“there’s no reason information should be preserved for 20 seconds”—A sort of aside on this comment. For an idealised classical mechanical (reversible) system the information will be preserved forever. Chaos, roughly speaking, is moving the same information into different significant figures (or relations between them). So an initial uncertainty in the 20th decimal place soon becomes an uncertainty in the first significant figure as the information moves about. All that information you had about the leading digits of precision describing the initial state is still there in some sense, it has been mapped into some bizarre constraints connecting different figures deep behind the decimals, and is practically useless.
So, in this sense the question (I think) is whether that remaining information you have tells you anything about which side of the box will have more molecules at t=20. So, instead of a 50⁄50 guess, does the information let you get to 60⁄40 or whatever.
My feeling is that it must be almost worthless. Lets say that information takes you from a 50⁄50 guess to a 50+s/50-s guess. My intuition is that if we plotted this “s” value as a function of time it is likely an exponential decay, and that 20 seconds feels like a very long time compared to the timescales involved in the molecular motion. At t=0 s will be very close to 50 (only if the perturbed molecule is within one angstrom of the dividing line between left and right will s be less than 50 at t=0). But at t=20 it has undergone many half-lives. So its probably 10^{-big number} after 20 seconds.
The prediction that the information would be significant implies an assumption that s does not exponentially decay with time, but is described by some other function (maybe a constant). So I think the core of the dispute might different assumptions on the shape of the s(t) function.
A little (perhaps pedantic) point of observation:
“there’s no reason information should be preserved for 20 seconds”—A sort of aside on this comment. For an idealised classical mechanical (reversible) system the information will be preserved forever. Chaos, roughly speaking, is moving the same information into different significant figures (or relations between them). So an initial uncertainty in the 20th decimal place soon becomes an uncertainty in the first significant figure as the information moves about. All that information you had about the leading digits of precision describing the initial state is still there in some sense, it has been mapped into some bizarre constraints connecting different figures deep behind the decimals, and is practically useless.
So, in this sense the question (I think) is whether that remaining information you have tells you anything about which side of the box will have more molecules at t=20. So, instead of a 50⁄50 guess, does the information let you get to 60⁄40 or whatever.
My feeling is that it must be almost worthless. Lets say that information takes you from a 50⁄50 guess to a 50+s/50-s guess. My intuition is that if we plotted this “s” value as a function of time it is likely an exponential decay, and that 20 seconds feels like a very long time compared to the timescales involved in the molecular motion. At t=0 s will be very close to 50 (only if the perturbed molecule is within one angstrom of the dividing line between left and right will s be less than 50 at t=0). But at t=20 it has undergone many half-lives. So its probably 10^{-big number} after 20 seconds.
The prediction that the information would be significant implies an assumption that s does not exponentially decay with time, but is described by some other function (maybe a constant). So I think the core of the dispute might different assumptions on the shape of the s(t) function.