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第136行： − + − We then define the extended Massieu function as follows:+ − − 然后我们将扩展的 Massieu 函数定义如下: 第148行：
第146行： − + − [ math > k { rm b } m = s-sum _ i (i _ i e _ i) , + − 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.+ − where <math>\ k_{\rm B}</math> is Boltzmann's constant, whence − 波尔兹曼常数是从哪里来的 + − − <math>\ k_{\rm B} \, dM = \sum_i (E_i \, dI_i).</math> − − <math>\ k_{\rm B} \, dM = \sum_i (E_i \, dI_i).</math> − − − The independent variables are the intensities. − − 自变量是强度。 + − − − Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system. − − 强度是全局值，有效的系统作为一个整体。当边界对系统施加不同的局部条件时，例如:。温度差) ，有密集的变量代表平均值和其他代表梯度或更高的矩。后者是热力学力量，驱动广泛性质的通量通过系统。 − − − It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not. − − 可以证明，无论是否处于平衡状态，勒壤得转换改变了定态扩展 Massieu 函数的最小条件下熵的最大条件(平衡时有效)。 − + − − − In thermodynamics one is often interested in a stationary state of a process, allowing that the stationary state include the occurrence of unpredictable and experimentally unreproducible fluctuations in the state of the system. The fluctuations are due to the system's internal sub-processes and to exchange of matter or energy with the system's surroundings that create the constraints that define the process. − − 在热力学中，人们经常对一个过程的定态感兴趣，允许系统状态的定态包括不可预测和实验上不可重复的涨落的发生。波动是由于系统的内部子过程和物质或能量与系统周围环境的交换所造成的，这些环境制约了这一过程。 + − If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy. The reproducible response of the system is then to increase the entropy back to its maximum by irreversible processes: the fluctuation cannot be reproduced with a significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323). The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system. − − 如果这个过程的定态是稳定的，那么不可复制的涨落就包含了局部瞬时的熵减。系统的可重复响应是通过不可逆过程将熵增加到最大值: 涨落不能以显著的概率水平再现。稳定定态的波动极小，除了临界点附近(Kondepudi 和 Prigogine 1998,323页)。稳定的定态有一个局部最大熵，是系统局部最可复制的状态。关于涨落的不可逆耗散有几个定理。这里“局部”是指相对于系统状态热力学坐标的抽象空间的局部。 + − If the stationary state is unstable, then any fluctuation will almost surely trigger the virtually explosive departure of the system from the unstable stationary state. This can be accompanied by increased export of entropy. − − 如果定态是不稳定的，那么任何波动几乎肯定会触发系统几乎爆炸性地偏离不稳定的定态。这可能伴随着熵输出的增加。 − +
→Basic concepts
该形式体系非常适合描述高频过程和小尺度材料。
该形式体系非常适合描述高频过程和小尺度材料。
==Basic concepts==
== Basic concepts 基本概念 ==
There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary [[Couette flow]], a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. [[Laser]] action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven [[complex fluids]], turbulent systems and glasses are other examples of non-equilibrium systems.
There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary [[Couette flow]], a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. [[Laser]] action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven [[complex fluids]], turbulent systems and glasses are other examples of non-equilibrium systems.
静态非平衡系统有许多案例，其中一些非常简单，例如，将一个系统限制在两个温度不同的恒温器之间，或者是常规'''<font color="#ff8000"> 库埃特流体Couette flow</font>'''运动模型的两个平板之间（封闭状态），该平板互相沿反方向运动，而平板壁上需要定义非平衡条件。另外'''<font color="#ff8000"> 激光作用Laser action</font>'''也是一个非平衡过程，但它依赖于从局部热力学平衡出发，因此超出了经典不可逆热力学的范围。在此，两个分子自由度（分子激光，振动和旋转的分子运动）之间维持着明显的温差，要求在一个很小的空间区域中存在两组“温度”组成部分，其中不包括局部热力学平衡，因为后者仅需要一个“温度”。另外声扰动或冲击波阻尼过程是非静态非平衡的过程。而驱动的'''<font color="#ff8000"> 复杂流体Complex fluids</font>'''，湍流系统和玻璃是非平衡系统的其他案例。
The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of [[extensive quantity|extensive quantiti]]es. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential [[Helmholtz free energy]] (''A'' = ''U'' - ''TS''), a [[Legendre transformation]] of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the [[Gibbs free energy]] (''G'' = ''U'' + ''PV'' - ''TS''), where the system's properties are determined both by the temperature and by the pressure.
The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of [[extensive quantity|extensive quantiti]]es. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential [[Helmholtz free energy]] (''A'' = ''U'' - ''TS''), a [[Legendre transformation]] of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the [[Gibbs free energy]] (''G'' = ''U'' + ''PV'' - ''TS''), where the system's properties are determined both by the temperature and by the pressure.
<math>\ k_{\rm B} M = S - \sum_i( I_i E_i),</math>
宏观系统的力学取决于广延量。这里需要强调的是，所有的系统都在与周围环境永久性地相互作用，从而导致不可避免的大量波动。热力学系统的平衡条件与熵的极限性质有关。如果允许波动的唯一广延量是其内部能量，而所有其他能量都严格保持恒定，则系统温度是可测量且有意义的。那么使用热力学势'''<font color="#ff8000"> 亥姆霍兹自由能Helmholtz free energy</font>'''（A = U-TS）（能量的'''<font color="#ff8000"> 勒让德变换Legendre transformation</font>'''）可以最方便地描述系统的属性。如果在能量波动后，系统的宏观尺寸（体积）能同样保持波动，则我们可以使用'''<font color="#ff8000"> 吉布斯自由能</font>'''（G = U + PV-TS），其中系统的特性既取决于温度又取决于压力。
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，并且可能会发生更大范围的波动。边界条件将特殊的强度变量强加给系统，例如温度梯度或扭曲的集体运动（剪切运动，涡旋等），通常称为热力学力。如果自由能在平衡热力学中非常有用，则必须要强调的是，没有任何定律能像热力学第二定律去定义平衡热力学中的熵那样，去定义能量的静态非平衡属性。这就是为什么在这种情况下，应考虑使用更广义的勒让德变换。这是扩展的马休势Massieu potential。
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:
根据定义，熵（S）是广延量集合的函数<math>E_i</math>。每个广延量都有一个共轭强化变量<math>I_i</math>（通过与该链接中给出的定义进行比较，在此使用了强化变量的受限定义）：
:<math> I_i = \frac{\partial{S}}{\partial{E_i}}.</math>
:<math> I_i = \frac{\partial{S}}{\partial{E_i}}.</math>
We then define the extended [[Massieu function]] as follows:
We then define the extended [[Massieu function]] as follows:
然后，我们定义扩展的'''<font color="#ff8000"> 马休函数Massieu function</font>'''，如下所示：
:<math>\ k_{\rm B} M = S - \sum_i( I_i E_i),</math>
:<math>\ k_{\rm B} M = S - \sum_i( I_i E_i),</math>
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'''<font color="#ff8000"> 玻尔兹曼常数Boltzmann's constant</font>'''，因此
:<math>\ k_{\rm B} \, dM = \sum_i (E_i \, dI_i).</math>
:<math>\ k_{\rm B} \, dM = \sum_i (E_i \, dI_i).</math>
The independent variables are the intensities.
The independent variables are the intensities.
其自变量是强度。
Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system.
Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system.
强度是全局值，对整个系统有效。当边界设定强加给系统不同的局部条件（例如，温度差异）时，将存在代表平均值的密集变量，还有一些代表梯度或更高矩的变量。后者是通过系统驱动广延特性通量的热动力。
It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not.
It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not.
可以证明，不管是否建立在平衡状态下，勒让德变换都在扩展的马休函数的最小状态下改变熵（在平衡时有效）的最大状态。
==Stationary states, fluctuations, and stability==
==Stationary states, fluctuations, and stability==