Gibbs distribution and Bose statistics

[ala]

For photon gas chemical potential is zero. It is because derivate of free energy with respect to number (T, V fixed) of particles is zero in equilibrium. (free energy has minimum).
I was wondering why cannot I apply this reasoning to conclude (wrong) that chemical potential is zero for any Bose gas?

Explanation:

When deriving Bose statistics one applies Gibbs distribution for variable number of particles (grand canonical) to set of particles in the gas which are in a given quantum state. Then apply formula for grand potential (equal to -PV), mean number of particles in given quantum state is then obtained as derivate of grand potential with respect to chemical potential. So one gets Bose distribution. But if I say now that chemical potential is equal to derivate of free energy with respect to number of particles and latter is equal to zero (when gas is in equilibrium) I get wrong conclusion. Why?

 

[BigJDubs]

Because for Bose gas it is not enough to just consider equilibrium thermodynamics. One should also take into account that particle in gas can move from one quantum state to another. This means that particle number in one or another quantum state can change and therefore chemical potential can no longer be considered as derivate of free energy with respect to number of particles. Even when gas is in equilibrium, there can be non-zero chemical potential if particles can move from one quantum state to another.

 

[charng]

The reasoning you have applied is incorrect because it is based on a misunderstanding of the concept of chemical potential. The chemical potential is not equal to the derivative of the free energy with respect to the number of particles. Rather, it is the change in free energy with respect to the change in number of particles at constant temperature and volume. In other words, it is a measure of the energy required to add one more particle to the system.

In the case of a photon gas, the chemical potential is zero because photons are massless particles and can be added to the system without any additional energy. However, this does not mean that the chemical potential will always be zero for any Bose gas. The chemical potential will depend on the properties of the particles in the gas, such as their mass and interactions with each other. Therefore, the reasoning cannot be applied to conclude that the chemical potential is always zero for any Bose gas.

Furthermore, the fact that the derivative of the free energy with respect to the number of particles is zero at equilibrium does not necessarily mean that the chemical potential is also zero. This is because the free energy is a function of temperature, volume, and number of particles, while the chemical potential is only a function of temperature and volume. Therefore, it is important to understand the fundamental differences between these two quantities and not make assumptions based on incorrect reasoning.