On Fri, Jan 10, 2014 at 12:20 PM, John Clark <johnkcl...@gmail.com> wrote:
> On Thu, Jan 9, 2014 Jesse Mazer <laserma...@gmail.com> wrote: > > >I never claimed Liouville's theorem was a "fundamental law of physics" in >> itself, >> > > Good, I agree. > > > rather it is derivable as a mathematical consequence of certain features >> of the fundamental laws. >> > > And of the initial conditions! > No, it doesn't depend on initial conditions. No matter what set of initial microstates you choose at time T0, if you evolve each one forward to get a new set of microstates at time T1, then the volume of phase space occupied by the microstates at T0 will be precisely equal to the volume of phase space occupied by the microstates at T1. Do you disagree? > > > Liouville's theorem is derived in deterministic classical mechanics. >> > > Then Liouville's theorem can only be approximately true. > It'd be precisely true in a possible universe where the laws of classical physics hold exactly. Of course in our universe they don't, but there is apparently a quantum analogue of Liouville's theorem, though I don't understand it as well--see https://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)#Quantum_Liouville_equation > > >> > It [Liouville's theorem] only becomes statistical if you interpret the >> original set of microstates as representing your own uncertainty >> > > But that's the only way you can interpret it because the laws of physics > insist that you will *always* be uncertain about the microstates, all you > know are purely statistical things about the system, like its temperature > and pressure. > In classical physics there is no limit in principle to your knowledge of the microstate. 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 (like a theory that says that particles have precise position and momentum at all times, even though you can't measure them both simultaneously) would this be less-than-complete information about the microstate. > > > This line of discussion got started because I was disputing your >> statement that we can derive the 2nd law in a *purely* logical way like >> 2+2=5, with no need to invoke knowledge about the laws of physics that was >> based on observation. This would imply that *any* logically possible >> mathematical laws of nature would obey the 2nd law. >> > > Yes, *any* logically possible mathematical law of nature must actually do > something, or it shouldn't be called a law. If the initial state of a > system is in a state of lowest possible entropy, and if one of those laws > goes to work on that state then the entropy of the system in that state > will NOT go down. And that is the second law of thermodynamics. > Do you think my "Toroidal Game of Life" (a finite grid of cells with the edges identified, giving it the topology of a torus) is a mathematically well-defined possible universe? 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? > > > If you did not mean to suggest that we can know a priori the 2nd law is >> true because it would be true in any logically possible universe whose >> behavior follows mathematical laws, please clarify. >> > > That is exactly what I meant to suggest, provided that the initial > conditions were of very low entropy. > 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. > > > But I thought you were talking about logically possible universes as >> well, not just our universe >> > > If the initial conditions were of high entropy then applying a law of > physics to that mess would be just as likely to decrease its entropy as > increase it, therefore the second law would not be true and time would have > no arrow; in fact the very concept of time would have no meaning in that > universe. > 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. For example, suppose we have 10,000 gas atoms in a box with no external forces acting on them, and we divide the box into two equal halves, and choose an initial macrostate where 5,100 atoms are in one half of the box and 4,900 atoms are in the other half. If the laws of physics applied to this initial macrostate were such that the ratio of atoms in each side was more likely to get *further* from 50:50 than 51:49 rather than closer to 50:50, that would be a clear violation of the 2nd law. Do you disagree? If you agree with that, then it's easy to construct a similar example for the "Toroidal Game of Life"--if you have a board with 10,000 squares and you start with a ratio of white:black of 51:49, I think the later state would likely be even more dominated by white squares, not closer to 50:50. For that matter, I think this would be true even if you start with a very low-entropy macrostate, like a ratio of white:black of 99:1; if only 1% of squares are black and the state is otherwise chosen randomly, most of those black squares are likely to have less than two black neighbors which means they will flip to white on the next time-step, in which case the ratio of white:black would go to some even lower-entropy value like 500:1. > > > the very fact that you were willing to discuss the Game of Life >> suggested this, 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! >> > > Well... you can make a Turing Machine from the Game of Life. And according > to the Bekenstein Bound > The Bekenstein Bound is itself just a property of the particular laws of physics in our universe, 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? > > >>> 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, >>> >> >> >> So the ratio of white squares to black is a finite number divided by >> infinity. >> > > > No, because I said that in this case the region of the grid being >> *simulated* could still be finite >> > > 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, 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 they are not neighbors with any black square (because the initial conditions of the ideal platonic infinite board were specified as a given finite pattern of black squares with all other squares being white). In any case, if you find this confusing we can restrict the discussion to the "Toroidal Game of Life", unless you disagree that it's a mathematically well-defined possible universe. 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