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 τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d 〈 F 〉 / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ 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 Δ 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 Δ S (square root of entropy fluctuations). For example, we obtain the time−energy-and−time−entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 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 τ F Δ S ≥ k B τ , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty Δ S .

Keywords

• uncertainty relations
• maximum entropy production
• steepest-entropy-ascent
• quantum thermodynamics
• second law of thermodynamics
• entropy
• nonequilibrium
• Massieu