Caveats and Extensions

The Maxwell-Boltzmann distribution does not always govern the distribution of energy in all systems.  How can this be?

  1. If energy is flowing into or out of the system, this moves the system out of equilibrium.  For example, a flask of gas exposed to a flame will not have a Maxwell-Boltzmann distribution of velocities.  Note that both the MB and MBS distributions include temperature as an input.  Therefore, systems whose temperature isn't well-defined can't have MB or MBS distributions.  A gas exposed to a heat source achieves different temperatures in different parts of the gas, and the temperature changes rapidly with time. This is how we know that MB and MBS distributions aren't applicable here.
  2. If heat was formerly flowing in or out of a system, then it will take a finite amount of time to return to equilibrium.  For example, if the flame is removed from the flask of gas described above, the system is not immediately describable by MB and MBS distributions.  If the system is left alone for a while (amount of time varies from one system to another), its parts will exchange energy and the MB/MBS distributions will be reestablished.
  3. If parts of the system cannot communicate, then they cannot achieve equilibrium.  If we have two closed flasks of gas at different temperatures, gas molecules from the two flasks cannot exchange energy with each other.  Therefore, each flask will come to equilibrium, but the two flasks will not equilibrate with each other. As an extension of this, if the components only communicate weakly, then equilibrium can take a long time to establish.  Consider a thermos bottle.  The molecules of air outside can exchange heat with the casing, which in turn can exchange heat with the hot beverage inside.  However, the insulation of the bottle makes this a VERY weak connection between inside and outside, and the beverage can take hours or days to approach room temperature.
  4. The Maxwell-Boltzmann probability distribution assumes that the particles are heavy enough that quantum-mechanical effects can be disregarded.  Systems with very light particles (gases of electrons or helium atoms, for example) equilibrate to other distributions because of quantum mechanical effects.  These distributions are called the Fermi-Dirac and Bose-Einstein distributions, and they are covered in statistical mechanics courses.  For most gases of molecules (heavier than helium), these effects can be disregarded.

© Andrew M. Rappe