wandering sets, part 2
In the summer of 2003, exactly one decade ago, I had a few months to spare before I started graduate school with no full time employment. So I spent that time reading and reviewing as many books on physics as I could. Partly to prepare for graduate school, partly for fun, and partly because I knew the first thing I would have to do when I got to California was take the PhD qualifying exams. One of the most fun parts for me was rereading the interesting parts of my undergrad textbook on statistical mechanics, and making sure that I understood entropy and related concepts very well.
The 2nd law of thermodynamics says that entropy always increases as time progresses. One of the most common popular conceptions of entropy is that it's sort of like a measure of order. It's not exactly true--rigorously, it's the number of microstates in a system that correspond to a given macrostate, but if you're reading this and not familiar with it at all, order should be the first thing you think of when you think of entropy. The visual picture of the 2nd law you should have is that... when eggs fall from a great height, they naturally break into little tiny pieces. But you never see the time-reversed process happening, little tiny splinters of eggshells gathering themselves up and coming together to form a whole egg. In this way, there is an "arrow" associated with time, it has a direction to it that progresses from low entropy states to high entropy states but not the reverse. Even though all of the microscopic laws of physics are 100% reversible, the macroscopic progression of states is irreversible.
One argument that advocates of intelligent design often make is that if natural processes always go from low entropy states (a lack of disorder) to high entropy states (more disorder) then there is no natural process that can explain the spontaneous emergence of order from nothing. Hence, they claim, there must have been some intelligent being who injected such order into the system, unnaturally. So one of my projects that summer was to understand what's wrong with this argument, and why it isn't the conclusion one draws if you actually understand what entropy is and how it works. For the most part, I succeeded in understanding this--to sum it up, it's mostly that they are forgetting the the earth radiates heat out into space, and heat is a form of disorder, it contributes to entropy. The earth absorbs energy from the sun, turns some of it into useful work, organizing things, and lets some of it messily spill out into the surrounding space. Entropy increases locally here on earth in that things look more organized every century here than they did the last, but this comes at the cost of dumping all this messy heat into outer space. (And lately, it seems, not all of that heat has even been making it out into space, some of it has been getting trapped in the atmosphere, warming the planet and causing disorganized things to happen like freak hurricanes and tsunamis! Probably another part of science that intelligent design advocates don't believe in.)
I say I succeeded for the most part. But after I made my own notes about everything, what energy, temperature, entropy, pressure, and volume were and how these macroscopic properties of our world arise from microscopic physical laws that determine the motion of individual particles... after I'd mapped out all that, I realized there was one missing piece. I didn't quite understand the definition of reversibility, and how to tell what kind of a system would be reversible and what kind wouldn't. I knew that reversible processes were those where entropy remains constant, rather than increasing. My textbook said that a reversible process was one which was "slow enough to stay in equilibrium, and without the presence of any "dissipative forces". I understood the first half of that definition. If you expand a box to twice its size instantly, you go out of equilibrium because the air inside expands very rapidly, and this has the effect of increasing the entropy (entropy is proportionate to volume, for a fixed temperature). By contrast, if you double the size by slowly and carefully expanding it, then you cool the gas as you expand it and the whole thing stays in equilibrium the whole time... there is no entropy increase, because miraculously, the entropy that would have been added by an increase in volume is cancelled out exactly by the decrease in temperature. The former process is irreversible, while the latter is reversible. But then there was this pesky second qualification they added to the definition... not only does it have to be slow, but it also has to lack "dissipative forces". But what the heck is a dissipative force? I searched all over for a definition of this but didn't find a good one that made sense to me.
I knew various examples of forces that I thought of as being dissipative... friction, electrical resistors, hurricanes. And all of these things had certain things in common... they all seem like messy kinds of things that mess up the nice clean elegant laws of physics you usually learn in college, where you can count on things like conservation of energy and path independence. Electrical resistors radiate heat, and friction gives off heat. Hurricanes (or tornadoes, cyclones, or any kind of vortex in a fluid) exchange various stuff including heat with its surroundings. But if giving off heat was all there was to it, why didn't they just say that the system can't exchange heat with its surroundings? Couldn't they have just said it has to be a closed system? And most importantly, what is it really about these processes that makes them the kind of things that give off heat, while other types of processes don't? And what about mixing processes--when you pour cream into a coffee cup, and let it sit for a while, it eventually mixes into the coffee. This is an irreversible process as it won't naturally unmix, and therefore it does increase entropy. But in what sense is this "dissipative"? Surely it was a slow process. Although perhaps it couldn't have been said to be in equilibrium before it got done mixing.
The concept of a wandering set purports to explain what exactly a dissipative system is, so I feel like it is the missing part I was looking for here. To be continued in part 3...
The 2nd law of thermodynamics says that entropy always increases as time progresses. One of the most common popular conceptions of entropy is that it's sort of like a measure of order. It's not exactly true--rigorously, it's the number of microstates in a system that correspond to a given macrostate, but if you're reading this and not familiar with it at all, order should be the first thing you think of when you think of entropy. The visual picture of the 2nd law you should have is that... when eggs fall from a great height, they naturally break into little tiny pieces. But you never see the time-reversed process happening, little tiny splinters of eggshells gathering themselves up and coming together to form a whole egg. In this way, there is an "arrow" associated with time, it has a direction to it that progresses from low entropy states to high entropy states but not the reverse. Even though all of the microscopic laws of physics are 100% reversible, the macroscopic progression of states is irreversible.
One argument that advocates of intelligent design often make is that if natural processes always go from low entropy states (a lack of disorder) to high entropy states (more disorder) then there is no natural process that can explain the spontaneous emergence of order from nothing. Hence, they claim, there must have been some intelligent being who injected such order into the system, unnaturally. So one of my projects that summer was to understand what's wrong with this argument, and why it isn't the conclusion one draws if you actually understand what entropy is and how it works. For the most part, I succeeded in understanding this--to sum it up, it's mostly that they are forgetting the the earth radiates heat out into space, and heat is a form of disorder, it contributes to entropy. The earth absorbs energy from the sun, turns some of it into useful work, organizing things, and lets some of it messily spill out into the surrounding space. Entropy increases locally here on earth in that things look more organized every century here than they did the last, but this comes at the cost of dumping all this messy heat into outer space. (And lately, it seems, not all of that heat has even been making it out into space, some of it has been getting trapped in the atmosphere, warming the planet and causing disorganized things to happen like freak hurricanes and tsunamis! Probably another part of science that intelligent design advocates don't believe in.)
I say I succeeded for the most part. But after I made my own notes about everything, what energy, temperature, entropy, pressure, and volume were and how these macroscopic properties of our world arise from microscopic physical laws that determine the motion of individual particles... after I'd mapped out all that, I realized there was one missing piece. I didn't quite understand the definition of reversibility, and how to tell what kind of a system would be reversible and what kind wouldn't. I knew that reversible processes were those where entropy remains constant, rather than increasing. My textbook said that a reversible process was one which was "slow enough to stay in equilibrium, and without the presence of any "dissipative forces". I understood the first half of that definition. If you expand a box to twice its size instantly, you go out of equilibrium because the air inside expands very rapidly, and this has the effect of increasing the entropy (entropy is proportionate to volume, for a fixed temperature). By contrast, if you double the size by slowly and carefully expanding it, then you cool the gas as you expand it and the whole thing stays in equilibrium the whole time... there is no entropy increase, because miraculously, the entropy that would have been added by an increase in volume is cancelled out exactly by the decrease in temperature. The former process is irreversible, while the latter is reversible. But then there was this pesky second qualification they added to the definition... not only does it have to be slow, but it also has to lack "dissipative forces". But what the heck is a dissipative force? I searched all over for a definition of this but didn't find a good one that made sense to me.
I knew various examples of forces that I thought of as being dissipative... friction, electrical resistors, hurricanes. And all of these things had certain things in common... they all seem like messy kinds of things that mess up the nice clean elegant laws of physics you usually learn in college, where you can count on things like conservation of energy and path independence. Electrical resistors radiate heat, and friction gives off heat. Hurricanes (or tornadoes, cyclones, or any kind of vortex in a fluid) exchange various stuff including heat with its surroundings. But if giving off heat was all there was to it, why didn't they just say that the system can't exchange heat with its surroundings? Couldn't they have just said it has to be a closed system? And most importantly, what is it really about these processes that makes them the kind of things that give off heat, while other types of processes don't? And what about mixing processes--when you pour cream into a coffee cup, and let it sit for a while, it eventually mixes into the coffee. This is an irreversible process as it won't naturally unmix, and therefore it does increase entropy. But in what sense is this "dissipative"? Surely it was a slow process. Although perhaps it couldn't have been said to be in equilibrium before it got done mixing.
The concept of a wandering set purports to explain what exactly a dissipative system is, so I feel like it is the missing part I was looking for here. To be continued in part 3...