Your finger will not get burned; it will suffer the cumulative damage resulting from an unusually high quantity of unrelated high-speed molecule attacks.
You could imagine a particle gun that shoots water molecules with the exact same speed distribution as hot water (carefully aligned so they don’t collide mid-beam), but all with the same direction—straight towards you.
The result of sticking your hand in such a beam would be roughly the same as putting it in hot water, ignoring the asymmetric momentum transfer. However, it is easy to see that you can extract useful energy from the beam.
Another pertinent example might be this: a metal shaft could spin so fast that its atoms’ velocity distribution could be the same as that of the (hotter!) gaseous form of the same metal. Yet the spinning of the shaft does not evaporate the metal.
Why? Because, to a typical observer of the shaft, its degrees of freedom are significantly more constrained. So, since the observer knows more about the shaft (that its atoms are in a solid lattice that moves in relative unison), that makes the shaft colder—and it allows you to extract more mechanical work from the shaft than if it were a hot gas with the same average particle velocities!
That is not in general possible. The speed at radius r is v = w r. Taking an arbitrary axisymmetric mass-distribution rho(r), we have a distribution of mass at speed v = w r that is U(r) = 2 pi h r^2 rho(r) and U(v) = 2 pi h v^2/w^2 rho(v/w) A monatomic gas at temperature T has a kinetic energy distribution of 2 Sqrt(E / Pi (kT)^3) exp(-E/kT) (dE), and a speed distribution of sqrt(2(m/kT)^3 / pi) v^2 exp(-m v^2 / 2 kT) (dv). By carefully tailoring rho(r) to an exponential, you can match this distribution (up to some finite cut-off, of course), at one specific match of angular speed w and temperature T.
This, of course, only matches speeds, not velocities, which will be a three-dimensional distribution. This spinning shaft of course, v_z = 0.
For a gas in a mass-sealed container, average vx, vy, vz are also zero, just as in the shaft, so this trivially matches.
You’re correct that I should have said speeds, and I should have mentioned that the shaft requires a special shape/density distribution, but the point stands: The molecular property distribution doesn’t by itself tell you how cold something is, or how much energy can be extracted—it’s also relevant how its degrees of freedom are constrained, which shows how your knowledge of (mutual information with) the system matters.
For a gas in a mass-sealed container, average vx, vy, vz are also zero, just as in the shaft, so this trivially matches.
Yes, but the average tells you little that’s meaningful: it’s equivalent to the overall velocity of the gas as whole.
I agree with your overall point. Temperature is not directly a property of the system, but of how we can characterize it, including what constraints there are on it. I just think that this wasn’t a great example for that point of view, precisely because you claimed agreement of distributions that doesn’t exist.
A better way to say explain this can use the shaft. A shaft has the very strong constraint that position and velocity are perfectly correlated. This constraint lets us extract virtually all the kinetic energy out. A gas that had the same distribution of velocities, but lacked this constraint, would be very hard to extract useful energy from. No simple arrangement could do much better than treating it as a thermalized gas with the same average kinetic energy (and it would quickly evolve so that the velocity distribution would match this).
I would call that ‘burned’. If I call ‘standing outside getting hit by lots of UV light’ sunburned then it seems fair to call getting hit by lots of high speed water molecules burned too.
the more uncertain you are about the velocity (not just speed)
Isn’t speed the same as velocity?
What is phase space? Is it the same as state space? You didn’t define it.
Does that make its thermodynamic entropy zero? Is the water colder, because we know more
about it?
Ignoring quantumness for the moment, the answer is: Yes! Yes it is!
I guess that if you stick your finger into the water it will still get burned, am I wrong?
And conversely, one subsystem cannot increase in mutual information with another subsystem,
without (a) interacting with it and (b) doing thermodynamic work.
It is not entirely clear to me how you arrived at this conclusion.
Your finger will not get burned; it will suffer the cumulative damage resulting from an unusually high quantity of unrelated high-speed molecule attacks.
You could imagine a particle gun that shoots water molecules with the exact same speed distribution as hot water (carefully aligned so they don’t collide mid-beam), but all with the same direction—straight towards you.
The result of sticking your hand in such a beam would be roughly the same as putting it in hot water, ignoring the asymmetric momentum transfer. However, it is easy to see that you can extract useful energy from the beam.
Another pertinent example might be this: a metal shaft could spin so fast that its atoms’ velocity distribution could be the same as that of the (hotter!) gaseous form of the same metal. Yet the spinning of the shaft does not evaporate the metal.
Why? Because, to a typical observer of the shaft, its degrees of freedom are significantly more constrained. So, since the observer knows more about the shaft (that its atoms are in a solid lattice that moves in relative unison), that makes the shaft colder—and it allows you to extract more mechanical work from the shaft than if it were a hot gas with the same average particle velocities!
That is not in general possible. The speed at radius r is v = w r. Taking an arbitrary axisymmetric mass-distribution rho(r), we have a distribution of mass at speed v = w r that is U(r) = 2 pi h r^2 rho(r) and U(v) = 2 pi h v^2/w^2 rho(v/w) A monatomic gas at temperature T has a kinetic energy distribution of 2 Sqrt(E / Pi (kT)^3) exp(-E/kT) (dE), and a speed distribution of sqrt(2(m/kT)^3 / pi) v^2 exp(-m v^2 / 2 kT) (dv). By carefully tailoring rho(r) to an exponential, you can match this distribution (up to some finite cut-off, of course), at one specific match of angular speed w and temperature T.
This, of course, only matches speeds, not velocities, which will be a three-dimensional distribution. This spinning shaft of course, v_z = 0.
For a gas in a mass-sealed container, average vx, vy, vz are also zero, just as in the shaft, so this trivially matches.
You’re correct that I should have said speeds, and I should have mentioned that the shaft requires a special shape/density distribution, but the point stands: The molecular property distribution doesn’t by itself tell you how cold something is, or how much energy can be extracted—it’s also relevant how its degrees of freedom are constrained, which shows how your knowledge of (mutual information with) the system matters.
Yes, but the average tells you little that’s meaningful: it’s equivalent to the overall velocity of the gas as whole.
I agree with your overall point. Temperature is not directly a property of the system, but of how we can characterize it, including what constraints there are on it. I just think that this wasn’t a great example for that point of view, precisely because you claimed agreement of distributions that doesn’t exist.
A better way to say explain this can use the shaft. A shaft has the very strong constraint that position and velocity are perfectly correlated. This constraint lets us extract virtually all the kinetic energy out. A gas that had the same distribution of velocities, but lacked this constraint, would be very hard to extract useful energy from. No simple arrangement could do much better than treating it as a thermalized gas with the same average kinetic energy (and it would quickly evolve so that the velocity distribution would match this).
Sure it does—it just requires a special shaft shaping/material. Hence the “a metal shaft could …”
Otherwise, we’re in agreement.
It’s a really great point; thank you.
I would call that ‘burned’. If I call ‘standing outside getting hit by lots of UV light’ sunburned then it seems fair to call getting hit by lots of high speed water molecules burned too.