Randomness, Chaos, Quantum mechanical probability functions

Can someone explain these 3 concepts into a unified framework.

1. 
   

   Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision of the experiment) physics but, the complexity allows us to treat the system as random. (i.e we can always run an exact physical simulation to predict the coin's fall)

   
   


2. 
   
   

   Chaotic randomness: randomness as seen in weather or fluid flow, where the system is highly sensitive to initial conditions and as a result no matter what the precision of measurement/simulation is, we cannot predict the out come.

   
   


3. 
   

   Quantum randomness: The inherent probability distributions that we study in QM, where the electron's position probablity distribution is set by QM.

   
   

how do the concepts of randomness in coin toss, randomness in chaos and randomness in QM fit into a unified picture. Can someone please elaborate ?

Answer

You pretty much know it already. "Random" is a broad word that we use to mean that we can't predict behavior. Each of the three cases of randomness that you cite is unpredictable for a different reason, though - that's the difference.

Dice are random because they are complicated, chaotic pendulums are random because we aren't good enough to measure their initial position perfectly, and quantum systems are random because they aren't deterministic.

Expanding on that:

The randomness of a coin toss or a dice roll is based on an imprecise model. In principle, if we knew everything about the coin (its initial position, the forces applied, the density of air that slows it down, etc) then you could predict whether it winds up heads or tails with certainty. In the real world, nobody bothers, since constructing this model is very difficult. It depends sensitively on the height you are flipping the coin from, its initial spot on your thumb, and so on.

Chaotic randomness is due to imprecision in initial measurements alone. Different initial conditions do not cause smooth changes in the final outcome. A good example is the chaotic motion of the planets - if we try to predict the position of Saturn in 500,000,000 years, we get a certain position based on where it is now and all of the forces acting on Saturn. But if we choose a slightly different initial condition - say, 10cm further along in its orbit to start - then we get a totally different answer potentially hundreds of thousands of km off. Then we look at an initial condition in between, 5cm further along - and the deviation is even worse - it's now millions of kilometers off! In other words, it chaotic randomness arises in systems where improving your accuracy of measurement does not help. The only way to get the "true" answer is to have the exact initial value.

Quantum randomness is due to fundamental laws of nature. Quantum particles behave randomly on their own because that is just what they do, axiomatically. There is no initial measurement which could even be exact. The outcome is fundamentally nondeterministic, not a limit based on our models or our measurements.

In some sense, quantum randomness guarantees that we can never "beat" chaos by getting a perfect initial measurement. But it arises from a fundamentally different origin.