At thermodynamic equilibrium, there is thermal, mechanical, and diffusive equilibrium. Does this imply:
$$d\mu = dT = dV = 0$$
$$dU = TdS - PdV \implies dU = TdS$$
Here, I know entropy is maximum, so perhaps $dS=0$ and hence $dU = 0$? I also don't think I can write $d\mu = 0$ as this may be sort of abusive treating of the chemical potential.