# 耗散结构

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A **dissipative system** is a thermodynamically 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 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.

Dissipative system 是一个热力学上的开放系统，在一个与热力学平衡交换能量和物质的环境中运行。龙卷风可以被认为是 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.

耗散结构是一种具有某种可重复的稳态动力学性质的 Dissipative system 结构。这种可复制的稳定状态可以通过系统的自然进化、技巧或两者的结合来达到。

## Overview

## Overview

概览

A dissipative structure is characterized by the spontaneous appearance of symmetry breaking (anisotropy) and the formation of complex, sometimes chaotic, structures where interacting particles exhibit long range correlations. Examples in everyday life include convection, turbulent flow, cyclones, hurricanes and living organisms. Less common examples include lasers, Bénard cells, droplet cluster, and the Belousov–Zhabotinsky reaction.^{[1]}

A dissipative structure is characterized by the spontaneous appearance of symmetry breaking (anisotropy) and the formation of complex, sometimes chaotic, structures where interacting particles exhibit long range correlations. Examples in everyday life include convection, turbulent flow, cyclones, hurricanes and living organisms. Less common examples include lasers, Bénard cells, droplet cluster, and the Belousov–Zhabotinsky reaction.

耗散结构是拥有属性对称性破缺(各向异性)的自发出现和复杂的，有时是混沌的结构的形成，在这些结构中，相互作用的粒子展现出长程关联。日常生活中的例子包括对流、湍流、旋风、飓风和生物体。较少见的例子包括激光、 b 细胞、液滴簇和别洛索夫-扎博廷斯基反应。

One way of mathematically modeling a dissipative system is given in the article on *wandering sets*: it involves the action of a group on a measurable set.

One way of mathematically modeling a dissipative system is given in the article on wandering sets: it involves the action of a group on a measurable set.

在游荡集合的文章中给出了一种数学建模 Dissipative system 的方法: 它涉及到一个可测集合上的一个群体的行为。

Dissipative systems can also be used as a tool to study economic systems and complex systems.^{[2]} 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.^{[3]}

Dissipative systems can also be used as a tool to study economic systems and complex systems. 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.

耗散系统也可以作为研究经济系统和复杂系统的工具。例如，一个包含纳米线自组装的 Dissipative system 被用作理解熵产生和生物系统鲁棒性之间关系的模型。

## Dissipative structures in thermodynamics

## 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.

耗散结构这个术语是由俄罗斯和比利时的物理化学家伊利亚 · 普里戈金创造的，他在这些结构上的开创性工作获得了1977年的诺贝尔化学奖。普里高金所考虑的耗散结构具有可视为热力学稳态的动力学机制，有时至少可以用合适的非平衡热力学中的极值定理来描述。

In his Nobel lecture,^{[4]} 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.^{[5]} 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.

In his Nobel lecture, 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. 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.

在他的诺贝尔演讲中，普里戈金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。我们可以假定系统的(广义)通量和力之间是线性关系。线性热力学的两个著名的结果是昂萨格互反关系和最小产生熵原理。在努力将这些结果推广到远离平衡的系统之后，发现它们在这个系统中不成立，并且得到了相反的结果。

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,^{[6]} 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, 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.^{[7]}

Recent research has seen reconsideration of Prigogine's ideas of dissipative structures in relation to biological systems.

最近的研究重新考虑了普里戈金的耗散结构思想与生物系统的关系。

## Dissipative systems in control theory

## Dissipative systems in control theory

控制论中的耗散系统

In systems theory the concept of dissipativity was first introduced by Willems,^{[8]} which describes dynamical systems by input-output properties. Considering a dynamical system described by its state [math]\displaystyle{ x(t) }[/math], its input [math]\displaystyle{ u(t) }[/math] and its output [math]\displaystyle{ y(t) }[/math], the input-output correlation is given a supply rate [math]\displaystyle{ 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]\displaystyle{ V(x(t)) }[/math] such that [math]\displaystyle{ V(0)=0 }[/math], [math]\displaystyle{ 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]\displaystyle{ x(t) }[/math], its input [math]\displaystyle{ u(t) }[/math] and its output [math]\displaystyle{ y(t) }[/math], the input-output correlation is given a supply rate [math]\displaystyle{ 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]\displaystyle{ V(x(t)) }[/math] such that [math]\displaystyle{ V(0)=0 }[/math], [math]\displaystyle{ V(x(t))\ge 0 }[/math] and

在系统论中，耗散性的概念是由 Willems 首先提出的，它通过输入输出的性质来描述动态系统。考虑一个由状态数学 x (t) / math 描述的动力系统，它的输入数学 u (t) / math 和输出数学 y (t) / math，输入输出相关性被给出一个供给率数学 w (u (t) ，y (t) / math。如果存在一个连续可微的存储函数数学 v (x (t)) / 这样的数学 v (0)0 / math，数学 v (x (t)) ge0 / math，那么系统相对于供给率是耗散的

- [math]\displaystyle{ \dot{V}(x(t)) \le w(u(t),y(t)) }[/math].
^{[9]}

[math]\displaystyle{ \dot{V}(x(t)) \le w(u(t),y(t)) }[/math].

Math dot { v }(x (t)) le w (u (t) ，y (t)) / math.

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]\displaystyle{ w(u(t),y(t)) = u(t)^Ty(t) }[/math].

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]\displaystyle{ 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。

The physical interpretation is that [math]\displaystyle{ V(x) }[/math] is the energy stored in the system, whereas [math]\displaystyle{ w(u(t),y(t)) }[/math] is the energy that is supplied to the system.

The physical interpretation is that [math]\displaystyle{ V(x) }[/math] is the energy stored in the system, whereas [math]\displaystyle{ w(u(t),y(t)) }[/math] is the energy that is supplied to the system.

物理解释是 math v (x) / math 是存储在系统中的能量，而 math w (u (t) ，y (t)) / math 是提供给系统的能量。

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.

这个概念与李雅普诺夫稳定性有很强的联系，在动力系统的可控性和可观测性的特定条件下，存储函数可以扮演 Lyapunov 函数的角色。

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模板:Clarify, 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模板:Clarify.^{[10]} 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。

## Quantum dissipative systems

## Quantum dissipative systems

量子耗散系统

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 – say, an oscillator – 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.

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 – say, an oscillator – 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，这些近似本质上不能描述耗散系统。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

## Applications on dissipative systems of dissipative structure concept

耗散结构概念在耗散系统中的应用

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^{[11]}^{[12]}, population dynamics and growth ^{[13]} ^{[14]}^{[15]} and chemomechanical structures^{[16]}^{[17]}^{[18]}

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, population dynamics and growth and chemomechanical structures

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、族群动态、生长和化学机械结构

## See also

## See also

参见

## Notes

## Notes

注释

- ↑ Li, HP (February 2014). "Dissipative Belousov–Zhabotinsky reaction in unstable micropyretic synthesis".
*Current Opinion in Chemical Engineering*.**3**: 1–6. doi:10.1016/j.coche.2013.08.007. - ↑ Chen, Jing (2015).
*The Unity of Science and Economics: A New Foundation of Economic Theory*. https://www.springer.com/us/book/9781493934645: Springer. - ↑ Hubler, Alfred; Belkin, Andrey; Bezryadin, Alexey (2 January 2015). "Noise induced phase transition between maximum entropy production structures and minimum entropy production structures?".
*Complexity*.**20**(3): 8–11. Bibcode:2015Cmplx..20c...8H. doi:10.1002/cplx.21639. - ↑ Prigogine, Ilya. "Time, Structure and Fluctuations".
*Nobelprize.org*. PMID 17738519. - ↑ Prigogine, Ilya (1945). "Modération et transformations irréversibles des systèmes ouverts".
*Bulletin de la Classe des Sciences, Académie Royale de Belgique*.**31**: 600–606. - ↑ Lemarchand, H.; Nicolis, G. (1976). "Long range correlations and the onset of chemical instabilities".
*Physica*.**82A**(4): 521–542. Bibcode:1976PhyA...82..521L. doi:10.1016/0378-4371(76)90079-0. - ↑ England, Jeremy L. (4 November 2015). "Dissipative adaptation in driven self-assembly".
*Nature Nanotechnology*.**10**(11): 919–923. Bibcode:2015NatNa..10..919E. doi:10.1038/NNANO.2015.250. PMID 26530021. - ↑ Willems, J.C. (1972). "Dissipative dynamical systems part 1: General theory" (PDF).
*Arch. Rational Mech. Anal*.**45**(5): 321. Bibcode:1972ArRMA..45..321W. doi:10.1007/BF00276493. hdl:10338.dmlcz/135639. - ↑ Arcak, Murat; Meissen, Chris; Packard, Andrew (2016).
*Networks of Dissipative Systems*. Springer International Publishing. ISBN 978-3-319-29928-0. - ↑ Bao, Jie; Lee, Peter L. (2007).
*Process Control - The Passive Systems Approach*. Springer-Verlag London. doi:10.1007/978-1-84628-893-7. ISBN 978-1-84628-892-0. https://www.springer.com/978-1-84628-892-0. - ↑ Lugiato, L. A.; Prati, F.; Gorodetsky, M. L.; Kippenberg, T. J. (28 December 2018). "From the Lugiato–Lefever equation to microresonator-based soliton Kerr frequency combs".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20180113. arXiv:1811.10685. Bibcode:2018RSPTA.37680113L. doi:10.1098/rsta.2018.0113. PMID 30420551. - ↑ Andrade-Silva, I.; Bortolozzo, U.; Castillo-Pinto, C.; Clerc, M. G.; González-Cortés, G.; Residori, S.; Wilson, M. (28 December 2018). "Dissipative structures induced by photoisomerization in a dye-doped nematic liquid crystal layer".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20170382. Bibcode:2018RSPTA.37670382A. doi:10.1098/rsta.2017.0382. PMC 6232603. PMID 30420545. - ↑ Zykov, V. S. (28 December 2018). "Spiral wave initiation in excitable media".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20170379. Bibcode:2018RSPTA.37670379Z. doi:10.1098/rsta.2017.0379. PMID 30420544. - ↑ Tlidi, M.; Clerc, M. G.; Escaff, D.; Couteron, P.; Messaoudi, M.; Khaffou, M.; Makhoute, A. (28 December 2018). "Observation and modelling of vegetation spirals and arcs in isotropic environmental conditions: dissipative structures in arid landscapes".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20180026. Bibcode:2018RSPTA.37680026T. doi:10.1098/rsta.2018.0026. PMID 30420548. - ↑ Gunji, Yukio-Pegio; Murakami, Hisashi; Tomaru, Takenori; Basios, Vasileios (28 December 2018). "Inverse Bayesian inference in swarming behaviour of soldier crabs".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20170370. Bibcode:2018RSPTA.37670370G. doi:10.1098/rsta.2017.0370. PMC 6232598. PMID 30420541. - ↑ Bullara, D.; De Decker, Y.; Epstein, I. R. (28 December 2018). "On the possibility of spontaneous chemomechanical oscillations in adsorptive porous media".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20170374. Bibcode:2018RSPTA.37670374B. doi:10.1098/rsta.2017.0374. PMC 6232597. PMID 30420542. - ↑ Gandhi, Punit; Zelnik, Yuval R.; Knobloch, Edgar (28 December 2018). "Spatially localized structures in the Gray–Scott model".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20170375. Bibcode:2018RSPTA.37670375G. doi:10.1098/rsta.2017.0375. PMID 30420543. - ↑ Kostet, B.; Tlidi, M.; Tabbert, F.; Frohoff-Hülsmann, T.; Gurevich, S. V.; Averlant, E.; Rojas, R.; Sonnino, G.; Panajotov, K. (28 December 2018). "Stationary localized structures and the effect of the delayed feedback in the Brusselator model".
*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**376**(2135): 20170385. arXiv:1810.05072. Bibcode:2018RSPTA.37670385K. doi:10.1098/rsta.2017.0385. PMID 30420547.

## References

## References

参考资料

- B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative Systems Analysis and Control. Theory and Applications. Springer Verlag, London, 2nd Ed., 2007.

- Davies, Paul
*The Cosmic Blueprint*Simon & Schuster, New York 1989 (abridged— 1500 words) (abstract— 170 words) — self-organized structures.

- Philipson, Schuster,
*Modeling by Nonlinear Differential Equations: Dissipative and Conservative Processes*, World Scientific Publishing Company 2009.

- Prigogine, Ilya,
*Time, structure and fluctuations*. Nobel Lecture, 8 December 1977.

- J.C. Willems. Dissipative dynamical systems, part I: General theory; part II: Linear systems with quadratic supply rates. Archive for Rationale mechanics Analysis, vol.45, pp. 321–393, 1972.

## External links

## External links

外部链接

- The dissipative systems model The Australian National University

Category:Thermodynamic systems

类别: 热力学系统

Category:Systems theory

范畴: 系统论

Category:Non-equilibrium thermodynamics

类别: 非平衡态热力学

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