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Pfft comments on Open Thread, Feb. 2 - Feb 8, 2015 - Less Wrong Discussion
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In a previous thread, I brought up the subject of entropy being subjective and got a lot of interesting responses. One point of contention was that if you know the positions and velocities of all the molecules in a hot cup of tea, then its temperature is actually at absolute zero (!). I realized that the explanation of this in usual terms is a bit clumsy and awkward. I'm thinking maybe if this could be explained in terms of reversible operations on strings of bits (abstracting away from molecules and any solid physical grounding), it might be easier to precisely see why this is the case. In other words, I'm looking for a dynamical systems interpretation of this idea. I googled a bit but couldn't find any accessible material on this. There's a book about dynamical systems approaches to thermodynamics but it's extremely heavy and does not seem to have been reviewed in any detail so I'm not even sure of the validity of the arguments. Anyone know of any accessible materials on ideas like this?
Isn't all this just punning on definitions? If the particle velocities in a gas are Maxwell-Boltzmann distributed for some parameter T, we can say that the gas has "Maxwell-Boltzmann temperature T". Then there is a separate Jaynes-style definition about "temperature" in terms of the knowledge someone has about the gas. If all you know is that the velocities follow a certain distribution, then the two definitions coincide. But if you happen to know more about it, it is still the case that almost all interesting properties follow from the coarse-grained velocity distribution (the gas will still melt icecubes and so on), so rather than saying that it has zero temperature, should we not just note that the information-based definition no longer captures the ordinary notion of temperate?
You are essentially right. The point is that 'average kinetic energy of particles' is just a special case that happens to correspond to the Jaynes-style definition, for some types of systems. But the Jaynes-style definition is the 'true' definition that is valid for all systems.
Again, as I mentioned in my previous replies, the gas will melt ice cubes, but <the gas + knowledge about the gas> is only in thermal equilibrium with 0 K ice cubes.
This claim seems dubious to me.
Like, the "original, naive" definition of thermal equilibrium is that two systems are out of equilibrium if, when put them in contact with each other, heat will flow from one to the other. If you have a 0K icecube one one hand and a gas and piece of RAM encoding that state of the gas on the other, then they certainly do not seem to be in equilibrium in this sense: when you remove the partitioning wall, the gas atoms will start bouncing against the cube, the ice atoms will start moving, and the energy of the ice cube atoms increases. Heat energy was transferred from one system to the other.
I am not claiming that there is some other temperature T such that an icecube at T would be in equilibrium with the system; rather, it seems the gas+RAM system is itself not in thermal equilibrium, and therefore does not have a temperature?
My more general point is that one can not just claim by fiat that the Jaynes-style definition is the "true" one; if there are multiple ones in play and they sometimes disagree, then one has see which one is more useful. Thermodynamics was originally motivated by heat energy flowing between different gases. It seems that in these (highly artificial) examples, the information-based definition no longer describes heat flow well, which would be a mark against it...
gas+RAM is not in thermal equilibrium with the ice cube, because a large enough stick of RAM to hold this information would itself have entropy, and a lot of it (far, far larger than the information it is storing). This is actually the reason why Maxwell demons are impossible in practice - storing the information becomes a very difficult problem, and the entropy of the system becomes entirely contained within the storage medium. If the storage medium is assumed to be immaterial (an implicit assumption which we are making in this example), then the total system entropy is 0 and it's at 0 K.
It is true for the same reason that Bayesian updating is the only true method for updating beliefs; any other method is either suboptimal or inconsistent or both. In fact it is the very same reason, because the entropy of a physical system is literally the entropy of the Bayesian posterior distribution of the parameters of the system according to some model.