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Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundary conditions impose on them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc.), often called thermodynamic forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the entropy in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential.

Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundary conditions impose on them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc.), often called thermodynamic forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the entropy in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential.

非平衡系统要复杂得多，它们可能存在更多广延量的波动。边界条件施加给它们某些强度量，如温度梯度或形变集体运动(剪切运动、涡旋等)，通常称为热力学力。如果自由能在平衡态热力学中非常有用，那么必须强调的是，没有像平衡态热力学中熵的热力学第二定律定律那样定义能量的静态非平衡性质的一般定律。这就是为什么在这种情况下，应该考虑一个更一般的勒让德变换。这就是拓展的马休势。

非平衡系统要复杂得多，它们可能存在更多广延量的涨落。边界条件施加给它们某些强度量，如温度梯度或形变集体运动(剪切运动、涡旋等)，通常称为热力学力。如果自由能在平衡态热力学中非常有用，那么必须强调的是，没有像平衡态热力学中熵的热力学第二定律那样定义能量的静态非平衡性质的一般定律。这就是为什么在这种情况下，应该考虑一个更一般的勒让德变换。这就是拓展的马休势。

By definition, the [[entropy]] (''S'') is a function of the collection of [[extensive quantity|extensive quantiti]]es <math>E_i</math>. Each extensive quantity has a conjugate intensive variable <math>I_i</math> (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

By definition, the [[entropy]] (''S'') is a function of the collection of [[extensive quantity|extensive quantiti]]es <math>E_i</math>. Each extensive quantity has a conjugate intensive variable <math>I_i</math> (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that:

where <math>\ k_{\rm B}</math> is [[Boltzmann's constant]], whence

where <math>\ k_{\rm B}</math> is [[Boltzmann's constant]], whence

其中 <math>\ k_{\rm B}</math> 是波尔兹曼常数，由此

其中 <math>\ k_{\rm B}</math> 是玻尔兹曼常数，由此

One can think here of two 'relaxation times' separated by order of magnitude. The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. The shorter is of the order of magnitude of times taken for a single 'cell' to reach local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning and other approaches have to be proposed, see for instance Extended irreversible thermodynamics. For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies at altitudes below about 60km where sound propagates, but not above 100km, where, because of the paucity of intermolecular collisions, sound does not propagate.

One can think here of two 'relaxation times' separated by order of magnitude. The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. The shorter is of the order of magnitude of times taken for a single 'cell' to reach local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning and other approaches have to be proposed, see for instance Extended irreversible thermodynamics. For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies at altitudes below about 60km where sound propagates, but not above 100km, where, because of the paucity of intermolecular collisions, sound does not propagate.

===Milne's definition in terms of radiative equilibrium===

===Milne's definition in terms of radiative equilibrium===

The second law of thermodynamics requires that the matrix <math>L</math> be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is symmetric. This fact is called the Onsager reciprocal relations.

The second law of thermodynamics requires that the matrix <math>L</math> be positive definite. Statistical mechanics considerations involving microscopic reversibility of dynamics imply that the matrix <math>L</math> is symmetric. This fact is called the Onsager reciprocal relations.

热力学第二定律要求矩阵<math>L</math>是正定的。统计力学动力学的微观可逆性的考虑暗示了矩阵是对称的。这个事实被称为昂萨格倒易关系。

热力学第二定律要求矩阵<math>L</math>是正定的。统计力学动力学的微观可逆性暗示了矩阵是对称的。这个事实被称为昂萨格倒易关系。

Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999) concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008) is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's 1931 origination of this subject. Other writers have also felt that prospects for general global extremal principles are clouded. Such writers include Glansdorff and Prigogine (1971), Lebon, Jou and Casas-Vásquez (2008), and Šilhavý (1997).

Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999) concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008) is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's 1931 origination of this subject. Other writers have also felt that prospects for general global extremal principles are clouded. Such writers include Glansdorff and Prigogine (1971), Lebon, Jou and Casas-Vásquez (2008), and Šilhavý (1997).

直到最近，这个领域中有用的极端原理的前景似乎还很模糊。Nicolis (1999)得出结论，大气动力学的一个模型有一个吸引子，它不是最大或最小耗散的范畴; 她说这似乎排除了全局组织原则的存在，并评论说，这在某种程度上是令人失望的; 她还指出，很难找到一个热力学上一致的形式的熵产生。另一位顶级专家对熵产生极值原理和能量耗散原理的可能性进行了广泛的讨论: Grandy (2008年)的第12章非常谨慎，发现在许多情况下难以定义“内部熵产生速率”，并发现有时为了预测一个过程的进程，一个叫做能量耗散速率的极值可能比熵产生的速率更有用; 这个量出现在昂萨格尔1931年创立的这个主题中。其他研究者也认为，一般的全局极值原理的前景是模糊的。这些作家包括格兰斯多夫和普里戈金(1971年)、莱邦、乔和卡萨斯-瓦斯奎斯(2008年) ，以及伊尔哈维(1997年)。

直到最近，这个领域中有用的极端原理的前景似乎还很模糊。Nicolis (1999)得出结论，大气动力学的一个模型有一个吸引子，它不是最大或最小耗散的范畴; 她说这似乎排除了全局组织原则的存在，并评论说，这在某种程度上是令人失望的; 她还指出，很难找到一个热力学上一致的形式的熵产生。另一位顶级专家对熵产生极值原理和能量耗散原理的可能性进行了广泛的讨论: Grandy (2008年)的第12章非常谨慎，发现在许多情况下难以定义“内部熵产生速率”，并发现有时为了预测一个过程的进程，一个叫做能量耗散速率的极值可能比熵产生的速率更有用; 这个量出现在昂萨格尔1931年创立的这个主题中。其他研究者也认为，一般的全局极值原理的前景是模糊的。这些作家包括格兰斯多夫和普里高津(1971年)、莱邦、乔和卡萨斯-瓦斯奎斯(2008年) ，以及伊尔哈维(1997年)。