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==Basic concepts==

==Basic concepts==

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.

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.

有许多固定的非平衡系统的例子，其中一些非常简单，例如在不同温度下被限制在两个恒温器之间的系统，或普通的 Couette 流动，两个平板壁之间的流体沿相反方向运动，并定义了壁上的非平衡条件。激光作用也是一个非平衡过程，但它依赖于脱离局部热力学平衡，因此超出了经典不可逆热力学的范围; 在这里，两个分子自由度(分子激光、振动和转动分子运动)之间保持了很大的温差，在一个很小的空间区域需要两个分量的‘温度’ ，排除了局部热力学平衡，这就要求只需要一个温度。声扰动或激波的阻尼是非平稳的非平衡过程。被驱动的复杂流体、湍流系统和玻璃是非平衡系统的其他例子。

有许多静态非平衡系统的例子，其中一些非常简单，例如被限制在两个不同温度恒温器之间的系统，或者常见的库埃特流动，两个沿相反方向运动的平板壁之间的流体，并定义了壁上的非平衡条件。激光作用也是一个非平衡过程，但它依赖于脱离局部热力学平衡，因此超出了经典不可逆热力学的范围; 这种情况下，两个分子自由度(分子激光，振动和转动分子运动)之间保持了很大的温差，这要求在一个很小的空间区域存在两个部分的“温度”，所以排除了局部热力学平衡，因为热力学平衡只需要一个温度。声扰动或激波的阻尼是非静态非平衡过程。被驱动的复杂流体、湍流系统和玻璃是非平衡系统的其他例子。

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 quantities. 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 quantities. 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.

宏观系统的力学依赖于大量的数据。应当强调的是，所有系统都与其周围环境永久地相互作用，从而造成大量不可避免的波动。热力学系统的平衡条件与熵的最大性质有关。如果唯一允许大范围波动的量是内能，而其它量都严格保持恒定，那么系统的温度就是可测量的，也是有意义的。系统的性质可以用热动力位能亥姆霍兹自由能(a = u-TS)来描述，它是能量的勒壤得转换。如果除了能量的波动，系统的宏观尺寸(体积)仍然保持波动状态，我们使用吉布斯自由能(g = u + PV-TS) ，其中系统的性质既由温度决定，也由压力决定。

宏观系统的力学依赖于大量的广延量。应当强调的是，所有系统都与其周围环境永久地相互作用，从而造成广延量不可避免的波动。热力学系统的平衡条件与熵最大的性质有关。如果唯一允许波动的广延量是内能，而其它量都严格保持恒定，那么系统的温度就是可测量和有意义的。系统的性质可以用热力学势能亥姆霍兹自由能(A = U - TS)来描述，它是能量的勒让德变换。如果除了能量的波动，系统的宏观尺寸(体积)也可以波动，我们使用吉布斯自由能(G = U + PV - TS)，其中系统的性质既由温度决定，也由压强决定。

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，它们可能经历更广泛的量的波动。边界条件强加给它们特殊的强度变量，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。如果自由能在平衡态热力学中非常有用，那么必须强调的是，在平衡态热力学中，定义能量的稳态非平衡性质的一般定律，并不像平衡态热力学中熵的热力学第二定律定律那样。这就是为什么在这种情况下，应该考虑一个更广泛的勒壤得转换。这就是延伸的马西欧势。

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。

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:

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)。

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)。

==Stationary states, fluctuations, and stability==

==Stationary states, fluctuations, and stability==