Standing waves of pressure (a.k.a. sound resonances) are macroscopic patterns with some persistence. Their amplitudes would be really low (≈kT of sound energy per resonance) in a random initial configuration, but if you knew every particle, you would still know the starting amplitude and phase of each standing wave, and could project it out, and one particle won’t appreciably affect the starting amplitude and phase I think.
So, do the standing waves have enough persistence to matter? How much do those standing waves decay after 20 seconds? I dunno. I know how to calculate all the frequencies, but I have no idea how to calculate the Q-factors—at least, not off the top of my head.
I’m also not sure which standing waves (if any?) bear on left-right difference in particle count.
All this is on the edge of my knowledge, so I could well be wrong. Insert “I thinks” and “from what I remembers” as appropriate throughout what follows.
If we start with non-interacting air molecules then the standing waves of pressure are the normal modes of the container. With non-interacting molecules the movement of a single molecule is not necessarily chaotic, whether it is or not depends on the shape of the container.
Assuming no loss (Q factor of infinity) then, knowing that the motion contains some contribution from a particular normal mode allows us to plot that normal mode (sine wave say) out to infinite future (and past) times. However, in a chaotic system it is required that the frequencies of the normal modes are approximately equally spaced. Their are no big gaps in the frequencies. I think the relevance of this to this question is that if all we know is that normal mode number 27 has some amplitude that sine wave we can infer out is added to all the other modes, which add white noise. (The mode spacing argument ensuring the noise is in fact white, and not colored noise that we could exploit to actually know something). So, assuming that mode 27 only has a typical amplitude we learn very little.
When we add collisions between the air molecules back in, then I believe it is chaotic for any shape of container. Here the true normal modes of the total system include molecule bumping, but the standing waves we know about from the non-interacting case are probably reasonably long-lived states.
Yeah, standing waves where what me and Thomas also most talked about when we had a long conversation about this. Seems like there would be a bunch, and they wouldn’t obviously decay that fast.
My guess is that it’s extremely unlikely that enough energy is concentrated in a standing wave at initialization for it to not dissipate in 20 seconds. By equipartition it should be extremely unlikely for energy to be concentrated in any degree of freedom in any physical system, but I don’t know enough physics to be confident that this argument applies.
Certainly there is no conservation of standing wave amplitude, because with two billiard balls waves can form and dissipate. The question is how long it takes for waves of the tiny amplitudes caused by initialization to dissipate.
Why are you guys talking about waves necessarily dissipating, wouldn’t there be an equal probability of waves forming and dissipating given that we are sampling a random initial configuration, hence in equilibrium w.r.t. formation/dispersion of waves?
If you look at a noise-driven damped harmonic oscillator, the autocorrelation of “oscillator state at time t1” and “oscillator state at time t2” cycles positive and negative at the oscillator frequency [if it’s not overdamped], but with an envelope that gradually decays to zero when t1 and t2 get far enough apart from each other.
This whole thing is time-symmetric—knowing the state at time 0 is unhelpful for guessing the state at very positive timestamps AND unhelpful for guessing the state at very negative timestamps.
But the OP question was about fixing an initial state and talking about later times, so I was talking in those terms, which is more intuitive anyway. I.e., as time moves forward, the influence of the initial state gradually decays to zero (because the wave is damped), while meanwhile the accumulated influence of the noise driver gradually increases.
Yes, it would be more correct to say the question is how long it takes for the probability distribution of the amplitude and phase of a given oscillation mode to be indistinguishable from that of any other random box of gas.
Yes by the equipartition theorem there’s an average of kT of energy in each standing wave mode at any given moment. Might be fun to calculate how many left-right atoms that corresponds to—I think that calculation should be doable. I imagine that for the fundamental mode, it would be comparable to the √(number of atoms in the box) difference that we expect for other reasons.
It’s continuous and exponential. If amplitude of standing wave mode N decays by a factor of 2 in X seconds, then it‚ it’s the same X whether the initial amplitude in that mode is macroscopic versus comparable-to-the-noise-floor. (Well, unless there are nonlinearities / anharmonicities, but that’s probably irrelevant in this context.) But meanwhile, noise is driving the oscillation too. So anyway, I think it really matters how X compares to 20 seconds, which again is something I don’t know.
Standing waves of pressure (a.k.a. sound resonances) are macroscopic patterns with some persistence. Their amplitudes would be really low (≈kT of sound energy per resonance) in a random initial configuration, but if you knew every particle, you would still know the starting amplitude and phase of each standing wave, and could project it out, and one particle won’t appreciably affect the starting amplitude and phase I think.
So, do the standing waves have enough persistence to matter? How much do those standing waves decay after 20 seconds? I dunno. I know how to calculate all the frequencies, but I have no idea how to calculate the Q-factors—at least, not off the top of my head.
I’m also not sure which standing waves (if any?) bear on left-right difference in particle count.
All this is on the edge of my knowledge, so I could well be wrong. Insert “I thinks” and “from what I remembers” as appropriate throughout what follows.
If we start with non-interacting air molecules then the standing waves of pressure are the normal modes of the container. With non-interacting molecules the movement of a single molecule is not necessarily chaotic, whether it is or not depends on the shape of the container.
Assuming no loss (Q factor of infinity) then, knowing that the motion contains some contribution from a particular normal mode allows us to plot that normal mode (sine wave say) out to infinite future (and past) times. However, in a chaotic system it is required that the frequencies of the normal modes are approximately equally spaced. Their are no big gaps in the frequencies. I think the relevance of this to this question is that if all we know is that normal mode number 27 has some amplitude that sine wave we can infer out is added to all the other modes, which add white noise. (The mode spacing argument ensuring the noise is in fact white, and not colored noise that we could exploit to actually know something). So, assuming that mode 27 only has a typical amplitude we learn very little.
When we add collisions between the air molecules back in, then I believe it is chaotic for any shape of container. Here the true normal modes of the total system include molecule bumping, but the standing waves we know about from the non-interacting case are probably reasonably long-lived states.
Yeah, standing waves where what me and Thomas also most talked about when we had a long conversation about this. Seems like there would be a bunch, and they wouldn’t obviously decay that fast.
My guess is that it’s extremely unlikely that enough energy is concentrated in a standing wave at initialization for it to not dissipate in 20 seconds. By equipartition it should be extremely unlikely for energy to be concentrated in any degree of freedom in any physical system, but I don’t know enough physics to be confident that this argument applies.
Certainly there is no conservation of standing wave amplitude, because with two billiard balls waves can form and dissipate. The question is how long it takes for waves of the tiny amplitudes caused by initialization to dissipate.
Why are you guys talking about waves necessarily dissipating, wouldn’t there be an equal probability of waves forming and dissipating given that we are sampling a random initial configuration, hence in equilibrium w.r.t. formation/dispersion of waves?
If you look at a noise-driven damped harmonic oscillator, the autocorrelation of “oscillator state at time t1” and “oscillator state at time t2” cycles positive and negative at the oscillator frequency [if it’s not overdamped], but with an envelope that gradually decays to zero when t1 and t2 get far enough apart from each other.
This whole thing is time-symmetric—knowing the state at time 0 is unhelpful for guessing the state at very positive timestamps AND unhelpful for guessing the state at very negative timestamps.
But the OP question was about fixing an initial state and talking about later times, so I was talking in those terms, which is more intuitive anyway. I.e., as time moves forward, the influence of the initial state gradually decays to zero (because the wave is damped), while meanwhile the accumulated influence of the noise driver gradually increases.
Yes, it would be more correct to say the question is how long it takes for the probability distribution of the amplitude and phase of a given oscillation mode to be indistinguishable from that of any other random box of gas.
Yes by the equipartition theorem there’s an average of kT of energy in each standing wave mode at any given moment. Might be fun to calculate how many left-right atoms that corresponds to—I think that calculation should be doable. I imagine that for the fundamental mode, it would be comparable to the √(number of atoms in the box) difference that we expect for other reasons.
It’s continuous and exponential. If amplitude of standing wave mode N decays by a factor of 2 in X seconds, then it‚ it’s the same X whether the initial amplitude in that mode is macroscopic versus comparable-to-the-noise-floor. (Well, unless there are nonlinearities / anharmonicities, but that’s probably irrelevant in this context.) But meanwhile, noise is driving the oscillation too. So anyway, I think it really matters how X compares to 20 seconds, which again is something I don’t know.