Editing Dissipative system (section)

== Dissipative structures in thermodynamics ==
Russian-Belgian physical chemist [[Ilya Prigogine]], who coined the term ''dissipative structure,'' received the [[Nobel Prize in Chemistry]] in 1977 for his pioneering work on these structures, which have dynamical regimes that can be regarded as thermodynamic steady states, and sometimes at least can be described by suitable [[extremal principles in non-equilibrium thermodynamics]].

In his Nobel lecture,<ref name="PrigogineNobel">{{cite journal|last1=Prigogine|first1=Ilya|title=Time, Structure and Fluctuations|url=https://www.nobelprize.org/nobel_prizes/chemistry/laureates/1977/prigogine-lecture.html|journal=Science|year=1978|volume=201|issue=4358|pages=777–785|doi=10.1126/science.201.4358.777|pmid=17738519|bibcode=1978Sci...201..777P |s2cid=9129799 }}</ref> Prigogine explains how thermodynamic systems far from equilibrium can have drastically different behavior from systems close to equilibrium. Near equilibrium, the ''local equilibrium'' hypothesis applies and typical thermodynamic quantities such as free energy and entropy can be defined locally. One can assume linear relations between the (generalized) flux and forces of the system. Two celebrated results from linear thermodynamics are the [[Onsager reciprocal relations]] and the principle of minimum [[entropy production]].<ref>{{cite journal|last1=Prigogine|first1=Ilya|title=Modération et transformations irréversibles des systèmes ouverts|journal=Bulletin de la Classe des Sciences, Académie Royale de Belgique|date=1945|volume=31|pages=600–606}}</ref> After efforts to extend such results to systems far from equilibrium, it was found that they do not hold in this regime and opposite results were obtained.

One way to rigorously analyze such systems is by studying the stability of the system far from equilibrium. Close to equilibrium, one can show the existence of a [[Lyapunov function]] which ensures that the entropy tends to a stable maximum. Fluctuations are damped in the neighborhood of the fixed point and a macroscopic description suffices. However, far from equilibrium stability is no longer a universal property and can be broken. In chemical systems, this occurs with the presence of [[autocatalytic]] reactions, such as in the example of the [[Brusselator]]. If the system is driven beyond a certain threshold, oscillations are no longer damped out, but may be amplified. Mathematically, this corresponds to a [[Hopf bifurcation]] where increasing one of the parameters beyond a certain value leads to [[limit cycle]] behavior. If spatial effects are taken into account through a [[reaction–diffusion equation]], long-range correlations and spatially ordered patterns arise,<ref name="LemarchandNicolis">{{cite journal|last1=Lemarchand|first1=H.|last2=Nicolis|first2=G.|title=Long range correlations and the onset of chemical instabilities|journal=Physica|date=1976|volume=82A|issue=4|pages=521–542|doi=10.1016/0378-4371(76)90079-0|bibcode=1976PhyA...82..521L}}</ref> such as in the case of the [[Belousov–Zhabotinsky reaction]]. Systems with such dynamic states of matter that arise as the result of irreversible processes are dissipative structures.

Recent research has seen reconsideration of Prigogine's ideas of dissipative structures in relation to biological systems.<ref name="England">{{cite journal|last1=England|first1=Jeremy L.|title=Dissipative adaptation in driven self-assembly|journal=Nature Nanotechnology|date=4 November 2015|volume=10|issue=11|pages=919–923|doi=10.1038/NNANO.2015.250|pmid=26530021|bibcode=2015NatNa..10..919E}}</ref>