The Nonlinear Library LW - Is a random box of gas predictable after 20 seconds? by Thomas Kwa
Jan 26, 2024
02:12
Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Is a random box of gas predictable after 20 seconds?, published by Thomas Kwa on January 26, 2024 on LessWrong.
This came up as a tangent from this question, which is itself a tangent from a discussion on The Hidden Complexity of Wishes.
Suppose we have a perfect cubical box of length 1 meter containing exactly 1 mol of argon gas at room temperature.
At t=0, the gas is initialized with random positions and velocities θ drawn from the Maxwell-Boltzmann distribution.
Right after t=0 we perturb one of the particles by 1 angstrom in a random direction X to get the state m(θ,X).
All collisions are perfectly elastic, so there is no viscosity [edit, this is wrong; even ideal gases have viscosity] and energy is conserved.
For each possible perturbation, we run physics forward for 20 seconds and measure whether there are more gas molecules in the left side or right side of the box at t=20 seconds (the number on each side will be extremely close to equal, but differ slightly). Do more than 51% of the possible perturbations result in the same answer? That is, if is_left is the predicate "more gas molecules on the left at t=20", is |PrX[is_left(m(θ,X))]0.5|>0.01?
This is equivalent to asking if an omniscient forecaster who knows the position and velocity of all atoms at t=0 except for 1 angstrom of uncertainty in 1 atom can know with >51% confidence which side has more gas molecules at t=20.
I think the answer is no, because multiple billiard balls is a textbook example of a chaotic system that maximizes entropy quickly, and there's no reason information should be preserved for 20 seconds. This is enough time for each atom to collide with others millions of times, and even sound waves will travel thousands of meters and have lots of time to dissipate.
@habryka thinks the answer is yes and the forecaster could get more than 99.999% accuracy, because with such a large number of molecules, there should be some structure that remains predictable.
Who is right?
Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org
This came up as a tangent from this question, which is itself a tangent from a discussion on The Hidden Complexity of Wishes.
Suppose we have a perfect cubical box of length 1 meter containing exactly 1 mol of argon gas at room temperature.
At t=0, the gas is initialized with random positions and velocities θ drawn from the Maxwell-Boltzmann distribution.
Right after t=0 we perturb one of the particles by 1 angstrom in a random direction X to get the state m(θ,X).
All collisions are perfectly elastic, so there is no viscosity [edit, this is wrong; even ideal gases have viscosity] and energy is conserved.
For each possible perturbation, we run physics forward for 20 seconds and measure whether there are more gas molecules in the left side or right side of the box at t=20 seconds (the number on each side will be extremely close to equal, but differ slightly). Do more than 51% of the possible perturbations result in the same answer? That is, if is_left is the predicate "more gas molecules on the left at t=20", is |PrX[is_left(m(θ,X))]0.5|>0.01?
This is equivalent to asking if an omniscient forecaster who knows the position and velocity of all atoms at t=0 except for 1 angstrom of uncertainty in 1 atom can know with >51% confidence which side has more gas molecules at t=20.
I think the answer is no, because multiple billiard balls is a textbook example of a chaotic system that maximizes entropy quickly, and there's no reason information should be preserved for 20 seconds. This is enough time for each atom to collide with others millions of times, and even sound waves will travel thousands of meters and have lots of time to dissipate.
@habryka thinks the answer is yes and the forecaster could get more than 99.999% accuracy, because with such a large number of molecules, there should be some structure that remains predictable.
Who is right?
Thanks for listening. To help us out with The Nonlinear Library or to learn more, please visit nonlinear.org