English: Chemical potential of classical ideal gas and quantum ideal gases (Fermi gas, Bose gas) as a function of temperature, for a fixed density of particles. This is for the case of non-relativistic (massive, slow) particles in three dimensions.

A few features can be seen:

• As temperature increase, the Fermi and Bose gases approach the classical gas line, gradually.
• The classical gas has an interesting non-monotonic behaviour, but the quantum gases are monotonic (note that in 1D, however, the Fermi gas is also non-monotonic).
• The Fermi gas is always higher chemical potential, and Bose gas is lower chemical potential, compared to classical ideal gas.
• The * marker indicates the upper temperature of Bose condensation. Moving below this temperature, the chemical potential is pinned to ε₀, which is the energy of a particle at rest (i.e. the per-particle potential energy).
• The Fermi gas at low T (far below Fermi temperature T_F) saturates to a value ε₀ + ε_F. This extra ε_F=kT_F is its Fermi energy.

The figure has been scaled in a way that the particle degeneracy factor, density, mass, etc. are all factored out and irrelevant.

This was generated using the same source code as File:Quantum ideal gas pressure 3d.svg.