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Consider, a box with $N$ particles each of mass $m$, arranged at time $t=0$ to be moving parallel to the $x$-direction, with mean velocity v and mean distance l. The particles are spheres of mean radius r. Assume, the collisions are elastic and head-on, collisions occur when two particles move along the same direction, the collision with walls are also elastic. It seems, that the system would preserve its initial condition for a long time under this setting, but it is not the case.

How do I show that a small force, perhaps due to gravitational force of some distant particle-$f$ can destroy the predictability of the system? and how do I find the approximate number of collisions of particles that would take for the system to lose its predictability?

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  • $\begingroup$ You say "mean velocity," "mean radius," and "mean distance". Are these quantities drawn from some random distribution? If so what distribution? Specifying this might make an it possible to find an interesting answer to your question. (Also, distance from what?) $\endgroup$
    – Daniel
    Aug 1, 2020 at 5:41

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Everything you've described, including the box of particles and possibly gravity or other classical forces, are fully deterministic. In other words, given exact knowledge of the initial conditions, you can in principle know everything about the system at any later time. Classically, uncertainty can only arise from imprecise knowledge of the initial conditions. You can demonstrate that the uncertainty in initial conditions will grow by numerically simulating the time evolution of the system with various initial conditions within the initial error bars.

Chaos theory is the study of deterministic laws highly sensitive to initial conditions. At first glance I thought nonlinear terms, the source of sensitivity to initial conditions in the butterfly effect, in the equation of motion would be necessary. However, uncertainty can increase through simple geometry, for example turning uncertainty in a linear measurement into uncertainty in an angular measurement during elastic collisions of round objects. A simplified example that comes to mind is the the Galton Board, which is similar to the game of Plinko. The Galton board shows the random walk of a single puck as it collides with pegs on multiple levels. Each collision sends the ball down to the next level with a certain chance of falling to the left or right of the peg it collides with. The final horizontal location of the ball has a binomial probability distribution, which, if several levels are used, approximates a normal distribution. The relation of the Galton board to the original question is to imagine all but a single ball acts like a peg with a fixed location. I imagine you can slowly transform the Galton board into the situation described in the question by turning pegs into additional balls and studying the final probability distribution of a single ball. I also imagine as the pegs get turned into additional balls, the distribution becomes uniform throughout the box (as much uncertainty as possible for the balls in the box).

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    $\begingroup$ "Classically, uncertainty can only arise from imprecise knowledge of the initial conditions." - Couldn't one also be uncertain about the Hamiltonian, e.g. a classical version of a random-bond model? Another model of uncertainty that's sometimes interesting is uncertainty in the time for which the system has evolved. These are isomorphic to uncertainty in initial conditions, but it can sometimes be more natural to think about. $\endgroup$
    – Daniel
    Aug 1, 2020 at 5:39
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    $\begingroup$ For example, if the radii of the spheres are random, it makes more sense to model that as uncertainty in the Hamiltonian rather than uncertainty in the initial condition. $\endgroup$
    – Daniel
    Aug 1, 2020 at 5:42
  • $\begingroup$ There are ways for unpredictability if you restrict yourself slightly further to Newtonian mechanics though en.wikipedia.org/wiki/Norton%27s_dome $\endgroup$
    – J Kusin
    Aug 1, 2020 at 22:44

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