Quantum equilibrium ($\rho =|\Psi {|}^2$)

The ensemble density of actual configurations equals the squared norm of the guiding wave function. For spinless particles with $v_k=j_k/|\Psi {|}^2$, the Schrödinger continuity equation is ${\partial }_t|\Psi {|}^2+{\sum }_k{\nabla }_k\cdot (|\Psi {|}^2{v}_k)=0$. A density transported by the same guiding field satisfies the same equation, so initial equality is preserved. This equivariance is a consistency property, not a proof that every possible initial ensemble is in equilibrium. The physical account of the equilibrium assumption and subsystem statistics requires an additional argument.