Exploring Randomness and Chaos in Classical & Quantum Physics

[Isaac0427]

TL;DR Summary

In the scenarios of a coin flip and the motion of gas molecules, what is really at play: true randomness or chaos?

Note: This question involves both classical and quantum physics, so I didn't know where to put it.

I'll start with the coin flip:
People often compare electron spins to a coin flip, citing that coin flips are random. I am wondering if that is a true analogy, or just another faulty analogy between classical and quantum physics. My thinking is that while coin flips appear random, if you are given the the torque on the coin, the trajectory of the wind, and the initial state of the coin (including its position and whatnot), one could determine with certainty the result of the flip. This would be like a chaotic (local) hidden variable behind the result of the coin flip, which is forbidden in the case of electron spins. Given this, are coin flips purely random (like a quantum system), or do they just appear random due to underlying classical physics?

Now, with the gas:
Though there is an element of quantum mechanics at play here, I'm wondering if the "random motion" of a gas assumed in the kinetic theory is truly randomness. Say you zero in on each molecule and measure its position to an uncertainty of one micrometer, and then measure its momentum to the lowest uncertainty allowed. Though these may be large uncertainties for the size of the molecule, in terms of a 20L container, we know what that molecule is doing pretty well. Is the randomness here a result of true quantum randomness, or is it just due to the number of molecules that are all sensitive to initial conditions? That is, if I had to an allowed level of uncertainty the positions and momenta of every molecule, could I predict how the molecules would distribute over time?

Also, I'm not positive if I am using the term chaos correctly, but I think I am. Please correct me if I'm wrong.

Thank you in advance!

 

[Filip Larsen]

The nature of true randomness is always a very good question. Allow me to comment a bit on the coin flip part.

Coin flipping, die tossing and other similar experiments that after a short transient settles into one of several fixed point attractors and which can be modeled exclusively with classical physics are usually considered deterministic. If the dynamics contains many hard non-linearities (like a rolling die interacting with a roulette-like surface) the inherent uncertainties in initial state can of course quickly be amplified to macroscopic scale making a stochastic model for the steady-state (i.e. a mapping of the basins of attraction for each fixed point) more useful.

Such transient systems are usually not considered to be a chaotic system even if they have sensitivity to initial conditions and folding dynamics, as they lack the property of having dense periodic orbits. For instance, tossing a die onto a flat stationary plate is not considered chaotic system, but if you add drive the plate with a continuous periodical motion such that the die is kept in motion, then it may be characterized as chaotic (but since the die never settles calling it a die tossing experiment would then be misleading, I guess). A simple example of a mechanical system that do exhibit chaos is the driven double pendulum.

 

[atyy]

I think it is better to phrase the question in classical physics, so that there is no "true" randomness. Restricting ourselves to deterministic systems, one may ask whether random-looking behaviour is due to chaos or not. It turns out that the question is hard to answer experimentally. In theoretically well-defined deterministic systems, one can answer the question, and these same systems show that it is hard to experimentally tell what the "true" underlying theory is for real systems.

https://www.nature.com/articles/29721
Experimental evidence for microscopic chaos
Gaspard et al. Nature 394: 865–868(1998)

The above article by Gaspard was an interesting article that attempted to experimentally demonstrate that certain random-looking behaviour was due to chaos. However, commentators were not convinced:
https://www.nature.com/articles/44762
https://arxiv.org/abs/chao-dyn/9904041

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.79.031909
Very long transients, irregular firing, and chaotic dynamics in networks of randomly connected inhibitory integrate-and-fire neurons
Rüdiger Zillmer, Nicolas Brunel, and David Hansel
Phys. Rev. E 79, 031909 – Published 18 March 2009

The article by Zillmer et al shows several theoretically defined deterministic systems, some chaotic and some not, but they all produce random-looking behaviour.

 

[atyy]

The separate question of how to determine whether something is operationally random in the quantum sense is addressed by

https://arxiv.org/abs/1708.00265
Certified randomness in quantum physics
Antonio Acín, Lluis Masanes

In the quantum case, it is allowed that there may be hidden variables such that the entire system is deterministic, but our ignorance of the state of the hidden variables means the system appears random to us. The system will be operationally random as long as no one has experimental control of the hidden variables. The above paper shows how one may certify such operational randomness.