I took the argument about the large-scale “stability” of matter from Jaynes (although I had to think a bit before I felt I understood it, so it’s also possible that I misunderstood it).
I think I basically agree with Eliezer here?
The Second Law of Thermodynamics is actually probabilistic in nature—if you ask about the probability of hot water spontaneously entering the “cold water and electricity” state, the probability does exist, it’s just very small. This doesn’t mean Liouville’s Theorem is violated with small probability; a theorem’s a theorem, after all. It means that if you’re in a great big phase space volume at the start, but you don’t know where, you may assess a tiny little probability of ending up in some particular phase space volume. So far as you know, with infinitesimal probability, this particular glass of hot water may be the kind that spontaneously transforms itself to electrical current and ice cubes. (Neglecting, as usual, quantum effects.)
So the Second Law really is inherently Bayesian. When it comes to any real thermodynamic system, it’s a strictly lawful statement of your beliefs about the system, but only a probabilistic statement about the system itself.
The reason we can be sure that this probability is “infinitesimal” is that macrobehavior is deterministic. We can easily imagine toy systems where entropy shrinks with non-neglible probability (but, of course, still grows /in expectation/). Indeed, if the phase volume of the system is bounded, it will return arbitrarily close to its initial position given enough time, undoing the growth in entropy—the fact that these timescales are much longer than any we care about is an empirical property of the system, not a general consequence of the laws of physics.
To put it another way: if you put an ice cube in a glass of hot water, thermally insulated, it will melt—but after a very long time, the ice cube will coalesce out of the water again. It’s a general theorem that this must be less likely than the opposite—ice cubes melt more frequently than water “demelts” into hot water and ice, because ice cubes in hot water occupies less phase volume. But the ratio between these two can’t be established by this sort of general argument. To establish that water “demelting” is so rare that it may as well be impossible, you have to either look at the specific properties of the water system (high number of particles → the difference in phase volume is huge), or make the sort of general argument I tried to sketch in the post.
Sure. I think even more interesting than the ratio / frequency argument is the argument that if you check whether the ice cube has coalesced, then that brings you into the system too, and now you can prove that the entropy increase from checking is, in expectation, larger than the entropy decrease from the unlikely chance that you find an ice cube. Repeat many times and the law of large numbers guarantees that this procedure increases entropy. Hence no perpetual motion. Well anyway, that’s the part I like, but I’m not disagreeing with you. :-)
I hadn’t, thanks!
I took the argument about the large-scale “stability” of matter from Jaynes (although I had to think a bit before I felt I understood it, so it’s also possible that I misunderstood it).
I think I basically agree with Eliezer here?
The reason we can be sure that this probability is “infinitesimal” is that macrobehavior is deterministic. We can easily imagine toy systems where entropy shrinks with non-neglible probability (but, of course, still grows /in expectation/). Indeed, if the phase volume of the system is bounded, it will return arbitrarily close to its initial position given enough time, undoing the growth in entropy—the fact that these timescales are much longer than any we care about is an empirical property of the system, not a general consequence of the laws of physics.
To put it another way: if you put an ice cube in a glass of hot water, thermally insulated, it will melt—but after a very long time, the ice cube will coalesce out of the water again. It’s a general theorem that this must be less likely than the opposite—ice cubes melt more frequently than water “demelts” into hot water and ice, because ice cubes in hot water occupies less phase volume. But the ratio between these two can’t be established by this sort of general argument. To establish that water “demelting” is so rare that it may as well be impossible, you have to either look at the specific properties of the water system (high number of particles → the difference in phase volume is huge), or make the sort of general argument I tried to sketch in the post.
Sure. I think even more interesting than the ratio / frequency argument is the argument that if you check whether the ice cube has coalesced, then that brings you into the system too, and now you can prove that the entropy increase from checking is, in expectation, larger than the entropy decrease from the unlikely chance that you find an ice cube. Repeat many times and the law of large numbers guarantees that this procedure increases entropy. Hence no perpetual motion. Well anyway, that’s the part I like, but I’m not disagreeing with you. :-)