“耗散系统 Dissipative system”的版本间的差异
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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 – 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'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 – 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'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 – 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, '''<font color="#32CD32"> i.e., an assembly of many oscillators in thermal equilibrium with a broad band spectrum, and trace (average) over the bath.</font>''' 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. |
| − | + | 由于'''<font color="#FFD700">量子力学 Quantum Mechanics</font>'''，以及所有经典的'''<font color="#FFD700">动力系统 Dynamical System</font>'''都严重依赖于时间可逆的'''<font color="#FFD700">哈密顿力学 Hamiltonian mechanics</font>'''，因此这些近似在本质上不能描述耗散系统。有人提出，原则上，人们可以将系统（例如，一个振荡器）弱耦合到'''<font color="#FFD700">浴bath</font>'''中，'''<font color="#32CD32">即在热平衡状态下具有宽带光谱的多个振荡器的组合，和浴上的迹(平均值)。</font>'''这就产生了一个主方程，这是一个较为普遍的情况下的特例，被称为'''<font color="#FFD700">林德布劳德方程Lindblad equation</font>'''，它是经典'''<font color="#FFD700">刘维尔方程Liouville equation</font>'''的量子等价物。众所周知，这个方程和它的量子对应物把时间作为一个可逆变量来积分，但耗散结构的基础认为时间具有不可逆且建设性的作用。 | |
== 耗散结构概念在耗散系统中的应用 Applications on dissipative systems of dissipative structure concept == | == 耗散结构概念在耗散系统中的应用 Applications on dissipative systems of dissipative structure concept == | ||
2020年8月11日 (二) 02:39的版本
此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。模板:More footnotes
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.
耗散系统 Dissipative system 是一种远离热力学平衡的热力学开放系统，运行在与之交换能量和物质的环境中。例如，龙卷风可以被认为是一个耗散系统。
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 Sissipative Structure是一种耗散系统，其 动力学性质dynamical régime在某种意义上是一种可重复的稳定状态。可以通过系统的自然进化、使用技巧来进化或两者的结合来达到这种可重复的稳定状态。
概览 Overview
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.
耗散结构的特点是自发出现对称性破坏（各向异性）和形成复杂的、有时是混沌的结构，在这些结构中，相互作用的粒子展现出长程关联的性质。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Bénard cells、液滴簇droplet cluster和BZ反应Belousov–Zhabotinsky reaction。
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.
在关于游荡集合 Wandering Sets的文章中给出了一种对耗散系统进行数学建模的方法：这种方法涉及到一个可测集上的群的作用。
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耗散系统被用作理解熵产生和生物系统鲁棒性之间关系的模型。
热力学中的耗散结构 Dissipative structures in 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.
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 Ilya Prigogine创造的，他由于在这些结构上的开创性工作获得了1977年的诺贝尔化学奖。Prigogine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，Prigogine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。因为在接近平衡时，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Onsager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 Lyapunov function，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Prigogine的耗散结构思想与生物系统的关系。
控制论中的耗散系统 Dissipative systems in control theory
Willems first introduced the concept of dissipativity in systems theory[8] to describe 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
- [math]\displaystyle{ \dot{V}(x(t)) \le w(u(t),y(t)) }[/math].[9]
Willems first introduced the concept of dissipativity in systems theory to describe dynamical systems by input-output properties. Considering a dynamical system described by its state 𝑥(𝑡), its input 𝑢(𝑡) and its output 𝑦(𝑡), the input-output correlation is given a supply rate 𝑤(𝑢(𝑡),𝑦(𝑡)). A system is said to be dissipative with respect to a supply rate if there exists a continuously differentiable storage function 𝑉(𝑥(𝑡)) such that 𝑉(0)=0, 𝑉(𝑥(𝑡))≥0 and𝑉˙(𝑥(𝑡))≤𝑤(𝑢(𝑡),𝑦(𝑡)).
Willems首先在系统理论中引入耗散性的概念，用输入输出特性来描述动力系统。一个由其状态𝑥（𝑡）、其输入𝑢（𝑡）和其输出𝑡（𝑡）所描述的动力系统，给出了输入输出关系式。如果存在一个连续可微的存储函数𝑉（𝑥（𝑡）），使得𝑉（𝑥（𝑡））≥0且𝑉（𝑥（𝑡））≤𝑢（𝑡），𝑦（𝑡））。
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 𝑤(𝑢(𝑡),𝑦(𝑡))=𝑢(𝑡)𝑇𝑦(𝑡).
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𝑤（𝑢（𝑡），𝑦（𝑡））=𝑢（𝑡）𝑇𝑦（𝑡）成立，则称系统为无源系统。
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 𝑉(𝑥) is the energy stored in the system, whereas 𝑤(𝑢(𝑡),𝑦(𝑡)) is the energy that is supplied to the system.
物理解释是，𝑉（𝑥）是储存在系统中的能量，而𝑤（𝑢（𝑡））是供给系统的能量。
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Lyapunov稳定性有很强的联系，其中存储函数可以在一定的能控性和可观测性条件下发挥李雅普诺夫函数的作用。
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.
粗略地说，耗散性理论对于线性系统的设计是有用的。耗散系统理论已经由V.M.Popov、J.C.Willems、D.J.Hill和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，一个基本的工具就是所谓的Kalman-Yakubovich-Popov引理，它联系了正实系统的状态空间和频域特性。
量子耗散系统 Quantum dissipative systems
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.
由于量子力学 Quantum Mechanics，以及所有经典的动力系统 Dynamical System都严重依赖于时间可逆的哈密顿力学 Hamiltonian mechanics，因此这些近似在本质上不能描述耗散系统。有人提出，原则上，人们可以将系统（例如，一个振荡器）弱耦合到浴bath中，即在热平衡状态下具有宽带光谱的多个振荡器的组合，和浴上的迹(平均值)。这就产生了一个主方程，这是一个较为普遍的情况下的特例，被称为林德布劳德方程Lindblad equation，它是经典刘维尔方程Liouville equation的量子等价物。众所周知，这个方程和它的量子对应物把时间作为一个可逆变量来积分，但耗散结构的基础认为时间具有不可逆且建设性的作用。
耗散结构概念在耗散系统中的应用 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[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的行为的机制，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population dynamics、生长和化学机械结构等。
参见 See also
- Complex system 复杂系统
- Autowave 自动波
- Autocatalytic reactions and order creation 自催化反应和订单创建
- Dynamical system 动力系统
- Autopoiesis 自生
- Relational order theories 关系顺序理论
- Loschmidt's paradox 洛施密特的悖论
- Viable System Theory 可行系统理论
注释 Notes
- ↑ 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.
参考资料 References
- 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.
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