Chapter 8 - Systems, States, and the Second Law

What constitutes a "good explanation" depends on what you are trying to achieve.  If you are trying to, say, launch a communications satellite then it matters a lot whether the earth is round.  (Indeed, the exact radius of the earth matters a lot.)  But if all you care about it finding your way to the corner grocery then the exact radius of the earth doesn't matter; as long as it's a lot bigger than the distance to the grocery store, you might as well behave as if it were flat.

Over the past 2000 years or so of human history there have been three major revolutions in what constitutes a "good explanation".  The earliest explanation of natural phenomena, one which goes back to pre-historic times, was that nature contains *agents*, complex entities with goals and desires with which one can communicate.  Humans are agents, but (the explanation goes) nature also contains non-human agents in the form of a wide variety of deities.  Because deities are agents, they have goals and desires of their own.  They can get angry, and when they do they sometimes take their anger out on humans.  But, again because deities are agents, you can communicate with them by means of prayer and rituals.  If you get it right, if you pray to the right gods and engage in the correct rituals, you may succeed in quelling their anger, and they will send rains to water your crops rather than plagues to kill your livestock.

The problem with this explanation is that deities are very cagey.  You can talk to them, but very rarely do they return the favor, and when they do it's usually in the form of a private communication, a voice inside someone's head that only they can hear, and different people hear different deities telling them different things at different times.  Figuring out how to reliably appease the gods is very, very hard.  Even today it provides employment for millions of priests, pastors, rabbis, imams, shamans, psychics, yogis, and job titles for people who will guide you through the mysterious ways of the gods in exchange for a small donation.  But again, different tour guides will lead you down different paths, which still leaves one with the problem of deciding which ones to follow.

The earliest record we have of people trying to solve this problem is in ancient Greece.  The Greeks had deities, of course, but they also had a tradition of trying to explain nature *without* invoking any agents, that is, to explain nature in purely naturalistic terms, as something that obeys *universal laws* that can be understood and agreed on by everyone.  By following this approach, the Greeks were able to figure out that, for example, the earth was round, that the moon was very far away, and the sun even further away than that.  Greek science culminated in the works of Aristotle and Ptolemy, which stood as the best known explanations for about 1500 years before they were replaced by classical mechanics in the 17th century, which in turn was displaced by relativity and quantum mechanics in the 20th.

The reason I'm belaboring all this is that I am about to make some factual claims, that is, I am going to tell you that some things are "true", and I want to very carefully circumscribe what I mean by that: to be "true" in the scientific sense is to be the best known explanation within a certain context, that is, for a certain range of phenomena or observations.  The context for which classical mechanics is "true" in that sense is, more or less, the world of the industrial revolution, that is, the world of steam engines and internal combustion engines.  

You might be tempted to think of steam engines as a relic of the past, but they are still very much relevant in today's world.  They might not be chuffing down railroad tracks any more, but steam engines are still very much alive in the centralized power plants that keep modern civilization humming along.  Any power plant that burns fuel to generate heat, including nuclear fuel, is, at its core, a steam engine.

So although Aristotle and Ptolemy are thoroughly obsolete and most of historical interest, classical mechanics is still very relevant today despite the fact that it is not 100% accurate.  It's also one hell of a lot easier to wrap your brain around than relativity and quantum mechanics, so it's a good place to start honing your scientific instincts.

Classical mechanics is based on the idea that the world is made of material objects that exist in particular places at particular times.  It is unfortunate and surprising that English doesn't have a better word for this.  The word "thing" *can* mean "material object" but it can also mean other things (!) like activities ("sunrise is a thing that happens every morning") or feelings ("happiness is a thing that money cannot buy").

Even "material object" is not quite right either, because we tend to think of "material objects" as being *solid*, and I don't want to restrict it to that.  Liquids and gasses are described quite well by classical mechanics.  In fact, the the vast majority of things (!) that happen in your day-to-day life can be explained by classical mechanics.

So if we're going to talk about classical mechanics, we need a word for the "things" that classical mechanics talks about.  As long as we're on a new-word kick I may as well introduce you to a word that actually *means* "the things that a scientific theory is able to talk about".  That word is "ontology".  Ontology is philosophy-speak for "the study of existence".  Every scientific theory has an ontology, that is, a description of the things that are taken to exist and hence can be talked about in that theory.  The ontology of Aristotelian mechanics included four elements: earth, air, water, and fire.  These in turn were described in terms of four "qualities": hot, cold, wet, and dry.  Fire was hot and dry.  Water was cold and wet.

The ontology of classical mechanics is *matter*, and a collection of matter is called a *system*.  Systems are collections of matter that exist at a particular place at a particular time, and have some kind of *identity* so that we can track it as time passes.  A baseball, for example, is a system, because we can say things like, "That baseball is over here" and then at same later time "That same baseball, which used to be over here, is now over there."  That baseball is a system.

The description of the location (and possibly other properties) of a system is called its *state*.  In the baseball example above, the baseball started out in the "over here" state and then at a later time it was in the "over there" state.  Of course, most of the time we will want to be more precise than "over here" and "over there", but it's not strictly necessary.  As long as everyone agrees where "over here" and "over there" are (or as long as everyone understands that these are placeholders to be filled in with actual locations later) then it's perfectly fine to be a little sloppy.

A system can be made of different parts, not all of which are necessarily connected to one another.  For example, we can talk about a *collection* of baseballs as if it were one system.  The state of that system would not necessarily be a single location (though it might be if all the baseballs were, say, gathered together in a bag).  In situations like that you will sometimes see the word "configuration" used instead of "state", but these two words mean the same thing.

It turns out that all material objects in our universe are made of atoms, of which 93 different kinds occur naturally, and a few dozen others can be created artificially.  This was hard-won knowledge.  It was 300 years between when Newton first invented classical mechanics and when the debate over whether or not atoms were real was finally settled in the early 20th century.  (The main argument against them was that they could not be seen.)  But as the existence and ubiquity of atoms is, in today's world, about as well-established a fact as you could hope to find, I'm just going to roll with that because it makes talking about this stuff a lot easier.

So... matter is made of atoms, which you can think of as teeny weeny little spheres kind of like baseballs (though a better mental image would be billiard balls or a marbles), that move around, and sometimes stick to each other, and sometimes bounce off each other.  Collections of atoms exist in three major states: solid, where all the atoms are stuck together and mostly don't move relative to each other, liquid, where the atoms have become unstuck but are still more or less in contact with each other, and gas, where the atoms are just bouncing around without sticking together at all.  Again, this is an oversimplification, an approximation to the truth, but a very, very good approximation, more than adequate to explain most every-day phenomena.

Apart from their locations, systems of atoms can have other properties as well.  In addition to their *location* at any given time we can also note whether or not that location is *changing*, i.e. whether or not the system is *moving*.  And the state of a system can include other properties as well.  Systems can have colors and textures and, most importantly, *temperatures*, which are connected to something we call *heat*: add heat (whatever that might actually be) to a system and it gets hotter, i.e. its temperature goes up.  Take heat away, and it cools, i.e. its temperature goes down.

One of the most profound consequences of classical mechanics is that there is a connection between heat and motion.  Indeed, heat is a *kind* of motion.  This is not at all obvious.  For a very long time people thought that heat was a *substance* called "caloric".  Hot things contained more caloric than cold things.  Caloric naturally "flowed" from hot things to cold things in the same way that water flowed downhill.

But this theory had problems.  You might want to stop and see if you can think of any yourself before reading on.

The most obvious problem with caloric was that it didn't seem to *weigh* anything.  If you took a rock and heated it up (presumably by adding caloric to it) the rock still weighed the same as when it was cold.  This was not a fatal flaw because there was another substance that was known to exist that also didn't seem to weigh anything: air.  Air, of course, *does* have weight (or, to be more precise, mass) but that is not so easy to show, and it took a very long time to sort out the whole mess.  But this chapter is not about history, so I'm just going to cut to the chase and tell you that after a lot of drama, mankind (because it was still a boy's club) figured out that heat is just atoms wiggling around in random patterns.  More wiggling, more heat.  Less wiggling, less heat.  No wiggling at all and you are at absolute zero, as cold as it gets.

(Note that this is somewhat of an oversimplification.  Indeed, classical mechanics itself turns out to be somewhat of an oversimplification.  But for day-to-day life, these approximations are very, very good.)

Because heat is a kind of motion, you can use motion to generate heat: you can boil water by churning it up in a blender, or start a fire by rubbing two sticks together.  This is the reason the brakes on your car get hot when you use them: they convert the motion of your car into heat in your brake rotors.

That you can turn motion into heat has been known since ancient times.  What took a lot longer to figure out is that you can also reverse the process and convert heat into motion.  This is what happens in a steam engine or an internal combustion engine, and it was the basis for the industrial revolution.  You burn some fuel to generate heat, and with the right arrangement of cleverly designed machinery, mainly a cylinder with a piston in it, some of the random wiggling in the hot material is harnessed to move the atoms in the piston all in the same direction at the same time, which turns out to be a tremendously useful trick.

It is worth describing this process is some detail because it is crucially important in today's world.  Pretty much all of technological civilization was driven by it (literally!) for several hundred years.  The basic idea is quite simple: you take a cylinder made of some sturdy material (typically iron or steel, but aluminum can be made to work too), and you put a piston inside that has a very close fit to the sides of the cylinder.  Put a hot gas on one side of the piston and a cold gas on the other side and the piston will move.  This is because the atoms on the hot side are moving faster than on the cold side, and so they hit the piston harder than on the cold side, and so the piston moves just as if it were being hit by countless teeny weeny jackhammers.

Note that the piston is actually being jackhammered on both sides.  If it is being jackhammered equally on both sides, it won't move.  It will only move if one side is being jackhammered harder than the other, i.e. if one side is hotter than the other.

Something else happens once the piston starts to move: when a hot atom hits the piston, it *bounces off*.  But if the piston is *moving* then the speed with which the atom bounces off will be *different* than the speed with which it hit.  If the piston is moving away from the atom, then the atom will be moving more slowly after it bounces, and likewise, if the piston is moving towards the atom, it will be faster afterwards.  Because the piston moves from the hot side towards the cold side, the hot side will cool down and the warm side will heat up.  Eventually, absent some kind of intervention, the two sides will reach the same temperature and the piston will stop moving.

To make this heat-to-motion converter practical we of course have to intervene.  We have to somehow move the piston back to its original position, get rid of the extra heat on what is supposed to be the cool side, and heat up the hot side so that the whole process can begin again.  Different kinds of engines accomplish this in different ways, generally involving complex combinations of valves and crankshafts and whatnot, but those details don't matter right now.  What matters is that to make the process work requires a *difference in temperature* -- a hot side, and a cold side -- and the process of converting heat into motion invariably *cools down the hot side and heats up the cold side* and the longest that this can possibly go on is until both sides are the same temperature.  Then the thing necessarily grinds to a halt.

And yet, when we look around, we don't see things grinding to a halt, we see continuous frenetic activity: cars driving, factories manufacturing, and even before that birds flying and rivers flowing since time immemorial.  So there *must* be something else going on, right?

This is the line of reasoning that in the early days of the industrial revolution drove the quest for a perpetual motion machine.  Nowadays these are the very paragon of the fool's errand, the unmistakable mark of the crackpot.  But back then it was not at all an unreasonable thing to pursue because nature appears to be a perpetual motion machine, at least to casual observation, and so that would seem to be an existence proof that it must be possible.

In fact, it's not even very difficult to imagine *how* it might be possible.  If we have a bunch of atoms all bouncing around, then sometimes when they bounce off each other they will end up with different speeds.  All we would need to do is separate the fast, hot atoms from the slow, cold ones.  With modern technology, surely we could build a little gadget that would measure the speed of atoms and sort them into fast and slow?  How hard could it be?

You may have already heard that the answer to this is that it is not merely hard, but actually impossible.  The reason it's impossible is something called the second law of thermodynamics, which is such a bedrock principle of science that it is often referred to simply as the Second Law, with no further qualification.

What does the Second Law actually say?  That turns out to be surprisingly challenging to pin down.  The introduction on Wikipedia gives two different rendering: "heat always moves from hotter objects to colder objects unless energy in some form is supplied to reverse the direction of heat flow."  and "Not all heat energy can be converted into work in a cyclic process."  But both of those are little more than using different words to say, "perpetual motion is impossible."  Neither one explains *why* perpetual motion is impossible.  Maybe it is possible, but mankind has simply not yet been sufficiently clever to figure out how to do it.

Some formulations of the Second Law state it in terms of something called "entropy", which is usually defined as a measure of the "disorder" of a system.  The Second Law stated in terms of entropy is, "Entropy always increases (unless energy is added)", which is to say, disorder always increases (unless energy is added).  But a simple experiment reveals that this can't possibly be right: take a bottle and fill half-full with water and the other half with oil.  The oil and water will spontaneously separate, that is, they will become ordered.  In order to make the oil and water disordered (mixed) you have to shake the bottle (or introduce an emulsifier, but we'll ignore that for now).  As soon as you stop shaking, the oil and water will separate again.

In fact, the *reason* that the oil and water separate is that they have different densities; water is heavier, so it sinks to the bottom.  But the same is true of hot and cold air!  Hot air rises; cool air sinks.  So if we (say) make a sufficiently tall enclosed container, we would expect the air to spontaneously separate into hot air rising to the top and cool air sinking to the bottom!  How tall is "sufficiently tall"?  Well, we'll have to do some research, for which we will need some funding.  Want to invest now and get in on the ground floor of the invention that reshapes human civilization by providing it with unlimited energy?  If so, please do get in touch.

I will eventually provide you with a proper explanation of why the Second Law is true, but that will be a long time coming.  It's one of the thorniest problems that mankind has ever grappled with, and even most scientists I think don't really have a deep understanding of it.  I'll wager that not one in ten can provide a proper (non-circular) explanation of why the tall-container-of-air idea I just described doesn't let you build a perpetual motion machine.  (If you want to try to figure it out yourself, here's a hint: note that as you go higher into the earth's atmosphere, at least for the first few miles, the air gets *colder*, not hotter.  Why would that be if hot air rises?)

I want to close out this chapter with three important observations.

First, turning heat into motion is horrifically complicated compared with turning motion into heat.  You can do the latter simply by rubbing your hands together, bit the former took humans millennia to figure out.  You have to be able to make cylinders and pistons that are fitted closely enough that they make a good seal but loosely enough that the piston can still move without too much friction.  Just that bit took a very long time.  But then you have to have all kinds of extra stuff to manage the generation of heat, make sure things don't melt or explode, and so on.  There is a reason that things look messy under the hood of a gasoline-powered car.  (There is also a reason that despite all of our technological prowess no one has ever build a perpetual motion machine that actually works.)

Second, the process of turning heat into motion cannot be 100% efficient.  Remember, that process depends on a *difference in temperature* somewhere, and an inherent part of the process is *heating up* the part that started out on the cold side of that difference.  So not all of the heat you started with on the hot side can get turned into motion.  Some of it has to be wasted.  By way of contrast, turning motion into heat can easily be done with 100% efficiency.  Brakes that bring your car to complete stop in seconds are much simpler than the car's engine.  In fact, you don't really need brakes at all; just turn off the engine and wait a while and you car will eventually stop, with all of the kinetic energy of the car's motion being turned into heat.

Third, in order to talk about this at all I had to introduce the concepts of "heat", which I described as a "kind of motion", specifically, a *disordered* motion where atoms are moving in all different directions.  The temperature of a system is a single number, but the system actually consists of a vast number of atoms, each with its own position and velocity.  How can it be that we can distill all of this complexity down to a single number?  Surely our description of the state of the system must lose some fidelity when we do that?  In other words, it seems like it should be possible to have two different systems at the same temperature and still have some discernible difference between them.

The answer to this is "yes and no".  There are actually at least two more numbers you need to know in order to have a complete description of a system (at least in a gaseous state): pressure and volume.  But that is still only three numbers compared to the the vast amount that would be needed to describe the position and velocity of every single atom.  If you have two systems made of the same kinds of atoms with the same temperature, pressure, and volume, no experiment you can do will let you tell which is which, despite the fact that these two systems are (almost certainly) not completely identical.  How is it possible that we can distill the description of so much complexity down to so few numbers without, apparently, losing any fidelity?

I'll leave you to ponder that, but I should warn you: answering that question is not nearly as easy as proving that the earth isn't flat.