Time-energy and time-entropy uncertainty relations in dissipative   quantum dynamics
  We derive exact relations and general inequalities that extend the usual time-energy uncertainty relations from the domain of unitary Hamiltonian dynamics to that of dissipative dynamics as described by a broad class of linear and nonlinear evolution equations for the density operator. For non-dissipative dynamics, by using the Schroedinger inequality instead of the Heisenberg-Robertson inequality, we obtain a general exact time-energy uncertainty relation which is sharper than the usual Mandelstam-Tamm-Messiah relation $\tau_F\Delta_H\ge \hbar/2$. For simultaneous unitary/dissipative dynamics, the usual time-energy uncertainty relation is replaced by a less restrictive relation that depends on the characteristic time of dissipation, $\tau$, and the uncertainty associated with the generalized nonequilibrium Massieu-function operator which defines the structure of the dissipative part of the assumed class of evolution equations. Within the steepest-entropy-ascent dissipative quantum dynamics of an isolated system introduced earlier by this author, we obtain the interesting time-energy and time-entropy uncertainty relation $(2\tau_F\Delta_H/ \hbar)^2+ (\tau_F\Delta_S/k_B\tau)^2 \ge 1$. We illustrate this result and various other inequalities by means of numerical simulations. 
Gian Paolo Beretta