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Probability is useful also when we have complete knowledge of the state of a model involving many particles (e.g. gas), but can't reliably integrate equations o

• lisper 4y ago
  > when we have complete knowledge of the state of a model involving many particles (e.g. gas) We cannot have complete knowledge of the state of a gas. We cannot even have complete knowledge of the state of a single particle because the positions and velocities of classical particles are real numbers but our knowledge is constrained to be finite, so there will always be some measurement error, and so chaos theory applies. (And physical reality is actually quantum, so we have the uncertainty principle at work as well.) > This problem is symmetric towards the future and towards the past. Only because you've carefully crafted this example to have this property; it is not true in general. You have constrained your knowledge of the present to exclude that which is needed to reliably extrapolate into the past. But it is possible to change your scenario and collect data that allows reliable extrapolation into the past, for example, by looking at the coin before it is flipped and remembering (or writing down) what you saw. But for the future this is not possible. That is the essential asymmetry. > See e.g. this Jaynes' paper I have a lot of respect for Jaynes so I'll take a look but it may be a while before I have the time to give it the attention it deserves. One thing I see right away that he gets right (well, mostly): in section 6 he observes that there is no such thing as "the entropy" of a system, which is correct. Entropy can only be measured relative to a state of incomplete knowledge. But calling it "anthropomorphic" is not right. It's not a function of human ignorance specifically, it's a function of ignorance in general.
  • effie 4y ago
    > ...it is not true in general. Agreed. But it is true for your original argument applied to a non-equilibrium state at any single time instant, while not assuming any more on the past or the future states. If we adopt further assumptions about the past states, but not about the future states, then we agree this problem of prediction/retrodiction is asymmetric between past and future and this may be the reason why the multiplicity argument does not have to result in past entropy being higher. In particular, there may be other facts on record about the past states and that would override/modify the multiplicity argument for the past. > it is possible to change your scenario and collect data that allows reliable extrapolation into the past, for example, by looking at the coin before it is flipped and remembering (or writing down) what you saw. But for the future this is not possible. That is the essential asymmetry. Yes, in any realistic scenario involving prediction and retrodiction at any time, we have some data on the past states (in some lucky cases, enough data for having certainty), but never data on the future states (we may have predictions, but those are not of the same quality as data). I agree. But in addition, you seem to argue that this asymmetry of availability of data is an essential fact to derivations of the 2nd law. Consider a scenario where we have an isolated system that is a mixture of miscible liquids that exists for ages but we have data on its state only for the last 2 years, and that data says its entropy was steadily rising the whole time. Now, given this data, was entropy increasing before the collection of data started, or was it decreasing? For short enough times around the edges, we can extrapolate the entropy curve based on collected data, but for long enough times, we can't. Using the multiplicity argument, including the data, we get that entropy of very distant past states was much larger than entropy of the first recorded state. Which contradicts 2nd law. Effect of that "essential asymmetry" has a time span that is not universal, but limited to times where collection of data was happening. Outside it vanishes, and using only that data and the multiplicity argument, the distant past becomes as cloudy(highly entropic) as the distant future is.
    • lisper 4y ago
      > it is true for your original argument No, it isn't. But we need to take this one step at a time. (And, I should also point out, this is not my argument. It's Boltzmann's and Shannon's.) > further assumptions about the past states They are not assumptions, they are observations. There is actual evidence that it is possible to remember the past but not the future. (I can't believe I'm actually having to persuade you of that. This is a pretty basic fact about reality.) > in some lucky cases, enough data for having certainty Nope, you can never be certain. Even the best measurements have error bounds, and our brains and recording devices have finite storage capacity. Our knowledge of the past can never be perfect. But -- and this is the important part -- our knowledge of the past can be vastly better than our knowledge of the future. > you seem to argue that this asymmetry of availability of data is an essential fact to derivations of the 2nd law I don't know if it's essential. I'm just saying that 1) it is a fact supported by evidence, not an assumption and 2) the second law follows from it (plus a few other things). It might be possible to derive the second law without it, I don't know, and I don't care because the actual fact of the matter is that this asymmetry between past and future does exist in point of actual fact. I don't care about hypothetical universes that behave differently from the one I inhabit. > Using the multiplicity argument Sorry, what is "the multiplicity argument"? > Effect of that "essential asymmetry" has a time span that is not universal, but limited to times where collection of data was happening. No, it doesn't matter whether data is actually being collected. What matters is that it is possible to collect data (about the past) because the past actually happened. Whether or not anyone actually bothered to make a record is irrelevant. If a tree falls in the forest, it actually falls even if no one makes a video of it.
• o_nate 4y ago
  That Jaynes paper is very cool. I'm surprised that Wolfram doesn't mention it in his overview of the literature on the second law.