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A '''dissipative system''' is a thermodynamically [[open system (systems theory)|open system]] which is operating out of, and often far from, [[thermodynamic equilibrium]] in an environment with which it exchanges [[energy]] and [[matter]]. A tornado may be thought of as a dissipative system.

A '''dissipative system''' is a thermodynamically [[open system (systems theory)|open system]] which is operating out of, and often far from, [[thermodynamic equilibrium]] in an environment with which it exchanges [[energy]] and [[matter]]. A tornado may be thought of as a dissipative system.

A '''dissipative structure''' is a dissipative system that has a dynamical régime that is in some sense in a reproducible [[steady state]]. This reproducible steady state may be reached by natural evolution of the system, by artifice, or by a combination of these two.

A '''dissipative structure''' is a dissipative system that has a dynamical régime that is in some sense in a reproducible [[steady state]]. This reproducible steady state may be reached by natural evolution of the system, by artifice, or by a combination of these two.

 

 

 

 

 

 

 

 

 

 

 

 

One way of mathematically modeling a dissipative system is given in the article on ''[[wandering set]]s'': it involves the action of a [[group (mathematics)|group]] on a [[measure (mathematics)|measurable set]].

One way of mathematically modeling a dissipative system is given in the article on ''[[wandering set]]s'': it involves the action of a [[group (mathematics)|group]] on a [[measure (mathematics)|measurable set]].

 

 

 

 

Dissipative systems can also be used as a tool to study economic systems and [[complex systems]].<ref>{{Cite book|title = The Unity of Science and Economics: A New Foundation of Economic Theory|last = Chen|first = Jing|publisher = Springer|year = 2015|isbn = |location = https://www.springer.com/us/book/9781493934645|pages = }}</ref> For example, a dissipative system involving [[self-assembly]] of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems.<ref>{{cite journal|last1=Hubler|first1=Alfred|last2=Belkin|first2=Andrey|last3=Bezryadin|first3=Alexey|title=Noise induced phase transition between maximum entropy production structures and minimum entropy production structures?|journal=Complexity|date=2 January 2015|volume=20|issue=3|pages=8–11|doi=10.1002/cplx.21639|bibcode=2015Cmplx..20c...8H}}</ref>

Dissipative systems can also be used as a tool to study economic systems and [[complex systems]].<ref>{{Cite book|title = The Unity of Science and Economics: A New Foundation of Economic Theory|last = Chen|first = Jing|publisher = Springer|year = 2015|isbn = |location = https://www.springer.com/us/book/9781493934645|pages = }}</ref> For example, a dissipative system involving [[self-assembly]] of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems.<ref>{{cite journal|last1=Hubler|first1=Alfred|last2=Belkin|first2=Andrey|last3=Bezryadin|first3=Alexey|title=Noise induced phase transition between maximum entropy production structures and minimum entropy production structures?|journal=Complexity|date=2 January 2015|volume=20|issue=3|pages=8–11|doi=10.1002/cplx.21639|bibcode=2015Cmplx..20c...8H}}</ref>

 

 

 

== Dissipative structures in thermodynamics ==

== 热力学中的耗散结构 ==

The term ''dissipative structure'' was coined by Russian-Belgian physical chemist [[Ilya Prigogine]], who was awarded the [[Nobel Prize in Chemistry]] in 1977 for his pioneering work on these structures. The dissipative structures considered by Prigogine 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]].

The term ''dissipative structure'' was coined by Russian-Belgian physical chemist [[Ilya Prigogine]], who was awarded the [[Nobel Prize in Chemistry]] in 1977 for his pioneering work on these structures. The dissipative structures considered by Prigogine 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]].

 

在他的诺贝尔演讲中<ref name="PrigogineNobel" />，普利高津解释了为什么远离平衡的热力学系统可以有与接近平衡的系统截然不同的行为。在接近平衡时，采用局部平衡假设，可以局部地定义典型的热力学量，如自由能和熵。假定系统的(广义)通量和力之间是线性关系，线性热力学中两个著名的结论是昂萨格互反关系（[[Onsager reciprocal relations]]）和最小产生熵原理<ref name=":1" />。在尝试将这些结果推广到远离平衡态的系统之后，发现这些结论在这个系统中不成立，并且得到了相反的结果。

 

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|website=Nobelprize.org|pmid=17738519}}</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.

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.

严格分析这类系统的一种方法是研究系统远离平衡点的稳定性。接近平衡点时，我们可以证明存在一个李亚普诺夫函数，它确保熵趋于稳定的最大值。波动在固定点附近被削弱，宏观描述对此系统的稳定性足够解释。然而，远离平衡的稳定性不再是一个普遍的性质。在化学系统中，自催化反应存在远离平衡点的情况，例如在布鲁塞尔子的例子中。如果系统驱动超过一定的阈值，振荡不再减弱，可能会被放大。数学上，这相当于一个霍普夫分岔，其中一个参数的增加超过某个值会导致极限环行为。如果通过反应扩散方程考虑空间效应，就会产生长程相关性和空间有序图案，例如 Belousov-Zhabotinsky 反应。具有这种不可逆过程所产生的动态物质状态的系统是耗散结构。

 

严格分析这类系统的一种方法是研究系统远离平衡点的稳定性。接近平衡点时，我们可以证明存在一个李亚普诺夫函数，它确保熵趋于稳定的最大值。波动在固定点附近被阻尼，宏观描述就足够了。然而，远离平衡的稳定性不再是一个普遍的性质，可以被打破。在化学系统中，这发生在自催化反应的存在时，例如在布鲁塞尔子的例子中。如果系统驱动超过一定的阈值，振荡不再阻尼，但可能被放大。数学上，这相当于一个霍普夫分岔，其中一个参数的增加超过某个值会导致极限环行为。如果通过反应扩散方程考虑空间效应，就会产生长程相关性和空间有序图案，例如 Belousov-Zhabotinsky 反应。具有这种不可逆过程所产生的动态物质状态的系统是耗散结构。

 

 

 

 

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>

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>

最近的一些研究重新考虑了普利高津的耗散结构与生物系统的关系。<ref name="England" />

 

 

 

 

 

 

 

 

 

In systems theory the concept of dissipativity was first introduced by [[Jan Camiel Willems|Willems]],<ref>{{cite journal |last1=Willems |first1=J.C. |title=Dissipative dynamical systems part 1: General theory |journal=Arch. Rational Mech. Anal. |date=1972 |volume=45 |issue=5 |page=321 |doi=10.1007/BF00276493 |bibcode=1972ArRMA..45..321W |hdl=10338.dmlcz/135639 |url=http://dml.cz/bitstream/handle/10338.dmlcz/135639/Kybernetika_41-2005-1_5.pdf }}</ref> which describes dynamical systems by input-output properties. Considering a dynamical system described by its state <math> x(t) </math>, its input <math>u(t)</math> and its output <math>y(t)</math>, the input-output correlation is given a supply rate <math> w(u(t),y(t))</math>. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function <math> V(x(t))</math> such that <math>V(0)=0</math>, <math>V(x(t))\ge 0 </math> and

 

In systems theory the concept of dissipativity was first introduced by Willems, which describes dynamical systems by input-output properties. Considering a dynamical system described by its state <math> x(t) </math>, its input <math>u(t)</math> and its output <math>y(t)</math>, the input-output correlation is given a supply rate <math> w(u(t),y(t))</math>. A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function <math> V(x(t))</math> such that <math>V(0)=0</math>, <math>V(x(t))\ge 0 </math> and

 

:<math> \dot{V}(x(t)) \le w(u(t),y(t))</math>.<ref>{{cite book |last1=Arcak |first1=Murat |last2=Meissen |first2=Chris |last3=Packard |first3=Andrew |title=Networks of Dissipative Systems |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-29928-0 }}</ref>

:<math> \dot{V}(x(t)) \le w(u(t),y(t))</math>.<ref>{{cite book |last1=Arcak |first1=Murat |last2=Meissen |first2=Chris |last3=Packard |first3=Andrew |title=Networks of Dissipative Systems |date=2016 |publisher=Springer International Publishing |isbn=978-3-319-29928-0 }}</ref>

作为耗散性的一个特例，如果上述耗散性不等式对于被动供给率<math> w(u(t),y(t)) = u(t)^Ty(t) </math> 成立，则系统被称为被动系统。

 

 

 

 

 

As a special case of dissipativity, a system is said to be passive if the above dissipativity inequality holds with respect to the passivity supply rate <math> w(u(t),y(t)) = u(t)^Ty(t) </math>.

 

 

作为耗散性的一个特例，如果上述耗散性不等式对于被动供给率数学 w (u (t) ，y (t)) u (t) ^ Ty (t) / math 成立，则系统被称为被动系统。

 

 

 

 

 

 

 

This notion has a strong connection with [[Lyapunov stability]], where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

This notion has a strong connection with [[Lyapunov stability]], where the storage functions may play, under certain conditions of controllability and observability of the dynamical system, the role of Lyapunov functions.

 

 

 

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by V.M. Popov, J.C. Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems. Dissipative systems are still an active field of research in systems and control, due to their important applications.

Roughly speaking, dissipativity theory is useful for the design of feedback control laws for linear and nonlinear systems. Dissipative systems theory has been discussed by V.M. Popov, J.C. Willems, D.J. Hill, and P. Moylan. In the case of linear invariant systems, this is known as positive real transfer functions, and a fundamental tool is the so-called Kalman–Yakubovich–Popov lemma which relates the state space and the frequency domain properties of positive real systems. Dissipative systems are still an active field of research in systems and control, due to their important applications.

耗散性理论对于线性和非线性系统的反馈控制律设计具有一定的参考价值。耗散系统理论已由 v.m。波波夫，J.C.Willems，d.j。希尔和 p. 莫伊兰。在线性不变系统的情况下，这被称为正实传递函数，一个基本的工具是所谓的 Kalman-Yakubovich-Popov 引理，它把正实系统的状态空间和频域性质联系起来。由于耗散系统的重要应用，它仍然是系统与控制研究的一个活跃领域。

耗散性理论对于线性和非线性系统的反馈控制律设计具有一定的参考价值。耗散系统理论已由[[Vasile M. Popov|V.M. Popov]], [[Jan Camiel Willems|J.C. Willems]], D.J. Hill, and P. Moylan讨论过。在线性不变系统的情况下，这被称为正实传递函数，一个基本的工具是用 Kalman-Yakubovich-Popov 引理，它把正实系统的状态空间和频域性质联系起来。由于耗散系统的重要应用，它仍然是系统与控制研究的一个活跃领域。

 

== 量子耗散系统 ==

 

 

 

 

== Quantum dissipative systems ==

 

== Quantum dissipative systems ==

 

{{main|Quantum dissipation}}

{{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.

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.

As quantum mechanics, and any classical dynamical system, relies heavily on Hamiltonian mechanics for which 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 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 irreversible and constructive role for time.

正如量子力学，和任何经典的动力系统，严重依赖于时间是可逆的哈密顿力学前提，这些近似本质上并不能描述耗散系统。有人提出，原则上，一个人可以弱耦合系统---- 说，一个振荡器---- 浴，也就是说，许多振荡器组合在一个宽带光谱的热平衡，和迹(平均值)在浴。这就产生了一个主方程，这个主方程是一个被称为林德布劳德方程方程的更一般设置的特殊情况，它是经典 Liouville 方程的量子等价物。这个方程的众所周知的形式及其量子对应物需要时间作为可逆变量进行积分，但耗散结构的基础对时间起着不可逆和建设性的作用。

==Applications on dissipative systems of dissipative structure concept==

The framework of dissipative structures as a mechanism to understand the behavior of systems in constant interexchange of energy has been successfully applied on different science fields and applications, as in optics<ref name=":2">{{cite journal |last1=Lugiato |first1=L. A. |last2=Prati |first2=F. |last3=Gorodetsky |first3=M. L. |last4=Kippenberg |first4=T. J. |title=From the Lugiato–Lefever equation to microresonator-based soliton Kerr frequency combs |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20180113 |doi=10.1098/rsta.2018.0113|pmid=30420551 |arxiv=1811.10685 |bibcode=2018RSPTA.37680113L }}</ref><ref name=":3">{{cite journal |last1=Andrade-Silva |first1=I. |last2=Bortolozzo |first2=U. |last3=Castillo-Pinto |first3=C. |last4=Clerc |first4=M. G. |last5=González-Cortés |first5=G. |last6=Residori |first6=S. |last7=Wilson |first7=M. |title=Dissipative structures induced by photoisomerization in a dye-doped nematic liquid crystal layer |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170382 |doi=10.1098/rsta.2017.0382|pmid=30420545 |pmc=6232603 |bibcode=2018RSPTA.37670382A }}</ref>, population dynamics and growth <ref name=":4">{{cite journal |last1=Zykov |first1=V. S. |title=Spiral wave initiation in excitable media |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170379 |doi=10.1098/rsta.2017.0379|pmid=30420544 |bibcode=2018RSPTA.37670379Z }}</ref> <ref name=":5">{{cite journal |last1=Tlidi |first1=M. |last2=Clerc |first2=M. G. |last3=Escaff |first3=D. |last4=Couteron |first4=P. |last5=Messaoudi |first5=M. |last6=Khaffou |first6=M. |last7=Makhoute |first7=A. |title=Observation and modelling of vegetation spirals and arcs in isotropic environmental conditions: dissipative structures in arid landscapes |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20180026 |doi=10.1098/rsta.2018.0026|pmid=30420548 |bibcode=2018RSPTA.37680026T }}</ref><ref name=":6">{{cite journal |last1=Gunji |first1=Yukio-Pegio |last2=Murakami |first2=Hisashi |last3=Tomaru |first3=Takenori |last4=Basios |first4=Vasileios |title=Inverse Bayesian inference in swarming behaviour of soldier crabs |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170370 |doi=10.1098/rsta.2017.0370|pmid=30420541 |pmc=6232598 |bibcode=2018RSPTA.37670370G }}</ref> and chemomechanical structures<ref name=":7">{{cite journal |last1=Bullara |first1=D. |last2=De Decker |first2=Y. |last3=Epstein |first3=I. R. |title=On the possibility of spontaneous chemomechanical oscillations in adsorptive porous media |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170374 |doi=10.1098/rsta.2017.0374|pmid=30420542 |pmc=6232597 |bibcode=2018RSPTA.37670374B }}</ref><ref name=":8">{{cite journal |last1=Gandhi |first1=Punit |last2=Zelnik |first2=Yuval R. |last3=Knobloch |first3=Edgar |title=Spatially localized structures in the Gray–Scott model |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170375 |doi=10.1098/rsta.2017.0375|pmid=30420543 |bibcode=2018RSPTA.37670375G }}</ref><ref name=":9">{{cite journal |last1=Kostet |first1=B. |last2=Tlidi |first2=M. |last3=Tabbert |first3=F. |last4=Frohoff-Hülsmann |first4=T. |last5=Gurevich |first5=S. V. |last6=Averlant |first6=E. |last7=Rojas |first7=R. |last8=Sonnino |first8=G. |last9=Panajotov |first9=K. |title=Stationary localized structures and the effect of the delayed feedback in the Brusselator model |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 December 2018 |volume=376 |issue=2135 |pages=20170385 |doi=10.1098/rsta.2017.0385|pmid=30420547 |arxiv=1810.05072 |bibcode=2018RSPTA.37670385K }}</ref>

==Applications on dissipative systems of dissipative structure concept==

耗散结构框架作为一种理解系统在能量不断交换中行为的机制，已经成功地应用于不同的科学领域和应用，如光学<ref name=":2" /><ref name=":3" />、人口动力学<ref name=":4" /><ref name=":5" /><ref name=":6" />和化学机械结构<ref name=":7" /><ref name=":8" /><ref name=":9" />。

== See also ==

== See also ==

{{Div col}}

{{Div col}}