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In Vol I, Chap.39-4 (temperature and kinetic energy),Feynman proved that, two gases in containers separated by a movable piston(which can probably be considered adiabatic)end up with an equilibrium where they are of the same pressure and temperature. Feynman stated the same pressure alone is not enough to keep the piston still steadily, as the molecules from the two gases are hitting the piston at different frequencies and thus, the piston jiggles. While I believe the two gases should be of the same pressure($n_{1}<m_{1}v_{1}^{2}>=n_{2}<m_{2}v_{2}^{2}>$) and the same frequency to hit the piston($n_{1}<v_{1}>=n_{2}<v_{2}>$),concluding that $m_1T_1=m_2T_2$.

However,Feynman stated that ” Let us now go back to the piston problem. We can give an argument, which shows that the kinetic energy of this piston must also be $\frac{1}{2}m_{1}v_{1}^{2}$ Actually, that would be the kinetic energy due to the purely horizontal motion of the piston,so, forgetting its up and down motion, it will have to be the same as $\frac{1}{2}m_{1}v_{1x}^2$,.Likewise, from the equilibrium on the other side, we can prove that the kinetic energy of the piston is $\frac{1}{2}m_{2}v_{2x}^{2}$. Although this is not in the middle of the gas, but is on one side of the gas, we can still make the argument, although it is a little more difficult, that the average kinetic energy of the piston and of the gas molecules are equal as a result of all the collisions. ”

Eventually Feynman concluded that the average kinetic energy of the two gases should be the same, and thus, $T_1=T_2$. So I am wondering how we can prove Feynman’s argument about the kinetic energy of the piston, and can we really conclude that $T_1$ should be equivalent to $T_2$, even though the piston is adiabatic?

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I do not know if you will be happy with the reassurance that everything is fine statistically with this quote of the abstract from Gruber and Piasecki:"Stationary motion of the adiabatic piston", Physica A 268 (1999) 412-423, but here it is. The actual calculations are in the body of the paper and it is not for the faint hearted, and I admit I have not gone through it... Gruber wrote several other papers examining various aspects of the adiabatic piston problem that are easier to read.

Abstract

We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures $T_1$ and $T_2$, separated by an adiabatic movable piston whose mass $M$ is much larger than the mass $m$ of the fluid particles. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter $\epsilon = \sqrt{m/M}$, and explicitly given up to order $\epsilon ^2$. In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the uctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where $M = m$.

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    $\begingroup$ Thank your for answering! I read part of the article and I guess it’s what I need. Maybe I’ll read some of the simpler articles before I’m ready to get through this one. Again, thank you for providing valuable learning materials! $\endgroup$
    – Aimikan
    Commented Mar 2, 2023 at 0:41
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    $\begingroup$ am glad to help $\endgroup$
    – hyportnex
    Commented Mar 2, 2023 at 0:42
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Here is what Feynman says on p 39-7:

Let us consider now what happens if we have two gases in containers separated by a movable piston as in Fig. 39-2.

There is no mention of the piston being adiabatic. Indeed, if the piston were adiabatic it would not possible to establish thermal equilibrium berween the two sides.

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    $\begingroup$ Thank you for answering! Such problem also came into my mind once, but in Feynman’s argument, the piston doesn’t seem to have to be able to conduct heat. Maybe things are different in nonequilibrium thermodynamics and quoting Gruber and Piasecki:"Stationary motion of the adiabatic piston" (the article provided by hyportnex)——“ In other words whether an adiabatic fixed wall becomes diathermic when it is allowed to move.”. I have not figured out whether it makes sense(as I haven’t read through the article), but I’ll accept hyportnex’s anwser for now. Again, thank you for answering. $\endgroup$
    – Aimikan
    Commented Mar 2, 2023 at 0:36
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    $\begingroup$ @ Aimikan What makes you think that Feynman's argument assumes the piston to be adiabatic? By definition adiabatic means no ability to exchange heat, hence no ability to equalize temperatures. The motion of the piston is of no relevance. The paper you are quoting talks about the motion of a movable adiabatic piston due to different temperatures on each side. $\endgroup$
    – Themis
    Commented Mar 2, 2023 at 18:23

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