{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T11:06:48Z","timestamp":1774264008203,"version":"3.50.1"},"reference-count":152,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,11]],"date-time":"2019-07-11T00:00:00Z","timestamp":1562803200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam\u2013Tamm\u2013Messiah time\u2013energy uncertainty relation      \u03c4 F   \u0394 H  \u2265 \u210f \/ 2     provides a general lower bound to the characteristic time      \u03c4 F  =  \u0394 F  \/  | d  \u27e8 F \u27e9  \/ d t |      with which the mean value of a generic quantum observable F can change with respect to the width     \u0394 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     \u0394 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     \u0394 S     (square root of entropy fluctuations). For example, we obtain the time\u2013energy-and\u2013time\u2013entropy uncertainty relation       ( 2  \u03c4 F   \u0394 H  \/ \u210f )  2  +   (  \u03c4 F   \u0394 S  \/  k  B   \u03c4 )  2  \u2265 1     where    \u03c4    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\u2013entropy uncertainty relation      \u03c4 F   \u0394 S  \u2265  k  B   \u03c4    , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty     \u0394 S    .<\/jats:p>","DOI":"10.3390\/e21070679","type":"journal-article","created":{"date-parts":[[2019,7,11]],"date-time":"2019-07-11T11:28:28Z","timestamp":1562844508000},"page":"679","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Time\u2013Energy and Time\u2013Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9302-2468","authenticated-orcid":false,"given":"Gian Paolo","family":"Beretta","sequence":"first","affiliation":[{"name":"Department of Mechanical and Industrial Engineering, Universit\u00e0 di Brescia, via Branze 38, 25123 Brescia, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1649","DOI":"10.1103\/PhysRev.122.1649","article-title":"Time in the quantum theory and the uncertainty relation for time and energy","volume":"122","author":"Aharonov","year":"1961","journal-title":"Phys. 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