On Sun, Jan 12, 2014 at 10:53 AM, John Clark <[email protected]> wrote:
> > On Fri, Jan 10, 2014 at 2:23 PM, Jesse Mazer <[email protected]> wrot > > >> > In classical physics there is no limit in principle to your knowledge >> of the microstate. >> > > Yes, 150 years ago every physicist alive thought that, today we know > better. > We know better than to think classical physics represents an exact description of our universe, but it certainly describes a logically possible mathematical universe (note that in the previous paragraph of that message of mine you are replying to, I said Liouville's theorem would "be precisely true in a possible universe where the laws of classical physics hold exactly"...for reference, that message is at https://groups.google.com/d/msg/everything-list/hJ9bNWqoAzI/73DulLV7iyEJ ) > > >> > And in quantum physics, there is nothing in principle preventing you >> from determining an exact quantum state for a system; only if you believe >> in some hidden-variables theory >> > > And if you believe in some hidden-variable theory, ANY hidden-variable > theory, then you know that if things are realistic AND local then Bell's > inequality can NEVER be violated; and that would be true in every corner of > the multiverse provided that basic logic and arithmetic is as true there > as here. But experiment has shown unequivocally that Bell's inequality IS > violated. So you tell me, what conclusions can a logical person can draw > from that? > It tells us that either we must use a nonlocal hidden variables interpretation like Bohmian mechanics, or that hidden variables are wrong. Did you understand that in the sentence above that you quoted, I was saying that "there is nothing in principle preventing you from determining an exact quantum state for a system" in the case that the conjecture of hidden variables is FALSE, not in the case that it's true? If there are no hidden variables, then you can in principle perform an exhaustive measurement on a system that will give you its exact state vector in Hilbert space, putting it in a "pure state" rather than a "mixed state". So, this contradicts your claim that "the laws of physics insist that you will *always* be uncertain about the microstates"--a pure quantum state *is* a "microstate" in quantum physics without hidden variables, a macrostate would be a mixed state. > > > Do you disagree that starting from a randomly-chosen initial state which >> is likely to have something close to a 50:50 ratio of black to white >> squares, the board is likely to evolve to a state dominated by white >> squares, which would have lower entropy if we define macrostates in terms >> of the black:white ratio? > > > You said it yourself, the rules of the Game of Life are NOT reversible, > that means there is more than one way for something to get into a given > state. And the present entropy of a system is defined by Boltzman as the > logarithm of the number of ways the system could have gotten into the state > it's in now, therefore every application of one of the fundamental rules of > physics in the Game of Life universe can only increase entropy. > You are failing to specify whether you mean "state" to refer to microstate or macrostate and thus speaking ambiguously. The fact that the rules of the Game of Life are not reversible means that there is more than one way for something to get into a given microstate (even with reversible laws there is more than one way to get into a given macrostate). The entropy is defined not in terms of some vague notion of the "number of ways the system could have gotten into" its present microstate, but rather as the number of possible microstates the system might be in at this moment given that we only know the macrostate it's in at this moment. If we define macrostates for the "Toroidal Game of Life" in terms of the ratio of black to white squares, then the entropy of a given macrostate has nothing to do with looking at the board's possible states in the past, it's just a question of looking at the number of possible precise patterns of black and white squares that the board might have on the *current* time-increment that would give it that ratio of black:white on the current time-increment. For example, suppose we consider a very small 2x2 board with only 4 cells, and I use 0s to represent white cells and 1s to represent black cells. Then if the current macrostate is "2 black:2 white", the number of possible microstates would be 6, shown below: 11 00 10 10 10 01 01 10 01 01 00 11 If the macrostate were "1 black:3 white" there would be 4 possible microstates (and same for "3 black:1 white"), so this macrostate has a lower entropy: 10 00 01 00 00 10 00 01 And if the macrostate is "0 black:4 white" there's only one possible microstate (same for "4 black:0 white"), so this is the lowest possible entropy for a macrostate: 00 00 It's not hard to see why this pattern would continue to hold for larger boards--macrostates with a ratio that's closer to 1:1 will have a higher entropy than macrostates with a ratio that's more uneven. Thus, if the dynamics of the game of life are such that if you start with a randomly-selected microstate in a given macrostate, and the result is that the board is more likely than not to end up in a later macrostate where the ratio of black to white is *more* uneven, then for that initial macrostate the entropy tends to *decrease*, so the second law doesn't work in this case. > > The 2nd law is not restricted to initial conditions of "very low >> entropy", it says that if the entropy is anything lower than the maximum it >> will statistically tend to increase, and if the entropy is at the maximum >> it is statistically more likely to stay at that value than to drop to any >> specific lower value. >> > > If the universe started out in a state of maximum entropy then any change > in it, that is to say any application of one of the fundamental laws of > physics will with certainty DECREASE that entropy. And If the universe > started out in a state of ALMOST maximum entropy then any application of > one of the fundamental laws of physics will PROBABLY decrease that entropy. > No, again you must distinguish between macrostates and microstates rather than talking ambiguously about "states". If it starts out in a macrostate of maximum entropy there may be a vast number of possible ways the dynamics can take the system through a series of subsequent microstates that all belong to the the same macrostate, and thus all have the same entropy. And statistically, the system should be more likely to end up in another microstate with the same entropy than to end up in a microstate corresponding to any *specific* macrostate with a lower value of entropy (note that I did use that word "specific" in my comment above...if you just want the probability it will end up in *any* of the various possible lower-entropy macrostates, in this case I'm not sure if the 2nd law says it's more likely to keep the same entropy, perhaps there could be examples where the number of microstates N0 associated with the maximum-entropy macrostate M0 is smaller than the sum N1 + N2 + N3 + ... of microstates associated with each lower-entropy macrostate M1 + M2 + M3 + ... ) > > If the initial conditions deviated from maximum entropy even slightly, >> the second law says that an increase in entropy should be more likely than >> a decrease. >> > That would depend on initial conditions, just how slight the slight > deviation from maximum entropy was. > > I'm pretty sure that any mathematical formulation of the 2nd law wouldn't make exceptions for "how slight the slight deviation from maximum entropy was". Look for example at http://books.google.com/books?id=me7kjAzH5AIC&lpg=PA3&pg=PA3#v=onepage&q&f=falsewhich gives a simple derivation showing that if a system starts in macrostate b and you want to compare the probability that it will later be found in either macrostate a or macrostate c, then "If the state a is more probable than the state c (i.e. it has more entropy, according to Boltzmann), then this says that a system in the state b is more likely to make the transition to a than to the state c. That is, transitions are more likely to be observed in the direction of increasing entropy." There are no qualifications there about how close b is to maximum entropy, although the author does note the qualification that the argument "applies only in the long time limit" (he then offers a "less rigorous but nevertheless illuminating argument for finite intervals"). > > no one claims it would apply to all logically possible mathematical > universes, so how is it relevant to this discussion about whether the 2nd > law would apply to all such possible universes? > That wasn't what I was responding to. You said: > "since even though it's possible our universe could be a cellular > automaton, I think we can be pretty confident it's not a 2-dimensional > cellular automaton like the Game of Life!" > And I gave reasons why I am not "pretty confident" OK, sorry I misunderstood. But there's another reason to be pretty confident that the Game of Life rules don't describe fundamental physics in our universe: they are local and realistic, so there'd be no way to construct a scenario in the Game of Life where Bell's inequality is consistently violated. > >> So the rules of the Game of Life apply to some of the cells in the >>> grid but do not apply to others. What rules govern which cells must obey >>> the rules and which cells can ignore the rules, that is to say who is >>> allowed to ignore the laws of physics in that universe? >>> >> >> > No, they apply to all squares in the ideal platonic infinite board >> whose behavior you want to deduce, >> > > Then ratios become meaningless. > Not if you are assuming an infinite universe that follows these rules everywhere, but are *not* trying to define the macrostate of the entire universe, but just of an isolated system occupying a finite region of this larger universe. As I said at the very end of an earlier message to you at https://groups.google.com/d/msg/everything-list/hJ9bNWqoAzI/V3M_oN_qnmUJ -- "This is no different than defining the entropy of an isolated finite system in an infinite universe by looking at all the possible configurations within a finite volume of space large enough so that the system lies wholly inside it, and ignoring everything outside that volume (which should be fine if the system is truly isolated from outside influences)." This is how macrostates are pretty much always defined in real-world statistical mechanics. > > > but there is no need to actually *simulate* any of the squares outside >> the region containing black squares, because you know by the rules >> governing the ideal platonic infinite board that those squares will stay >> all-white as long as long as they are not neighbors with any black square >> > > I think you've got your colors backward because a solid block of active > cells does not stay a solid block. > It's just a matter of convention, but white is normally used for "dead" cells, black (or some other color) is normally used for "live" ones. This is how it's illustrated at https://en.wikipedia.org/wiki/Conway's_Game_of_Life and http://mathworld.wolfram.com/GameofLife.html for example, and if you do a google image search for "Conway Game of Life" and you'll see the majority of images fit this pattern. But never mind the point is that the pattern of active cells is constantly > expanding and shrinking in a unpredictable way (that is to say the only way > to know what it will do is watch it and see). Many Game of Life patterns > expand to infinity, so the shape and size of any closed figure you draw and > say you're only going to count cells inside that figure to obtain a ratio > would be entirely arbitrary. > That's why, when I initially brought up the idea of simulating a finite area of an ideal infinite board, I said that you would need the size of the grid area being simulated to be able to expand so it on each time-increment it would always be larger than the furthest region with black cells in each direction. See my message at https://groups.google.com/d/msg/everything-list/hJ9bNWqoAzI/h9oP-1L3eNMJwhere I said "Another alternative would be to imagine you do have an infinite grid, but with a starting state where there are only a finite pattern of black squares surrounded by an infinite number of white squares, then you can expand the size of the simulated grid if the region of black squares approaches its border, so that the grid always remains larger than the region of black squares". Jesse -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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