Hi I know the chemical potential of the phonon and photon is equal to zero. And I know the reason is the number of these particle isn't conserved.
But I don't know why the number of the phonon isn't conserved? Isn't the number of phonons 3N?
Thanks for your help.
The number of modes of the phonons is $3N$. However, for each mode, the number of phonons is not conserved.
An analog is that we have $3N$ different harmonic oscillators, and each of them can be in a generic excited state, labeled by an integer $n$. Here each harmonic oscillator is the analog of a mode of phonon, and for each harmonic oscillator, being in the excited state labeled by $n$ is the analog of having $n$ phonons for this mode.
$3N$ is the number of available vibrational modes, where $N$ is the number of atoms in the solid. Each mode can have a non-negative number of phonons, so the total number of phonons is not $3N$. In thermal equilibrium, the average number of phonons in each mode is given by Bose-Einstein distribution with zero chemical potential.
Consider a 1D harmonic oscillator. The eigenstates of the oscillator are characterized by the "the number of phonons" $n$, and the energy eigenvalues are $\hbar \omega_0(\frac 1 2 + n)$ where $\omega_0$ is the classical frequency of the oscillator. To use the phonon terminology, there is only one vibrational mode occupied by $n$ phonons. $n$ can be any non-negative number.
