Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam−Tamm−Messiah time−energy uncertainty relation <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi&g...
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| Format: | Article |
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MDPI AG
2019-07-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/21/7/679 |
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doaj-87881de7d1524f5fa5739500267bbba9 |
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| record_format |
Article |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Gian Paolo Beretta |
| spellingShingle |
Gian Paolo Beretta Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations Entropy uncertainty relations maximum entropy production steepest-entropy-ascent quantum thermodynamics second law of thermodynamics entropy nonequilibrium Massieu |
| author_facet |
Gian Paolo Beretta |
| author_sort |
Gian Paolo Beretta |
| title |
Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations |
| title_short |
Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations |
| title_full |
Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations |
| title_fullStr |
Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations |
| title_full_unstemmed |
Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations |
| title_sort |
time–energy and time–entropy uncertainty relations in nonequilibrium quantum thermodynamics under steepest-entropy-ascent nonlinear master equations |
| publisher |
MDPI AG |
| series |
Entropy |
| issn |
1099-4300 |
| publishDate |
2019-07-01 |
| description |
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam−Tamm−Messiah time−energy uncertainty relation <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>H</mi> </msub> <mo>≥</mo> <mi>ℏ</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> provides a general lower bound to the characteristic time <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi> <mi>F</mi> </msub> <mo>=</mo> <msub> <mo>Δ</mo> <mi>F</mi> </msub> <mo>/</mo> <mrow> <mo>|</mo> <mi mathvariant="normal">d</mi> <mrow> <mo>〈</mo> <mi>F</mi> <mo>〉</mo> </mrow> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>|</mo> </mrow> </mrow> </semantics> </math> </inline-formula> with which the mean value of a generic quantum observable <i>F</i> can change with respect to the width <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>F</mi> </msub> </semantics> </math> </inline-formula> of its uncertainty distribution (square root of <i>F</i> fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>H</mi> </msub> </semantics> </math> </inline-formula> (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 <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>S</mi> </msub> </semantics> </math> </inline-formula> (square root of entropy fluctuations). For example, we obtain the time−energy-and−time−entropy uncertainty relation <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>H</mi> </msub> <mo>/</mo> <mi>ℏ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>S</mi> </msub> <mo>/</mo> <msub> <mi>k</mi> <mstyle displaystyle="false" scriptlevel="2"> <mi mathvariant="normal">B</mi> </mstyle> </msub> <mi>τ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>S</mi> </msub> <mo>≥</mo> <msub> <mi>k</mi> <mstyle displaystyle="false" scriptlevel="2"> <mi mathvariant="normal">B</mi> </mstyle> </msub> <mi>τ</mi> </mrow> </semantics> </math> </inline-formula>, meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>S</mi> </msub> </semantics> </math> </inline-formula>. |
| topic |
uncertainty relations maximum entropy production steepest-entropy-ascent quantum thermodynamics second law of thermodynamics entropy nonequilibrium Massieu |
| url |
https://www.mdpi.com/1099-4300/21/7/679 |
| work_keys_str_mv |
AT gianpaoloberetta timeenergyandtimeentropyuncertaintyrelationsinnonequilibriumquantumthermodynamicsundersteepestentropyascentnonlinearmasterequations |
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1725340152337793024 |
| spelling |
doaj-87881de7d1524f5fa5739500267bbba92020-11-25T00:27:23ZengMDPI AGEntropy1099-43002019-07-0121767910.3390/e21070679e21070679Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master EquationsGian Paolo Beretta0Department of Mechanical and Industrial Engineering, Università di Brescia, via Branze 38, 25123 Brescia, ItalyIn the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam−Tamm−Messiah time−energy uncertainty relation <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>H</mi> </msub> <mo>≥</mo> <mi>ℏ</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> provides a general lower bound to the characteristic time <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi> <mi>F</mi> </msub> <mo>=</mo> <msub> <mo>Δ</mo> <mi>F</mi> </msub> <mo>/</mo> <mrow> <mo>|</mo> <mi mathvariant="normal">d</mi> <mrow> <mo>〈</mo> <mi>F</mi> <mo>〉</mo> </mrow> <mo>/</mo> <mi>d</mi> <mi>t</mi> <mo>|</mo> </mrow> </mrow> </semantics> </math> </inline-formula> with which the mean value of a generic quantum observable <i>F</i> can change with respect to the width <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>F</mi> </msub> </semantics> </math> </inline-formula> of its uncertainty distribution (square root of <i>F</i> fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>H</mi> </msub> </semantics> </math> </inline-formula> (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 <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>S</mi> </msub> </semantics> </math> </inline-formula> (square root of entropy fluctuations). For example, we obtain the time−energy-and−time−entropy uncertainty relation <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>H</mi> </msub> <mo>/</mo> <mi>ℏ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>S</mi> </msub> <mo>/</mo> <msub> <mi>k</mi> <mstyle displaystyle="false" scriptlevel="2"> <mi mathvariant="normal">B</mi> </mstyle> </msub> <mi>τ</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mi>τ</mi> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>τ</mi> <mi>F</mi> </msub> <msub> <mo>Δ</mo> <mi>S</mi> </msub> <mo>≥</mo> <msub> <mi>k</mi> <mstyle displaystyle="false" scriptlevel="2"> <mi mathvariant="normal">B</mi> </mstyle> </msub> <mi>τ</mi> </mrow> </semantics> </math> </inline-formula>, meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty <inline-formula> <math display="inline"> <semantics> <msub> <mo>Δ</mo> <mi>S</mi> </msub> </semantics> </math> </inline-formula>.https://www.mdpi.com/1099-4300/21/7/679uncertainty relationsmaximum entropy productionsteepest-entropy-ascentquantum thermodynamicssecond law of thermodynamicsentropynonequilibriumMassieu |