The Paradox of Gas Density: Why Is the Center Not Higher?

[ramollari]

Suppose you have an ideal gas in a closed container. The gas molecules will be moving at random at very high speeds.
Now I've figured out that the probability of a gas molecule for being in the center of the container is higher than the probability of being near the sides. Then should we also conclude that the density (and pressure) of the gas at the center is higher than that near the sides? This conclusion doesn't seem to hold.

 

[Galileo]

How's the probability of a particle being near the center higher?
If the probability density of a particle's position is uniform, then you'd have more chance of finding it near the sides, since the volume of a region with the sides as an outer boundary is generally much larger than a volume that is kept near the center. This doesn't affect the density, though.

 

[ramollari]

Take the particle isolated. If it is hit by other particles, then it has more probability to be found near the centre.
Take a look at the following simulation with Brownian motion, by setting m1/m2 = 1. http://home.wanadoo.nl/perpetual/brownian.html The container area colours blue (particle's path) near the center much faster.

 

[ethane]

On a side note, the effects of gravity would lead to a particle overall being more likely to be found near the bottom of a standing container. Aside from that, I have not enough experience with physics probability to comment further.

 

[Galileo]

Take the particle isolated. If it is hit by other particles, then it has more probability to be found near the centre.
Take a look at the following simulation with Brownian motion, by setting m1/m2 = 1. http://home.wanadoo.nl/perpetual/brownian.html The container area colours blue (particle's path) near the center much faster.

That's a nice link.

But it's not correct to say that the probability of finding a particle near the center is higher because the blue line crosses near the center more often. What you need to look at is the position of the red particle. It's presence in the box at any instance is equally likely to be anywhere.
Look at all the gray dots. They are uniformly distributed over the box too.

 

[Doc Al]

That's a nice link.

Seems a bit cranky to me.

But it's not correct to say that the probability of finding a particle near the center is higher because the blue line crosses near the center more often. What you need to look at is the position of the red particle. It's presence in the box at any instance is equally likely to be anywhere.
Look at all the gray dots. They are uniformly distributed over the box too.

Right. I think the animation was probably written to illustrate brownian motion and has several built-in assumptions which break down when you put m1/m2 = 1. If it were accurate, then for m1/m2 = 1, the red dot would move exactly like any other dot. But it doesn't. While the gray dots are uniformly distributed and move at some average speed, the red dot zooms all over the place. I don't think so.

 

[mapper]

put n=200 and m1/m2 = 1.