Time-energy and time-entropy uncertainty relations in nonequilibrium quantum thermodynamics under steepest-entropy-ascent nonlinear master equations

Abstract

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam-Tamm-Messiah time-energy uncertainty relation τFΔH ≥ h/2 provides a general lower bound to the characteristic time τF = ΔF/|d〈F〉/dt| with which the mean value of a generic quantum observable F can change with respect to the width DF 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 ΔH (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 ΔS (square root of entropy fluctuations). For example, we obtain the time-energy-and-time-entropy uncertainty relation (τFΔH ≥ h)2 + (τFΔH ≥ h)2 ≥ 1 where t 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 τFΔS ≥ kBτ, meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty ΔS.