That means that in 1D, particle interactions with each other cannot redistribute the kinetic energy and thus equalize the energy spectrum and thus make the system evolve towards the thermodynamic equilibrium. Only collisions with the floor or ceiling will be able to do that.A.T. said:Two identical particles simply swap their velocities in elastic 1D collision. So if you don't label them, and don't look, then afterwards you cannot tell if there was a collision, or if they just passed through each other. Therefore in a very thin, one particle wide cylinder of gas, the velocity distribution is not affected by whether the particles interact or just pass through each other.
The one-particle-wide-cylinder I was considering has a floor and ceiling, just like your 3D box. An the 1D math shows that particle-particle-interaction makes no difference to energy redistribution, just like you found for your 3D box (if I understand you correctly).Petr Matas said:That means that in 1D, particle interactions with each other cannot redistribute the kinetic energy and thus equalize the energy spectrum and thus make the system evolve towards the thermodynamic equilibrium. Only collisions with the floor or ceiling will be able to do that.
Maybe. But 1D seems to get the same (qualitative) result as you got for the 3D case, no?Petr Matas said:The situation is quite different in higher dimensions.
I meant something else: Interactions push the system towards thermodynamic equilibrium (they only don't for identical particles in 1D, as you have shown) by redistributing energy. If the system has not reached equilibrium yet, then the interactions will be important. In my analysis, I assume that the system is already in equilibrium. Once the equilibrium has been reached, further interactions will make no difference.A.T. said:An the 1D math shows that particle-particle-interaction makes no difference to energy redistribution, just like you found for your 3D box (if I understand you correctly).
So, for example, the equilibrium profile of density in a vessel should be independent on whether the vessel contains gas in a low density (and they rarely collide with each other, compared to walls) or high density (many collisions between wall and wall). And the collision cross-section should matter for the process of setting up equilibrium - but not for the equilibrium once established. Right?Petr Matas said:I meant something else: Interactions push the system towards thermodynamic equilibrium (they only don't for identical particles in 1D, as you have shown) by redistributing energy. If the system has not reached equilibrium yet, then the interactions will be important. In my analysis, I assume that the system is already in equilibrium. Once the equilibrium has been reached, further interactions will make no difference.
You're right. The density ##\rho(z)## as a function of altitude ##z## will be the same up to a multiplicative constant. We should just remember that we are talking about an ideal gas.snorkack said:the equilibrium profile of density in a vessel should be independent on whether the vessel contains gas in a low density (and they rarely collide with each other, compared to walls) or high density
Did you mean many collisions between particle and particle?snorkack said:high density (many collisions between wall and wall)
Exactly.snorkack said:And the collision cross-section should matter for the process of setting up equilibrium - but not for the equilibrium once established. Right?
Ah, I see that it was badly written.Petr Matas said:Did you mean many collisions between particle and particle?
I see, no problem. Thinking about this property, there is also a related quantity called mean free path.snorkack said:Ah, I see that it was badly written.
I meant that a particle has many collisions with other particles between collisions of the particle with a wall and a collision of the same particle with a different wall.