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wandering sets part 10: mystery solved!

When I wrote part 9 I was thinking I was pretty confused and I was not sure I had really made any progress on answering the basic question that I had set out to answer with this series. But I'm pleased to announce that within 24 hours after writing that one, I started thinking and piece by piece, it just all came together. I think I have pretty much solved the mystery, aside from some minor threads that might still need to be wrapped up. (Just didn't get a chance to write it down until today.) That was way faster than I'd imagined it would take.

The main thing I wasn't seeing is how mixing (whether the ordinary process of two gasses mixing in a box, or the more esoteric quantum measurement process) relates to wandering sets. And the lynchpin that was missing, that holds everything together, and explains how mixing relates to wandering sets, is "what is the identity of the attractor?"

I realized that if I could pinpoint what the attractor was in the case of mixing, then I would see why mixing is a wandering set (and hence, a dissipative process). Soon after I asked myself that question, the answer became pretty obvious. The attractor in the case of mixing--and indeed, in any case where you're transitioning from a non-equilibrium state to thermodynamic equilibrium--is the macrostate with maximal entropy. In other words, the macrostate that corresponds to "thermodynamic equilibrium".

I think the reason I wasn't seeing this is because I was thinking too much about the microstates. But from the point of view of a microscopic description of physics, any closed system is always conservative--all of the physics is completely reversible. You can only have dissipation in two ways. One is fairly trivial and uninteresting, and that's if the system is open and energy is being sucked out of it. Sucking out energy from a system reduces its state space, so from within that open system, ignoring the outside, you start in any corner of a higher dimensional space and then you get pulled into an attractor that represents the states which have lower total energy. If energy keeps getting sucked out, it will eventually all leave and you'll just be left in the ground state (which would in that case be the attractor).

But there's a much more interesting kind of dissipation, and that's when you course grain a system. If you don't care about some of the details of the microscopic state, but you only care about the big picture, then you can use an approximate description of the physics, you can just keep track of the macrostate. And that's where the concept of entropy comes into play, and that's when even closed systems can involve dissipation. There's no energy escaping anywhere, but if you start in a state that's not in thermodynamic equilibrium, such as two gasses that aren't mixed at all, or that are only halfway mixed, or only partially mixed anywhere in between... from the point of view of the macrostate space, you'll gradually get attracted towards the state of maximal entropy. So it's the macrostate phase space that is where the wandering sets comes in, in this case. Not the microstates! The physics of the evolution of the macrostate involves a dissipative action, meaning it contains wandering sets; and it is an irreversible process because you don't have the microstate information that would be required in order to know how to reverse the process.

So how does this work in the case of a quantum measurement? It's really the same thing, just another kind of mixing process. Let's say you have a quantum system that is just a single spin (a "qubit") interacting with a huge array of spins comprising the "environment". Before this spin interacts, it's in a superposition of spin-up and spin-down. It is in a pure state, similar to the state where two gasses are separated by a partition. Then you pull out the partition (in the quantum case, you allow the qubit to interact with its environment, suddenly becoming entangled with all of the other spins). In either case, this opens up a much larger space, increasing the dimensionality of the microstate space. Now in order to describe the qubit, you need a giant matrix of correlations between it and all of the other spins. As with the mixing case I described earlier, you could use a giant multidimensional Rubik's cube to do this. The only difference is that classically, each dimension would be a single bit "1" or "0", while this quantum mechanical mixing process involves a continuous space of phases (sort of ironic that quantization in this case makes something discrete into something continuous). If this is confusing, just remember that a qubit can be in any superposition of 1 and 0, and therefore it takes more information to describe it's state than a classical bit requires.

But after the interaction, we just want to know what state the qubit is in--we don't really care about all of these extra correlations with the environment, and they are random anyway. They are the equivalent of thermal noise, non-useful energy. So therefore, we shift from our fine grained description to a more course grained one. We define the macrostate as just the state of the single qubit, but averaged over all of the possibilities for the environmental spins. Each one involves a sum over its up and its down state. And if we sum over all of those different spins, that's accomplished by taking the trace of the density matrix, which I mentioned in part 9. Tracing over the density matrix is how you course grain the system, averaging over the effects of the environment. As with the classical mixing case, putting this qubit in contact with the environment suddenly puts it in a non-equilibrium state. But if you let it settle down for a while, it will quickly reach equilibrium. And the equilibrium state, the one with the highest entropy, is one where all of the phases introduced are essentially random, ie there are no special extra correlations between them. So the microstate space is a lot larger, but there is one macrostate that the whole system is attracted to. And in that macrostate, when you trace over the spins in the environment, you wind up with a single unique state for the qubit that was measured. And that state is a "mixed state", it's no longer a coherent superposition between "0" and "1" but a classical probability distribution between "0" and "1". The off diagonal elements of the density matrix have gone to zero. So while the microstate space has increased in dimensionality, the macrostate space has actually *decreased*! This is why I was running into so much confusion. There's both an increase in dimensionality AND a decrease in dimensionality, it just depends on whether you're asking about the space of microstates or the space of macrostates.

Mystery solved!

I'm very pleased with this. While I sort of got the idea a long time ago listening to Nima Arkani-Hamed's lecture on this, and I got an even better idea from reading Leonard Susskind's book, it really is all clear to me now. And I have to thank wandering sets for this insight (although in hindsight, I should have been able to figure it out without that).

I would like to say "The End" here, but I must admit there is one thread from the beginning--Maxwell's Demon--which I never actually wrapped up. I suspect that my confusion there, about why erasure of information corresponds to entropy increase, and exactly how it corresponds, is directly related to my confusion between macrostate and microstate spaces. So I will write a tenative "The End" here, but may add some remarks about that in another post if I think of anything more interesting to say. Hope you enjoyed reading this series as much as I enjoyed writing it!

The End

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