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 ${\tau }_F{\Delta }_H\ge \hslash /2$ provides a general lower bound to the characteristic time ${\tau }_F={\Delta }_F/|\mathrm{d}⟨F⟩/dt|$ 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 ${(2{\tau }_F{\Delta }_H/\hslash )}^2+{({\tau }_F{\Delta }_S/k_{\mathrm{B}}\tau )}^2\ge 1$ 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 ${\tau }_F{\Delta }_S\ge k_{\mathrm{B}}\tau$ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty ${\Delta }_S$ .