Q:

Classical indeterminacy and measurement uncertainty

In one of his lectures, Feynman argues that classical physics is essentially indeterminate. Here I have tried to distill his original argument which was related to collisions of (classical) atoms in the water droplets of a waterfall [1]:
For small particles of characteristic size $R$ whose positions and momenta are initially measured with uncertainty ("error") $\left[\Delta x\right]_0$ and $\left[\Delta p\right]_0$ respectively, there will be an uncertainty in position $\left[\Delta x\right]_t$ occurring some time $t$ later. Let $\tau$ be some time for which $\left[\Delta x\right]_\tau >> R$, then the following should hold:
$$\tau \propto -\log(\left[\Delta x\right]_0)$$
I'd like to know an elementary argument as to why this is, or if there's some physical principal connecting initial measurement error logarithmically to $\tau$ (the time after which the system is practically indeterminate)?

[1] Feynman Lectures Volume III, Chapter 2-10:
"Given an arbitrary accuracy, no matter how precise, one can find a time long enough that we cannot make predictions valid for that long a time...The time goes, in fact, only logarithmically with the error, and it turns out that in only a very, very tiny time we lose all our information. If the accuracy is taken to be one part in billions and billions and billions - no matter how many billions we wish, provided we do stop somewhere - then we can find a time less than the time it took to state the accuracy - after which we can no longer predict what is going to happen!"

A:

Feynmann is talking about chaotic motion.
Very roughly, in a chaotic system if two trajectories differ by some amount $\Delta x$ the the difference between them increases exponentially with time:
$$ \Delta x(t) \approx \Delta x(0) e^{t/\lambda} $$
where $\lambda$ is a timescale characteristic of the system called the Lyapunov time.
The equation you quote is just the above equation rearranged to give:
$$ t \approx \lambda \log\left(\frac{\Delta x(t)}{\Delta x(0)}\right) $$
Chaotic systems are deterministic, but for any real system we can never measure the initial conditions precisely so in practice there is some experimental error $\Delta x(0)$ that is never zero, and this error increases exponentially with time. That makes it impossible to predict the motion of real chaotic systems for more than a limited time. Weather forecasting  is the canonical example of this.

