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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的关系。

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的关系。

== Dissipative structures in thermodynamics ==

== 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]].

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。

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, 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.

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.

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.

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。

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.

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

 

 

 

 

== Dissipative systems in control theory ==

== Dissipative systems in control theory ==