On Maxwell-Boltzmann distribution for unevenly distributed gases Among the assumptions required for the applicability of Maxwell-Boltzmann statistics, there is one that assumes the particles to be evenly distributed in the volume $V$ they occupy.
Suppose now that they have within the same volume $V$ a density distribution $\rho(\mathbf{x}, t)$ instead. Is there a way to adapt the Maxwell-Boltzmann distribution of velocities
$$
{\displaystyle f(v)~d^{3}v=\left({\frac {m}{2\pi kT}}\right)^{3/2}\,e^{-{\frac {mv^{2}}{2kT}}}~d^{3}v}?
$$
to this specific case?
 A: Maxwell-Boltzmann distribution (and a big part of what is studied in introductory statistical mechanics) refers to a gas in equilibrium. Maxwell distribution specifically deals with the case where no external potential is present, in which case the equilibrium distribution of gas has uniform density. Boltzmann generalizes this to the case when external potential is present, and the density might be non-uniform in this case, $$W(\mathbf{p},\mathbf{x})\propto e^{-\frac{H(\mathbf{p},\mathbf{x})}{k_BT}}.$$
However, general non-unform density is not necessarily in thermal equilibrium and thus not necessarily obeys MB. In this case one needs to use non-equilibrium statistical mechanics (kinetic equation, etc.) - although this one also often uses an assumption of local equilibrium.
A: Yes, you can adapt the Maxwell-Boltzmann distribution to a case of non uniform spatial distribution of particles as long as the process is isothermal. This means that mixing should have no effect on temperature.  This condition is satisfied for ideal gases and also for ideal solutions. In both cases if components start with arbitrary spatial distribution but at the same temperature, temperature will be maintained during mixing and both components will satisfy the same MB distribution.
A: What is the point of evenly distributed or not, when result is in statistical form then there is variation. This distribution although assume thermal equilibirium to have only one variable as speed or mechanical energy, otherwise this distribution itself meant there is no thermal equilibirium. At equilibirium there is no distribution. But it is hard to have equilibirium if density is more or source has low power.
