Physics:Quantum Relaxation and thermalization - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:27Z

Restore missing Quantum reference definitions

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 </math>
-so the microscopic dynamics are reversible.<ref name="Polkovnikov2011"/> Yet experimentally relevant observables may still relax because of dephasing between many energy eigenstates, redistribution of correlations, and the practical inaccessibility of detailed phase information in local measurements.<ref name="GogolinEisert2016"/><ref name="Polkovnikov2011"/>
+so the microscopic dynamics are reversible.<ref name="Polkovnikov2011">{{cite journal
 A thermal equilibrium state is commonly represented by the density operator
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 </math>
-where <math>\beta = 1/(k_B T)</math> and <math>Z</math> is the partition function.<ref name="DAlessio2016"/>
+where <math>\beta = 1/(k_B T)</math> and <math>Z</math> is the partition function.<ref name="DAlessio2016">{{cite journal
 ==Equilibration versus thermalization==
-Equilibration and thermalization are related but distinct concepts.<ref name="GogolinEisert2016"/> A system '''equilibrates''' when expectation values of observables become nearly time-independent for most late times, even if the exact state continues to evolve unitarily. It '''thermalizes''' when those stationary values coincide with predictions from an appropriate thermal ensemble determined by conserved quantities such as energy.<ref name="RigolNature2008"/>
+Equilibration and thermalization are related but distinct concepts.<ref name="GogolinEisert2016"/> A system '''equilibrates''' when expectation values of observables become nearly time-independent for most late times, even if the exact state continues to evolve unitarily. It '''thermalizes''' when those stationary values coincide with predictions from an appropriate thermal ensemble determined by conserved quantities such as energy.<ref name="RigolNature2008">{{cite journal
 Thus a system can relax without becoming thermal in the ordinary Gibbs sense. This distinction is central in modern nonequilibrium quantum theory.<ref name="DAlessio2016"/><ref name="GogolinEisert2016"/>
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 ==Systems resisting thermalization==
-Not all systems thermalize rapidly or at all. Integrable systems relax to GGEs rather than conventional Gibbs states.<ref name="RigolIntegrable2007"/> Many-body localized systems can retain memory of their initial conditions in local observables for arbitrarily long times, preventing ordinary thermalization.<ref name="NandkishoreHuse2015"/> Other unusual nonthermal behavior appears in systems with dynamical constraints, quantum scars, or special symmetry structures.<ref name="NandkishoreHuse2015"/><ref name="DAlessio2016"/>
+Not all systems thermalize rapidly or at all. Integrable systems relax to GGEs rather than conventional Gibbs states.<ref name="RigolIntegrable2007"/> Many-body localized systems can retain memory of their initial conditions in local observables for arbitrarily long times, preventing ordinary thermalization.<ref name="NandkishoreHuse2015">{{cite journal
 ==Physical interpretation==
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 {{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
-==References==
+= References =
 {{reflist|3}}
 {{Author|Harold Foppele}}

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2026-05-22T11:33:50Z

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2026-05-22T10:49:22Z

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2026-05-20T12:26:38Z

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2026-05-20T09:05:28Z

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-'''Quantum relaxation and thermalization''' describe how a system prepared out of equilibrium evolves toward a stationary state and, under suitable conditions, toward a state consistent with [[Physics:Quantum Statistical mechanics|quantum statistical mechanics]].<ref name="DAlessio2016">{{cite journal
+'''Relaxation and thermalization''' quantum relaxation and thermalization describe how a system prepared out of equilibrium evolves toward a stationary state and, under suitable conditions, toward a state consistent with quantum statistical mechanics. Relaxation refers to the approach of observables toward long-time stationary values, while thermalization means that those values are described by thermal ensembles such as the canonical or microcanonical ensemble. In general, the natural tendency of many-body systems is toward thermal equilibrium, increasing entropy and erasing accessible memory of the initial preparation, although important exceptions exist in integrable, localized, or specially constrained systems. Relaxation and thermalization in quantum systems describe how nonequilibrium states evolve toward stationary behavior through dephasing, interactions, and coupling to an environment. For an isolated quantum system, the full state evolves unitarily according to the Schrödinger equation,
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	In general, the natural tendency of many-body systems is toward ~~[[Physics:~~thermal equilibrium~~\|thermal equilibrium]]~~, increasing entropy and erasing accessible memory of the initial preparation, although important exceptions exist in integrable, localized, or specially constrained systems.<ref name="Polkovnikov2011">{{cite journal		In general, the natural tendency of many-body systems is toward thermal equilibrium, increasing entropy and erasing accessible memory of the initial preparation, although important exceptions exist in integrable, localized, or specially constrained systems.<ref name="Polkovnikov2011">{{cite journal
	\|last1=Polkovnikov		\|last1=Polkovnikov
	\|first1=Anatoli		\|first1=Anatoli

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	==Magnetic resonance as a relaxation example==		==Magnetic resonance as a relaxation example==
	A concrete and experimentally important example of quantum relaxation occurs in ~~[[Physics:Magnetic resonance imaging\|~~magnetic resonance imaging]] (MRI) and ~~[[Physics:Nuclear magnetic resonance spectroscopy\|~~nuclear magnetic resonance spectroscopy]] (NMR), where an observable nuclear spin polarization is created by an external magnetic field and then perturbed by resonant radiofrequency pulses.<ref name="Rinck">{{cite book		A concrete and experimentally important example of quantum relaxation occurs in magnetic resonance imaging (MRI) and nuclear magnetic resonance spectroscopy (NMR), where an observable nuclear spin polarization is created by an external magnetic field and then perturbed by resonant radiofrequency pulses.<ref name="Rinck">{{cite book
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	\|first1=Peter A.		\|first1=Peter A.
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	==Bloch equations==		==Bloch equations==
	A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by [[Felix Bloch]].<ref name="Bloch1946">{{cite journal		A phenomenological description of magnetic resonance relaxation is provided by the Bloch equations, introduced by Felix Bloch.<ref name="Bloch1946">{{cite journal
	\|last=Bloch		\|last=Bloch
	\|first=F.		\|first=F.
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	Quantum relaxation and thermalization are important in:		Quantum relaxation and thermalization are important in:

	* ~~[[Physics:Magnetic resonance imaging\|~~magnetic resonance imaging]] and ~~[[Physics:Nuclear magnetic resonance spectroscopy\|~~nuclear magnetic resonance spectroscopy]]		* magnetic resonance imaging and nuclear magnetic resonance spectroscopy
	* ultracold atomic gases after quenches		* ultracold atomic gases after quenches
	* condensed-matter systems out of equilibrium		* condensed-matter systems out of equilibrium
	* quantum simulators and quantum information devices		* quantum simulators and quantum information devices
	* open quantum optical systems		* open quantum optical systems
	* studies of ergodicity, integrability, and ~~[[Physics:Many-body localization\|~~many-body localization]]<ref name="Rinck"/><ref name="Polkovnikov2011"/><ref name="GogolinEisert2016"/><ref name="NandkishoreHuse2015"/>		* studies of ergodicity, integrability, and many-body localization<ref name="Rinck"/><ref name="Polkovnikov2011"/><ref name="GogolinEisert2016"/><ref name="NandkishoreHuse2015"/>

	==See also==		==See also==

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