= Finite-dimensional Lindbladians have stationary states
{c}
{title2=$\exists\rho_*:\ \mathcal L(\rho_*)=0$}
For any initial <density operator>, average its time-independent <Lindblad equation> trajectory as $\bar\rho_T=T^{-1}\int_0^T\rho(t)\,dt$. These averages stay in the compact convex set of <density operators>. Along a convergent subsequence, $\mathcal L(\bar\rho_T)=(\rho(T)-\rho(0))/T\to0$, so the limit is stationary. Thus a finite-dimensional physical <Affine Bloch equation> has at least one physical equilibrium, even though an arbitrary affine real equation can have none.
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