Time-Energy and Time-Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam-Tamm-Messiah time-energy uncertainty relation $\tau_{F}\Delta_H\ge \hbar/2$ provides a general lower bound to the characteristic time $\tau_F =\Delta_F/|{\rm d} \langle F\rangle/dt|$ with which the mean value of a generic quantum observable $F$ can change with respect to the width $\Delta_F$ of its uncertainty distribution (square root of $F$ fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty $\Delta_H$ (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty $\Delta_S$ (square root of entropy fluctuations). For example, we obtain the time-energy--and--time-entropy uncertainty relation $(2\tau_{F}\Delta_H/ \hbar)^2+(\tau_{F}\Delta_S/{k_{\rm\scriptscriptstyle B}}\tau)^2 \ge 1$ where $\tau$ is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time-entropy uncertainty relation $\tau_{F}\Delta_S\ge {k_{\rm\scriptscriptstyle B}}\tau$, meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty $\Delta_S$.