# Why is the partial differentiation of density operator with respect to time zero, for an ensemble in thermal equilibrium?

Sakurai initially says that density operator evolves with time because state kets evolve with time. But for an ensemble in thermal equilibrium, its partial differentiation is zero.

As far as I know, thermal equilibrium means the temperature is same everywhere and no heat flows between the particles. How is it anyway related to time? Why does the density operator stop evolving?

At thermal equilibrium, the density matrix can be written as $$$$\hat{\rho}_{\text{eq}} = \frac{e^{-\beta \hat{H}}}{\mathcal{Z}},$$$$ where $$\mathcal{Z} = \text{Tr}\ e^{-\beta \hat{H}}$$ is the partition function. The states are distributed thermally, and the density operator can be written diagonally in the basis of energy eigenstates, so it commutes with the Hamiltonian. Since the time evolution of the density operator is proportional to its commutator with the Hamiltonian, evolution stops if the commutator vanishes.