wandering sets, part 1
There's a connected set of issues that rests at the very heart of physics, which I've always thought was not very clearly explained in any of my classes. I had a very vague understanding of it while I was an undergrad, and hoped that I'd be able to sharpen it up in graduate school. To some extent, I did, but I found that even after 6 years of graduate school in physics, these issues were still never quite cleared up in my classes, and I never had enough time while working on other things to read enough on my own about them to fill in all the gaps in my understanding. I have always had the impression that these issues *are* well understood, at least by somebody, but that not many physics professors do understand them fully and they tend to just avoid talking about them in actual classes, or talk about them in a very superficial way. I do think my adviser was among those who understood them, but I always felt a bit shy about wasting his time on questions that were purely for my own curiosity and unrelated to any research we were working on. Especially ones that would take a long time to sort out.
Recently, I've started thinking about them again, but through an unexpected route. A couple months ago, OkCupid added support for bitcoin, so as we got closer to the release of it, several of the people I work with would tend to get into idle conversations about bitcoin during the day. A few of them I got involved in, and in one of them we got on to the subject of bitcoin mining. One guy asked the group if it would be worth it to dedicate his server at home to bitcoin mining. This is where your computer works on various difficult mathematical tasks, like factoring large numbers, and helping to merge block chains into each other, and as a reward you collect newly minted bitcoins that are conjured into existence. One guy commented immediately that it would never be worth it with a standard PC, you'd have to buy very expensive specialized hardware to do it. The original asker of the question objected that he wasn't using his server for anything else anyway, but the cynic replied that with a standard PC, the cost of running the hardware in an increased electric bill was greater than the payoff. (An idle server draws less electricity than one engaged in difficult mathematical computation.) This got us onto a tangent, about whether you could cool the computer down to a point where it didn't cost as much to run because it wasn't dissipating as much heat. Naturally, we went from there to discussing why computing necessarily dissipates heat, and what the ultimate physical limits of that process is. He mentioned that only quantum computers were reversible and didn't dissipate heat. I was first made aware of this fact in 1998 during a gradate quantum computing class that I took as an undergrad at Georgia Tech, where we read some relevant papers on the subject. But while I have long had a superficial understanding for why this was true, and I can repeat the standard things that people say about it, I admitted to him that I never quite understood it fully. He then started talking rapidly about the many worlds interpretation of quantum mechanics, and how the so-called "measurement problem" is really just decoherence and thermodynamics. He said he thought it was fairly straightforward, and this is the point where I mentioned that I had a PhD in theoretical physics and had spent many years working on related topics, but felt that there was just always a missing piece for me conceptually surrounding entropy, Maxwell's demon, reversibility, and dissipation. He shrugged and said "well, it seems straightforward, but I guess if you've studied this more then there must be more to the story than I'm aware of." I told him I didn't want to get into discussing interpretations of quantum mechanics, because it was too complex a subject, and I probably couldn't say anything more on reversibility because I didn't know how exactly to articulate what was missing from my understanding of things.
Every few years I pick up this topic, or one of a related set of topics that ties into it, and try to understand it, and I always make a bit more progress. I think the last bit of big progress I made was in reading Leonard Susskind's book on black holes and information, which helped me understand both black hole complimentarity and more about the density matrix and how entropy and information work in quantum mechanics. But this conversation renewed my interest again, so for the past month or so my mind has occasionally drifted back to it, and a couple days ago I managed to stumble onto the Wikipedia page for "Wandering Sets" which I think is a huge part of the missing piece for me. For some reason, wandering sets were never mentioned in any of my undergrad or graduate classes, and yet they seem absolutely crucial to understanding what the word "dissipation" means. Until yesterday, I had honestly never even heard of them. It's no wonder that I went through undergrad and graduate school always feeling frustrated when professors would use the word "dissipation" without giving any definition for it and just assuming that we all knew what it meant. Unfortunately, I feel like the Wikipedia page is poorly written and there are two facts they mention which seem obviously contradictory to me.
I will explain in part 2 what I've learned so far, and how wandering sets are related to the ergodic theorem, Liouville's theorem, and pretty much every foundational area of physics. And then I'll go into what I still don't understand about it--perhaps after this I need to find the right book that covers this stuff. It seems very weird to me that they always gloss over it in physics classes.
Recently, I've started thinking about them again, but through an unexpected route. A couple months ago, OkCupid added support for bitcoin, so as we got closer to the release of it, several of the people I work with would tend to get into idle conversations about bitcoin during the day. A few of them I got involved in, and in one of them we got on to the subject of bitcoin mining. One guy asked the group if it would be worth it to dedicate his server at home to bitcoin mining. This is where your computer works on various difficult mathematical tasks, like factoring large numbers, and helping to merge block chains into each other, and as a reward you collect newly minted bitcoins that are conjured into existence. One guy commented immediately that it would never be worth it with a standard PC, you'd have to buy very expensive specialized hardware to do it. The original asker of the question objected that he wasn't using his server for anything else anyway, but the cynic replied that with a standard PC, the cost of running the hardware in an increased electric bill was greater than the payoff. (An idle server draws less electricity than one engaged in difficult mathematical computation.) This got us onto a tangent, about whether you could cool the computer down to a point where it didn't cost as much to run because it wasn't dissipating as much heat. Naturally, we went from there to discussing why computing necessarily dissipates heat, and what the ultimate physical limits of that process is. He mentioned that only quantum computers were reversible and didn't dissipate heat. I was first made aware of this fact in 1998 during a gradate quantum computing class that I took as an undergrad at Georgia Tech, where we read some relevant papers on the subject. But while I have long had a superficial understanding for why this was true, and I can repeat the standard things that people say about it, I admitted to him that I never quite understood it fully. He then started talking rapidly about the many worlds interpretation of quantum mechanics, and how the so-called "measurement problem" is really just decoherence and thermodynamics. He said he thought it was fairly straightforward, and this is the point where I mentioned that I had a PhD in theoretical physics and had spent many years working on related topics, but felt that there was just always a missing piece for me conceptually surrounding entropy, Maxwell's demon, reversibility, and dissipation. He shrugged and said "well, it seems straightforward, but I guess if you've studied this more then there must be more to the story than I'm aware of." I told him I didn't want to get into discussing interpretations of quantum mechanics, because it was too complex a subject, and I probably couldn't say anything more on reversibility because I didn't know how exactly to articulate what was missing from my understanding of things.
Every few years I pick up this topic, or one of a related set of topics that ties into it, and try to understand it, and I always make a bit more progress. I think the last bit of big progress I made was in reading Leonard Susskind's book on black holes and information, which helped me understand both black hole complimentarity and more about the density matrix and how entropy and information work in quantum mechanics. But this conversation renewed my interest again, so for the past month or so my mind has occasionally drifted back to it, and a couple days ago I managed to stumble onto the Wikipedia page for "Wandering Sets" which I think is a huge part of the missing piece for me. For some reason, wandering sets were never mentioned in any of my undergrad or graduate classes, and yet they seem absolutely crucial to understanding what the word "dissipation" means. Until yesterday, I had honestly never even heard of them. It's no wonder that I went through undergrad and graduate school always feeling frustrated when professors would use the word "dissipation" without giving any definition for it and just assuming that we all knew what it meant. Unfortunately, I feel like the Wikipedia page is poorly written and there are two facts they mention which seem obviously contradictory to me.
I will explain in part 2 what I've learned so far, and how wandering sets are related to the ergodic theorem, Liouville's theorem, and pretty much every foundational area of physics. And then I'll go into what I still don't understand about it--perhaps after this I need to find the right book that covers this stuff. It seems very weird to me that they always gloss over it in physics classes.