A post by a friend of mine reminded me of the following problem: coin tossing is considered to be the epitome of randomness. On the other hand it is not clear that this process is not deterministic if one considers a coin as a rigid body. Diaconis et al.
famously showed that the probability that the coin (cought by hand) will rest on the side which is up at the initial moment is 0.51. They speculated that bouncing may even the chance, but that has not been shown. So, where the indeterminism comes from? Is it in the dynamic behavior of the coin itself (some kind of dynamic instability) or something else? The recent study ("Dynamics of coin tossing is predictable," doi: 10.1016/j.physrep.2008.08.003 suggests that it is fully deterministic. ...The analysis of the dynamical behavior of roulette goes back to Poincare. His results suggest that as the roulette ball is spun more and more vigorously the outcome number is independent of the initial conditions (initial conditions are washed out). For a large number of trials the numbers become close to the uniform distribution.
...All initial conditions are mapped into one of the final configurations. The initial conditions which are mapped onto heads configuration create heads basin of attraction while the initial conditions mapped onto tails configuration create tails basin of attraction. The boundary which separates heads and tails basins consists of initial conditions mapped onto the coin standing on the edge configuration. For an infinitely thin coin this set is a set of zero measure and thus with probability one the coin ends up either heads or tails. For the finite thinness of the coin this measure is not zero but the probability of edge configuration to be stable is low. An American 25 cents coin lands on the edge about one time in 6000 tosses.
...From the point of view of the dynamical systems the outcome of the tossing coin should be deterministic. As the initial conditions - final configuration mapping is strongly nonlinear one can expect deterministic unpredictability due to the sensitive dependence on the initial conditions or fractal basin boundaries. In other words one can pose the question, is anything chaotic in the dynamics of the tossed coin which can produce a random like behavior?
...A [Newtonian] mechanical model of coin tossing is constructed to examine whether the initial states leading to heads or tails are distributed uniformly in phase space. We give arguments supporting the statement that the outcome of the coin tossing is fully determined by the initial conditions, i.e. no dynamical uncertainties due to the exponential divergence of initial conditions or fractal basin boundaries occur.
...For the realistic coin in which the distance between the center of the mass and the geometrical center is small, it is sufficient to consider a simplified model of the ideal thin coin. The air resistance causes the deviation of the trajectory of the mass center from vertical axis and damps the rotation of the coin. When the distance of the free fall is small the effect of the air resistance can be neglected. During the free fall the sensitive dependence on the initial conditions has not been observed. The process of the coin bouncing on the floor has a significant influence on the final state (heads or tails). It has been observed that the successive impacts introduce sensitive dependence on the initial conditions leading to transient chaotic behavior.
...The basins of attraction of heads and tails (the sets of the initial conditions leading to both outcomes) show that the boundaries between heads and tails domains are smooth. This allow us to state our main result; there exists an open set of initial conditions for which the outcome of the coin tossing is predictable. In practice although heads and tails boundaries are smooth the distance of a typical initial condition from a basin boundary is so small that practically any finite uncertainty in initial conditions can lead to the uncertainty of the result of tossing. This is especially visible in the case of the coin bouncing on the floor, when with the increase of the number of impacts the basin boundaries become more complicated. In this case one can consider the tossing of a coin as an approximately random process.
...If the outcome of the long sequence of coin tosses is to give random results, it can only be because the initial conditions vary from toss to toss. In the previous section we show numerically that for each initial condition there exists the accuracy e for which the final state is predictable. In this section we try to explain why for practically small (but not infinitely small) e the coin tossing procedure can approximate the random process. A sequence of coin tosses will be random if the uncertainty e is large in comparison to the width W of the stripes characterizing the basins of attraction so that the condition e>W is essential for the outcome to be random. Uncertainty e depends on the mechanism of coin tossing while the quantity W is determined by the parameters of the coin.
So there is not much randomness in the coin tossing per se, even with bouncing off a smooth floor, as the coin, actually, is a mediocre amplifier of dynamical uncertainty. The randomness is introduced purely by the hand of a tosser. We are bad at controlling our motions.