Editing Dissipative system (section)

== Quantum dissipative systems ==
{{main|Quantum dissipation}}
As [[quantum mechanics]], and any classical [[dynamical system]], relies heavily on [[Hamiltonian mechanics]] for which [[Time reversibility|time is reversible]], these approximations are not intrinsically able to describe dissipative systems. It has been proposed that in principle, one can couple weakly the system &ndash; say, an oscillator &ndash; to a bath, i.e., an assembly of many oscillators in thermal equilibrium with a broad band spectrum, and trace (average) over the bath. This yields a [[master equation]] which is a special case of a more general setting called the [[Lindblad equation]] that is the quantum equivalent of the classical [[Liouville's theorem (Hamiltonian)|Liouville equation]]. The well-known form of this equation and its quantum counterpart takes time as a reversible variable over which to integrate, but the very foundations of dissipative structures imposes an [[H-theorem|irreversible]] and constructive role for time.

Recent research has seen the quantum extension<ref name="Valente">{{cite journal|last1=Valente|first1=Daniel|last2=Brito|first2=Frederico|last3=Werlang|first3=Thiago|title=Quantum dissipative adaptation|journal=Communications Physics|date=19 January 2021|volume=4|issue=11|page=11 |doi=10.1038/s42005-020-00512-0 |arxiv=2111.08605 |bibcode=2021CmPhy...4...11V |doi-access=free}}</ref> of [[Jeremy England]]'s theory of dissipative adaptation<ref name="England"/> (which generalizes Prigogine's ideas of dissipative structures to far-from-equilibrium statistical mechanics, as stated above).