In the grand canonical ensemble a system can exchange particles with a reservoir so its number of particles is not fixed. So what does it mean that $\mu=0$ implies that the number $N$ of particles is not conserved, given that $N$ is always not conserved in the grand canonical ensemble?
I've read some posts about chemical potential (in particular when $\mu=0$) but I haven't found the answer to this question.
Answer
In the grand canonical ensemble, the number of particles is not fixed. Particles are continuously exchanged with a reservoir. The number of particles is not conserved but fluctuates whatever the value of the chemical potential $\mu$. The latter can be interpreted as the energy cost when a new particle is introduced in the system.
In relativistic quantum theories, new particles can appear in the system without coming from the reservoir and without any energy cost. For example, the number of particle may change due to the spontaneous production of a pair particle/anti-particle from a photon. In a photon gas, a photon can be absorbed by an atom and two photons may be emitted with the same total energy. In all these cases, the energy cost is zero so the chemical potential $\mu$ is necessarily equal to zero.
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