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 τF ∆H ≥h/2 provides a general lower bound to the characteristic time τF = ∆F /|d F /dt| with which the mean value of a generic quantum observable F can change with respect to the width ∆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 ∆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 ∆S (square root of entropy fluctuations). For example, we obtain the time-energy-and-time-entropy uncertainty relation (2τF ∆H /h) 2 + (τF ∆S/kBτ ) 2 ≥ 1 where τ 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 τF ∆S ≥ kBτ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty ∆S.

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