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 [Formula: see text] provides a general lower bound to the characteristic time [Formula: see text] with which the mean value of a generic quantum observable F can change with respect to the width [Formula: see text] 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 [Formula: see text] (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 [Formula: see text] (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation [Formula: see text] where [Formula: see text] 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 [Formula: see text] , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty [Formula: see text].