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→Correlation between statistical negentropy and Gibbs' free energy 统计学负熵与吉布斯自由能的关联
[[Willard Gibbs’ 1873 available energy (free energy) graph, which shows a plane perpendicular to the axis of v (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface of dissipated energy. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy) respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC its available energy (Gibbs energy) and its capacity for entropy (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume) respectively.]]
[[Willard Gibbs’ 1873 available energy (free energy) graph, which shows a plane perpendicular to the axis of v (volume) and passing through point A, which represents the initial state of the body. MN is the section of the surface of dissipated energy. Qε and Qη are sections of the planes η = 0 and ε = 0, and therefore parallel to the axes of ε (internal energy) and η (entropy) respectively. AD and AE are the energy and entropy of the body in its initial state, AB and AC its available energy (Gibbs energy) and its capacity for entropy (the amount by which the entropy of the body can be increased without changing the energy of the body or increasing its volume) respectively.]]
[ Willard Gibbs 的1873可用能量(自由能)图，它显示了一个垂直于 v (体积)轴和通过点 a 的平面，a 表示物体的初始状态。Mn 是表面耗散能的截面。Q 和 q 是平面0和0的截面，因此分别与(内能)和(熵)轴平行。Ad 和 AE 分别是物体初始状态的能量和熵，AB 和 AC 分别是物体的有效能(吉布斯能)和熵能(在不改变物体能量或增加物体体积的情况下增加物体熵的量)
[ Willard Gibbs 1873年的可用能量(自由能)图，它显示了一个垂直于 v (体积)轴和通过点 a 的平面，a 表示物体的初始状态。Mn 是表面耗散能的截面。Q 和 q 是平面0和0的截面，因此分别与(内能)和(熵)轴平行。Ad 和 AE 分别是物体初始状态的能量和熵，AB 和 AC 分别是物体的有效能(吉布斯能)和熵能(在不改变物体能量或增加物体体积的情况下增加物体熵的量)
There is a physical quantity closely linked to [[Thermodynamic free energy|free energy]] ([[free enthalpy]]), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1873, [[Josiah Willard Gibbs|Willard Gibbs]] created a diagram illustrating the concept of free energy corresponding to [[free enthalpy]]. On the diagram one can see the quantity called [[capacity for entropy]]. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume.<ref>Willard Gibbs, [http://www.ufn.ru/ufn39/ufn39_4/Russian/r394b.pdf A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces], ''Transactions of the Connecticut Academy'', 382–404 (1873)</ref> In other words, it is a difference between maximum possible, under assumed conditions, entropy and its actual entropy. It corresponds exactly to the definition of negentropy adopted in statistics and information theory. A similar physical quantity was introduced in 1869 by [[François Jacques Dominique Massieu|Massieu]] for the [[isothermal process]]<ref>Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. ''C. R. Acad. Sci.'' LXIX:858–862.</ref><ref>Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. ''C. R. Acad. Sci.'' LXIX:1057–1061.</ref><ref>Massieu, M. F. (1869), ''Compt. Rend.'' '''69''' (858): 1057.</ref> (both quantities differs just with a figure sign) and then [[Max Planck|Planck]] for the [[Isothermal process|isothermal]]-[[Isobaric process|isobaric]] process.<ref>Planck, M. (1945). ''Treatise on Thermodynamics''. Dover, New York.</ref> More recently, the Massieu–Planck [[thermodynamic potential]], known also as ''[[free entropy]]'', has been shown to play a great role in the so-called entropic formulation of [[statistical mechanics]],<ref>Antoni Planes, Eduard Vives, [http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html Entropic Formulation of Statistical Mechanics], Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona</ref> applied among the others in molecular biology<ref>John A. Scheilman, [http://www.biophysj.org/cgi/reprint/73/6/2960.pdf Temperature, Stability, and the Hydrophobic Interaction], ''Biophysical Journal'' '''73''' (December 1997), 2960–2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA</ref> and thermodynamic non-equilibrium processes.<ref>Z. Hens and X. de Hemptinne, [https://arxiv.org/pdf/chao-dyn/9604008 Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures], Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium</ref>
There is a physical quantity closely linked to [[Thermodynamic free energy|free energy]] ([[free enthalpy]]), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1873, [[Josiah Willard Gibbs|Willard Gibbs]] created a diagram illustrating the concept of free energy corresponding to [[free enthalpy]]. On the diagram one can see the quantity called [[capacity for entropy]]. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume.<ref>Willard Gibbs, [http://www.ufn.ru/ufn39/ufn39_4/Russian/r394b.pdf A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces], ''Transactions of the Connecticut Academy'', 382–404 (1873)</ref> In other words, it is a difference between maximum possible, under assumed conditions, entropy and its actual entropy. It corresponds exactly to the definition of negentropy adopted in statistics and information theory. A similar physical quantity was introduced in 1869 by [[François Jacques Dominique Massieu|Massieu]] for the [[isothermal process]]<ref>Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. ''C. R. Acad. Sci.'' LXIX:858–862.</ref><ref>Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. ''C. R. Acad. Sci.'' LXIX:1057–1061.</ref><ref>Massieu, M. F. (1869), ''Compt. Rend.'' '''69''' (858): 1057.</ref> (both quantities differs just with a figure sign) and then [[Max Planck|Planck]] for the [[Isothermal process|isothermal]]-[[Isobaric process|isobaric]] process.<ref>Planck, M. (1945). ''Treatise on Thermodynamics''. Dover, New York.</ref> More recently, the Massieu–Planck [[thermodynamic potential]], known also as ''[[free entropy]]'', has been shown to play a great role in the so-called entropic formulation of [[statistical mechanics]],<ref>Antoni Planes, Eduard Vives, [http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html Entropic Formulation of Statistical Mechanics], Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona</ref> applied among the others in molecular biology<ref>John A. Scheilman, [http://www.biophysj.org/cgi/reprint/73/6/2960.pdf Temperature, Stability, and the Hydrophobic Interaction], ''Biophysical Journal'' '''73''' (December 1997), 2960–2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA</ref> and thermodynamic non-equilibrium processes.<ref>Z. Hens and X. de Hemptinne, [https://arxiv.org/pdf/chao-dyn/9604008 Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures], Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium</ref>
There is a physical quantity closely linked to free energy (free enthalpy), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1873, Willard Gibbs created a diagram illustrating the concept of free energy corresponding to free enthalpy. On the diagram one can see the quantity called capacity for entropy. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume. In other words, it is a difference between maximum possible, under assumed conditions, entropy and its actual entropy. It corresponds exactly to the definition of negentropy adopted in statistics and information theory. A similar physical quantity was introduced in 1869 by Massieu for the isothermal process (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process. More recently, the Massieu–Planck thermodynamic potential, known also as free entropy, has been shown to play a great role in the so-called entropic formulation of statistical mechanics, applied among the others in molecular biology and thermodynamic non-equilibrium processes.
There is a physical quantity closely linked to free energy (free enthalpy), with a unit of entropy and isomorphic to negentropy known in statistics and information theory. In 1873, Willard Gibbs created a diagram illustrating the concept of free energy corresponding to free enthalpy. On the diagram one can see the quantity called capacity for entropy. This quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume. In other words, it is a difference between maximum possible, under assumed conditions, entropy and its actual entropy. It corresponds exactly to the definition of negentropy adopted in statistics and information theory. A similar physical quantity was introduced in 1869 by Massieu for the isothermal process (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process. More recently, the Massieu–Planck thermodynamic potential, known also as free entropy, has been shown to play a great role in the so-called entropic formulation of statistical mechanics, applied among the others in molecular biology and thermodynamic non-equilibrium processes.
存在一个与自由能(自由焓)密切相关的物理量，它具有熵的单位并且与统计学和信息论中我们所知的负熵同构。1873年，威拉德·吉布斯创建了一个图表，说明了自由能对应于自由焓的概念。在图表上，我们可以看到称为熵的容量的物理量。这个量表示在不改变内能或增加体积的情况下，可以增加的熵值。换句话说，它是在假定条件下可能的最大熵值与实际熵之间的差异。它正好符合统计学和信息论中负熵的定义。1869年，Massieu 在等温过程(两个量只有一个图形符号不同)中引入了一个类似的物理量，然后 Planck 引入到等温-等压过程中。最近，Massieu–Planck 热力学势，也被称为自由熵，已被证明在所谓的统计力学熵表述中发挥了重要作用，应用于分子生物学和热力学非平衡过程。
where:
where:
其中：
::<math>S</math> is [[entropy]]
::<math>S</math> is [[entropy]]
<math>S</math> is entropy
<math>S</math> is entropy
<math>S</math> 是熵
::<math>J</math> is negentropy (Gibbs "capacity for entropy")
::<math>J</math> is negentropy (Gibbs "capacity for entropy")
<math>J</math> is negentropy (Gibbs "capacity for entropy")
<math>J</math> is negentropy (Gibbs "capacity for entropy")
<math>J</math> 是负熵(吉布斯“熵的容量”)
::<math>\Phi</math> is the [[Free entropy|Massieu potential]]
::<math>\Phi</math> is the [[Free entropy|Massieu potential]]
<math>\Phi</math> is the Massieu potential
<math>\Phi</math> is the Massieu potential
<math>\Phi</math> 是 Massieu 势
::<math>Z</math> is the [[Partition function (statistical mechanics)|partition function]]
::<math>Z</math> is the [[Partition function (statistical mechanics)|partition function]]
<math>Z</math> is the partition function
<math>Z</math> is the partition function
<math>Z</math> 是配分函数
::<math>k</math> the [[Boltzmann constant]]
::<math>k</math> the [[Boltzmann constant]]
<math>k</math> the Boltzmann constant
<math>k</math> the Boltzmann constant
<math>k</math> 是波兹曼常数
In particular, mathematically the negentropy (the negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (in physics interpreted as the free energy).
In particular, mathematically the negentropy (the negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (in physics interpreted as the free energy).
特别地，数学上的负熵(负熵函数，在物理学中解释为自由熵)是 LogSumExp 的凸共轭(在物理学中解释为自由能)。
==Brillouin's negentropy principle of information==
==Brillouin's negentropy principle of information==