Changing distribution when going through a potential

[TheCanadian]

If we had a a sample of atoms in thermal equilibrium at a temperature, T, it would approximately follow a Maxwell-Boltzmann distribution and be isotropic. But if we now subject these atoms to a force in one direction (e.g. gravity, perhaps near a dense object), it will take the system out of equilibrium. But is there a (simple) way to now continue considering the distribution function for these atoms subject to this force? Can the Maxwell-Boltzmann distribution be adopted in form to one where the atoms are anisotropic with net velocity/acceleration in a particular direction? Or are there no well-defined distributions in such a case of being out equilibrium, and the characteristics of the sample would have to be considered at the level of each atom?

 

[DrClaude]

@Chestermiller has much more experience on this subject than I do, but here is my understanding.

First, you say that an external force will take the system out of equilibrium. This is not necessarily the case, and gravity is a good example here. You can still consider that air is locally at equilibrium, even though you have density and temperature gradients on large scales.

There are some cases where the external interaction will lead to a steady state instead of an equilibrium condition. In that case, you need to consider the specifics of the system. For instance, I know of cases in laser cooling where you get different temperatures in different directions (e.g., following a mostly Maxwell-Boltzmann distribution along and perpendicular to the laser axis, but with different temperatures) to distributions that are non-Gaussian (look up, for instance, Lévy statistics).

 

[Chestermiller]

@Chestermiller has much more experience on this subject than I do, but here is my understanding.

First, you say that an external force will take the system out of equilibrium. This is not necessarily the case, and gravity is a good example here. You can still consider that air is locally at equilibrium, even though you have density and temperature gradients on large scales.

There are some cases where the external interaction will lead to a steady state instead of an equilibrium condition. In that case, you need to consider the specifics of the system. For instance, I know of cases in laser cooling where you get different temperatures in different directions (e.g., following a mostly Maxwell-Boltzmann distribution along and perpendicular to the laser axis, but with different temperatures) to distributions that are non-Gaussian (look up, for instance, Lévy statistics).

I agree with everything in your 2nd paragraph regarding static equilibrium. I'm not familiar with how the velocity distribution is perturbed when there is flow, although the statistical thermodynamics of this has been studied in deriving the viscosity of gases under flow conditions.

 

[TheCanadian]

Interesting, thank you for those examples of external interactions resulting in a steady state.

First, you say that an external force will take the system out of equilibrium. This is not necessarily the case, and gravity is a good example here. You can still consider that air is locally at equilibrium, even though you have density and temperature gradients on large scales.

Well if the force was large enough that it would be out of equilibrium even at lower scale (e.g. a volume of gas at equilibrium at a temperature $T_i$ now placed a short distance, $r$, from a small spherical mass of $10^{30}$ kg, how could one adopt/transform the initial distribution function?

I am also curious about how this system with regions in local equilibrium could allow one to consider the larger system. For example, what would the larger system's distribution be if locally it can be described by a Maxwell-Boltzmann distribution?