Cutter Ketch said:
I believe he means that in a gas at equilibrium the probability for the net displacement of any single molecule over a fixed time is normally distributed.
Ah yes OK, I probably should have been able to get what was meant but it was the
Glenn G said:
a body of gas and plot where all they have got to
that threw me off.
Yes for a single particle when the fixed time is less than mean free time or if the walls of the container are at infinity.
The main point I was trying to make earlier hints at why this is:
muscaria said:
For a non interacting gas this corresponds to kinetic energy and thus determines the average speed and the distribution of speeds about this average, which is a gaussian distribution.
I was making reference to the Maxwell-Boltzmann distribution i.e that the speed of a particle is normally distributed.. which then leads to a normal distribution of displacement for fixed time.
Cutter Ketch said:
(I can't say that it is always true, but I suppose the central limit theorem makes it hard for it not to be true)
Here the normal distribution arises from the fact that the energy of a non-interacting gas is purely kinetic and therefore quadratic in the velocities. So the probability of measuring a particle with energy ##\epsilon=mv^2/2## is proportional to
$$exp\left[-\frac{mv_x^2+mv_y^2+mv_z^2}{2k_BT}\right]$$
where ##v=\sqrt{v_x^2+v_y^2+v_z^2}##.
Glenn G said:
I imagine deviations from the mean would depend on temperature, mean free path and speed of gas molecules. Any other?
As with all statistical mechanics observables, the spread from the average occurs due to energy fluctuating between the system and heat bath during thermal equilibrium and it is the temperature ##\textit{alone}## which determines this fluctuation (pressure doesn't play any role for instance). Increasing the energy of a system through work (pressure, E-M fields etc..) shifts energy levels upwards, whereas increasing energy by adding heat and raising the temperature doesn't change the energy levels but shifts the occupation towards higher energy levels: work changes the energy levels and leaves distribution unchanged, heat changes the distribution and leaves the energy levels unchanged.
So the deviation from the mean you are asking about, should be entirely fixed by the temperature of the system - for a given particle mass that is.. You can see this is the case from the probability distribution above. You can find everything you want to know from the Boltzmann distribution, e.g. the mean speed (and therefore mean displacement), the spread from the mean etc..