5. Systems with variable particle number

First we tackled isolated systems, then we considered systems in contact with a heat bath at temperature $T$. Now we consider systems which are also in diffusive contact with a particle reservoir at chemical potential $\mu$. In this case the Boltzmann distribution is modified and is called the Gibbs distribution.

This is useful in itself. However just as using the Boltzmann distribution freed us from the constraint that the total energy of all the particles had to add up to a given total energy, and allowed us to consider each particle independently, so using the Gibbs distribution frees us from the constraint that the total numbers of particles in each energy level has to add up to a fixed total, and allows us to treat each energy level independently. Once again we will use the fact that fluctuations in a macroscopic system are negligible to draw conclusions for isolated systems as well.