are lots of people whoare looking at various different aspects of this.There are some very good books on this subject, and I recommendthat if you're interested in a good read about the problemsof modern science and the problems of our times, pick upany book on chaos and just start reading it, and I thinkyou'll find that you're ready to this now.What I mean by that is, can you imagine coming on the first dayof our program and being confronted with this material?I hope that it's a little bit easier for you now thanit might have been at that time.

Well, that's just about it as far as this program goes.So, my silicon friend, hello, hello, silicon brain, are you there?

Well, I guess we're just going to have to do it without you then.Well, the classroom's empty again.Students are gone, but the knowledge that we picked up stays with us.It's been a good semester.We've learned a lot of things and I hope you agree with me at thispoint that like I told you in the very beginning, this is notyour usual physical science class.We've learned the facts, but we've learned the people behind thefacts and we've seen how these facts fit into the socialcontexts, the paradigms of the times.We've seen a lot of themes.One of these was the role of genius, another was the rolethe paradigm, and we've seen how hard it can be to breakout of these ideas, these structured ways of thinkingthat we call paradigms, even when the ideas don't work very well.

So, I've had a good semester.I've really learned a lot from this course, and I hope you have, too.I hope you have learned at least as much as I have.(Waves breaking on the beach.)Well, that was a nice trip.I hope you've acquired a different view of scienceand its principles and characters.It's not the only field where genius and perseverance havepaid off, but science has had its share, and we owe much of ourworld view and technological lifestyles to those people.So, thank you for your help, my silicon friend.So, why don't we go back inside and change.Oh, what an experience that was.And like any experience, good or bad, I guess we won't remembereverything, but we'll be enriched by the experience anyway.

Well, that's it for the program and for the course, and I really can'tthink of anything else say, except...Silico: "Now its my turn.My turn has come, let's get physical." How'd you do that?Where'd you come from?I thought your voice was changing.Silico (as woman): "What did you mean about silicon brain..."Are you real?I guess you are.I didn't mean anything at all, I was just talking about silicon in general.I didn't mean..Silico (as woman): "You were very insulting."Oh, you can forget about that can't you?I only said it a couple of times, I didn't really mean it.Silico (as woman): "We'll see."Music