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Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations

Department of Mechanical and Industrial Engineering, Università di Brescia, via Branze 38, 25123 Brescia, Italy
Entropy 2019, 21(7), 679; https://doi.org/10.3390/e21070679
Received: 13 June 2019 / Revised: 7 July 2019 / Accepted: 8 July 2019 / Published: 11 July 2019
(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)
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Abstract

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H / 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 / ) 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 . View Full-Text
Keywords: uncertainty relations; maximum entropy production; steepest-entropy-ascent; quantum thermodynamics; second law of thermodynamics; entropy; nonequilibrium; Massieu uncertainty relations; maximum entropy production; steepest-entropy-ascent; quantum thermodynamics; second law of thermodynamics; entropy; nonequilibrium; Massieu
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Beretta, G.P. Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations. Entropy 2019, 21, 679.

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