Have you seen Eliezer’s 2008 post on the 2nd law? His perspective matches my own, and I was delighted to see it written up so nicely. (Eliezer might have gotten it from Jaynes? Not sure. I reinvented that wheel, for my part, it’s quite possible that Eliezer did too.) That style of argument convinces me that the 2nd law cannot depend on any empirical claims like “matter is stable at large scales”. It should just depend on conservation of phase space (Liouville’s theorem in classical mechanics, unitarity in quantum mechanics). And it depends on human brains being part of the same universe as everything else, and subject to the same laws.
The entropy of a given macrostate is the uncertainty about the microstate of an observer who knows only the macrostate. In general, you have more information than this.
I basically agree. There’s a nice description here of getting a subregion of phase space that looks like ever-finer filamentary threads spread all around, or as I put it “it often turns out that you wind up with useless information about the microstate, i.e. information that cannot be translated into a “magic recipe” for undoing the apparent disorder … In that case, you might as well just forget that information and accept a higher entropy.”
There are cases where it’s not obvious what the macrostate is, and you need to think more concretely about what exactly you can and can’t do with the microstate information. My example here was a light beam whose polarization is set by a pseudorandom code that changes every nanosecond. Another example would be shining light through a ground-glass diffuser: It looks like a random, high-entropy, diffuse beam … But if you have an exact copy of the original diffuser, and a phase-conjugate mirror, you can unwind the randomness and magically get the original low-entropy beam back.
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. :-)
Have you seen Eliezer’s 2008 post on the 2nd law? His perspective matches my own, and I was delighted to see it written up so nicely. (Eliezer might have gotten it from Jaynes? Not sure. I reinvented that wheel, for my part, it’s quite possible that Eliezer did too.) That style of argument convinces me that the 2nd law cannot depend on any empirical claims like “matter is stable at large scales”. It should just depend on conservation of phase space (Liouville’s theorem in classical mechanics, unitarity in quantum mechanics). And it depends on human brains being part of the same universe as everything else, and subject to the same laws.
I basically agree. There’s a nice description here of getting a subregion of phase space that looks like ever-finer filamentary threads spread all around, or as I put it “it often turns out that you wind up with useless information about the microstate, i.e. information that cannot be translated into a “magic recipe” for undoing the apparent disorder … In that case, you might as well just forget that information and accept a higher entropy.”
There are cases where it’s not obvious what the macrostate is, and you need to think more concretely about what exactly you can and can’t do with the microstate information. My example here was a light beam whose polarization is set by a pseudorandom code that changes every nanosecond. Another example would be shining light through a ground-glass diffuser: It looks like a random, high-entropy, diffuse beam … But if you have an exact copy of the original diffuser, and a phase-conjugate mirror, you can unwind the randomness and magically get the original low-entropy beam back.
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. :-)