Negentropy: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 05:59, 1 August 2024

In information theory and statistics, negentropy is used as a measure of distance to normality. The concept and phrase "negative entropy" was introduced by Erwin Schrödinger in his 1944 popular-science book What is Life?^[1] Later, French physicist Léon Brillouin shortened the phrase to néguentropie (negentropy).^[2]^[3] In 1974, Albert Szent-Györgyi proposed replacing the term negentropy with syntropy. That term may have originated in the 1940s with the Italian mathematician Luigi Fantappiè, who tried to construct a unified theory of biology and physics. Buckminster Fuller tried to popularize this usage, but negentropy remains common.

In a note to What is Life? Schrödinger explained his use of this phrase.

... if I had been catering for them [physicists] alone I should have let the discussion turn on free energy instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to energy for making the average reader alive to the contrast between the two things.

Information theory

In information theory and statistics, negentropy is used as a measure of distance to normality.^[4]^[5]^[6] Out of all distributions with a given mean and variance, the normal or Gaussian distribution is the one with the highest entropy. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes if and only if the signal is Gaussian.

Negentropy is defined as

J(p_{x})=S(\varphi _{x})-S(p_{x})\,

where $S(\varphi _{x})$ is the differential entropy of the Gaussian density with the same mean and variance as $p_{x}$ and $S(p_{x})$ is the differential entropy of $p_{x}$ :

S(p_{x})=-\int p_{x}(u)\log p_{x}(u)\,du

Negentropy is used in statistics and signal processing. It is related to network entropy, which is used in independent component analysis.^[7]^[8]

The negentropy of a distribution is equal to the Kullback–Leibler divergence between $p_{x}$ and a Gaussian distribution with the same mean and variance as $p_{x}$ (see Differential entropy § Maximization in the normal distribution for a proof). In particular, it is always nonnegative.

Correlation between statistical negentropy and Gibbs' free energy

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.^[9] 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^[10]^[11]^[12] (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process.^[13] 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,^[14] applied among the others in molecular biology^[15] and thermodynamic non-equilibrium processes.^[16]

J=S_{\max }-S=-\Phi =-k\ln Z\,

where:

S

is entropy

J

is negentropy (Gibbs "capacity for entropy")

\Phi

is the Massieu potential

Z

is the partition function

k

the Boltzmann constant

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).

Brillouin's negentropy principle of information

In 1953, Léon Brillouin derived a general equation^[17] stating that the changing of an information bit value requires at least $kT\ln 2$ energy. This is the same energy as the work Leó Szilárd's engine produces in the idealistic case. In his book,^[18] he further explored this problem concluding that any cause of this bit value change (measurement, decision about a yes/no question, erasure, display, etc.) will require the same amount of energy.

Notes

^ Schrödinger, Erwin, What is Life – the Physical Aspect of the Living Cell, Cambridge University Press, 1944
^ Brillouin, Leon: (1953) "Negentropy Principle of Information", J. of Applied Physics, v. 24(9), pp. 1152–1163
^ Léon Brillouin, La science et la théorie de l'information, Masson, 1959
^ Aapo Hyvärinen, Survey on Independent Component Analysis, node32: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science
^ Aapo Hyvärinen and Erkki Oja, Independent Component Analysis: A Tutorial, node14: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science
^ Ruye Wang, Independent Component Analysis, node4: Measures of Non-Gaussianity
^ P. Comon, Independent Component Analysis – a new concept?, Signal Processing, 36 287–314, 1994.
^ Didier G. Leibovici and Christian Beckmann, An introduction to Multiway Methods for Multi-Subject fMRI experiment, FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.
^ Willard Gibbs, A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Transactions of the Connecticut Academy, 382–404 (1873)
^ Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. C. R. Acad. Sci. LXIX:858–862.
^ Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. C. R. Acad. Sci. LXIX:1057–1061.
^ Massieu, M. F. (1869), Compt. Rend. 69 (858): 1057.
^ Planck, M. (1945). Treatise on Thermodynamics. Dover, New York.
^ Antoni Planes, Eduard Vives, Entropic Formulation of Statistical Mechanics Archived 2008-10-11 at the Wayback Machine, Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona
^ John A. Scheilman, Temperature, Stability, and the Hydrophobic Interaction, Biophysical Journal 73 (December 1997), 2960–2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA
^ Z. Hens and X. de Hemptinne, Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures, Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium
^ Leon Brillouin, The negentropy principle of information, J. Applied Physics 24, 1152–1163 1953
^ Leon Brillouin, Science and Information theory, Dover, 1956

[1] Schrödinger, Erwin, What is Life – the Physical Aspect of the Living Cell, Cambridge University Press, 1944

[2] Brillouin, Leon: (1953) "Negentropy Principle of Information", J. of Applied Physics, v. 24(9), pp. 1152–1163

[3] Léon Brillouin, La science et la théorie de l'information, Masson, 1959

[4] Aapo Hyvärinen, Survey on Independent Component Analysis, node32: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science

[5] Aapo Hyvärinen and Erkki Oja, Independent Component Analysis: A Tutorial, node14: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science

[6] Ruye Wang, Independent Component Analysis, node4: Measures of Non-Gaussianity

[7] P. Comon, Independent Component Analysis – a new concept?, Signal Processing, 36 287–314, 1994.

[8] Didier G. Leibovici and Christian Beckmann, An introduction to Multiway Methods for Multi-Subject fMRI experiment, FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.

[9] Willard Gibbs, A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Transactions of the Connecticut Academy, 382–404 (1873)

[10] Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. C. R. Acad. Sci. LXIX:858–862.

[11] Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. C. R. Acad. Sci. LXIX:1057–1061.

[12] Massieu, M. F. (1869), Compt. Rend. 69 (858): 1057.

[13] Planck, M. (1945). Treatise on Thermodynamics. Dover, New York.

[14] Antoni Planes, Eduard Vives, Entropic Formulation of Statistical Mechanics Archived 2008-10-11 at the Wayback Machine, Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona

[15] John A. Scheilman, Temperature, Stability, and the Hydrophobic Interaction, Biophysical Journal 73 (December 1997), 2960–2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA

[16] Z. Hens and X. de Hemptinne, Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures, Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium

[17] Leon Brillouin, The negentropy principle of information, J. Applied Physics 24, 1152–1163 1953

[18] Leon Brillouin, Science and Information theory, Dover, 1956

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

@@ Line 1: / Line 1: @@
+{{Short description|Measure of distance to normality}}
-The '''negentropy''', also '''negative entropy''', '''syntropy''', '''extropy''', '''ectropy''' or '''entaxy''',<ref>Wiener, Norbert</ref> of a [[living system]] is the [[entropy]] that it exports to keep its own entropy low; it lies at the intersection of [[entropy and life]]. The concept and phrase "negative entropy" was introduced by [[Erwin Schrödinger]] in his 1944 popular-science book ''[[What is Life? (Schrödinger)|What is Life?]]''<ref>Schrödinger, Erwin, ''What is Life - the Physical Aspect of the Living Cell'', Cambridge University Press, 1944</ref> Later, [[Léon Brillouin]] shortened the phrase to ''negentropy'',<ref>Brillouin, Leon: (1953) "Negentropy Principle of Information", ''J. of Applied Physics'', v. '''24(9)''', pp. 1152-1163</ref><ref>Léon Brillouin, ''La science et la théorie de l'information'', Masson, 1959</ref> to express it in a more "positive" way: a living system imports negentropy and stores it.<ref>Mae-Wan Ho, [http://www.i-sis.org.uk/negentr.php What is (Schrödinger's) Negentropy?], Bioelectrodynamics Laboratory, Open university Walton Hall, Milton Keynes</ref> In 1974, [[Albert Szent-Györgyi]] proposed replacing the term ''negentropy'' with ''syntropy''. That term may have originated in the 1940s with the Italian mathematician [[Luigi Fantappiè]], who tried to construct a unified theory of [[biology]] and [[physics]]. [[Buckminster Fuller]] tried to popularize this usage, but ''negentropy'' remains common.
+{{Distinguish|text = [[Entropy and life#Negative entropy|Negative entropy]]}}
+{{Redirect|Syntropy||Syntropy (software)}}
+In [[information theory]] and [[statistics]], '''negentropy''' is used as a measure of distance to normality.  The concept and phrase "'''negative entropy'''" was introduced by [[Erwin Schrödinger]] in his 1944 popular-science book ''[[What is Life? (Schrödinger)|What is Life?]]''<ref>Schrödinger, Erwin, ''What is Life – the Physical Aspect of the Living Cell'', Cambridge University Press, 1944</ref> Later, [[French people|French]] [[physicist]] [[Léon Brillouin]] shortened the phrase to ''néguentropie'' (negentropy).<ref>Brillouin, Leon: (1953) "Negentropy Principle of Information", ''J. of Applied Physics'', v. '''24(9)''', pp. 1152–1163</ref><ref>Léon Brillouin, ''La science et la théorie de l'information'', Masson, 1959</ref> In 1974, [[Albert Szent-Györgyi]] proposed replacing the term ''negentropy'' with '''''syntropy'''''. That term may have originated in the 1940s with the Italian mathematician [[Luigi Fantappiè]], who tried to construct a unified theory of [[biology]] and [[physics]]. [[Buckminster Fuller]] tried to popularize this usage, but ''negentropy'' remains common.
 In a note to ''[[What is Life?]]'' Schrödinger explained his use of this phrase.
-{{cquote|[...] if I had been catering for them [physicists] alone I should have let the discussion turn on ''[[Thermodynamic free energy|free energy]]'' instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to ''[[energy]]'' for making the average reader alive to the contrast between the two things.}}
+{{cquote|... if I had been catering for them [physicists] alone I should have let the discussion turn on ''[[Thermodynamic free energy|free energy]]'' instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to ''[[energy]]'' for making the average reader alive to the contrast between the two things.}}
-Indeed, negentropy has been used by biologists as the basis for purpose or direction in life, namely cooperative or moral instincts.<ref>[[Jeremy Griffith]]. 2011. ''What is the Meaning of Life?''. In ''The Book of Real Answers to Everything!'' ISBN 9781741290073. From http://www.worldtransformation.com/what-is-the-meaning-of-life/</ref>
-In 2009, Mahulikar & Herwig redefined negentropy of a dynamically ordered sub-system as the specific entropy deficit of the ordered sub-system relative to its surrounding chaos.<ref>Mahulikar, S.P. & Herwig, H.: (2009) "Exact thermodynamic principles for dynamic order existence and evolution in chaos", ''Chaos, Solitons & Fractals'', v. '''41(4)''', pp. 1939-1948</ref> Thus, negentropy has SI units of (J kg<sup>-1</sup> K<sup>-1</sup>) when defined based on specific entropy per unit mass, and (K<sup>−1</sup>) when defined based on specific entropy per unit energy. This definition enabled: ''i'') scale-invariant thermodynamic representation of dynamic order existence, ''ii'') formulation of physical principles exclusively for dynamic order existence and evolution, and ''iii'') mathematical interpretation of Schrödinger's negentropy debt.
 ==Information theory==
-In [[information theory]] and [[statistics]], negentropy is used as a measure of distance to normality.<ref>Aapo Hyvärinen, [http://www.cis.hut.fi/aapo/papers/NCS99web/node32.html Survey on Independent Component Analysis, node32: Negentropy], Helsinki University of Technology Laboratory of Computer and Information Science</ref><ref>Aapo Hyvärinen and Erkki Oja, [http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb/node14.html Independent Component Analysis: A Tutorial, node14: Negentropy], Helsinki University of Technology Laboratory of Computer and Information Science</ref><ref>Ruye Wang, [http://fourier.eng.hmc.edu/e161/lectures/ica/node4.html Independent Component Analysis, node4: Measures of Non-Gaussianity]</ref> Out of all [[Distribution (mathematics)|distributions]] with a given mean and variance, the normal or [[Gaussian distribution]] is the one with the highest entropy. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes [[if and only if]] the signal is Gaussian.
+In [[information theory]] and [[statistics]], negentropy is used as a measure of distance to normality.<ref>Aapo Hyvärinen, [http://www.cis.hut.fi/aapo/papers/NCS99web/node32.html Survey on Independent Component Analysis, node32: Negentropy], Helsinki University of Technology Laboratory of Computer and Information Science</ref><ref>Aapo Hyvärinen and Erkki Oja, [http://www.cis.hut.fi/aapo/papers/IJCNN99_tutorialweb/node14.html Independent Component Analysis: A Tutorial, node14: Negentropy], Helsinki University of Technology Laboratory of Computer and Information Science</ref><ref>Ruye Wang, [http://fourier.eng.hmc.edu/e161/lectures/ica/node4.html Independent Component Analysis, node4: Measures of Non-Gaussianity]</ref> Out of all [[Distribution (mathematics)|distributions]] with a given mean and variance, the normal or [[Gaussian distribution]] is the one with the highest [[entropy]]. Negentropy measures the difference in entropy between a given distribution and the Gaussian distribution with the same mean and variance. Thus, negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes [[if and only if]] the signal is Gaussian.
 Negentropy is defined as
-:<math>J(p_x) = S(\phi_x) - S(p_x)\,</math>
+:<math>J(p_x) = S(\varphi_x) - S(p_x)\,</math>
-where <math>S(\phi_x)</math> is the [[differential entropy]] of the Gaussian density with the same [[mean]] and [[variance]] as <math>p_x</math> and <math>S(p_x)</math> is the differential entropy of <math>p_x</math>:
+where <math>S(\varphi_x)</math> is the [[differential entropy]] of the Gaussian density with the same [[mean]] and [[variance]] as <math>p_x</math> and <math>S(p_x)</math> is the differential entropy of <math>p_x</math>:
-:<math>S(p_x) = - \int p_x(u) \log p_x(u) du</math>
+:<math>S(p_x) = - \int p_x(u) \log p_x(u) \, du</math>
-Negentropy is used in [[statistics]] and [[signal processing]]. It is related to network [[Information entropy|entropy]], which is used in [[Independent Component Analysis]].<ref>P. Comon, Independent Component Analysis - a new concept?, ''Signal Processing'', '''36''' 287-314, 1994.</ref><ref>Didier G. Leibovici and Christian Beckmann, [http://www.fmrib.ox.ac.uk/analysis/techrep/tr01dl1/tr01dl1/tr01dl1.html An introduction to Multiway Methods for Multi-Subject fMRI experiment], FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.</ref>
+Negentropy is used in [[statistics]] and [[signal processing]]. It is related to network [[Information entropy|entropy]], which is used in [[independent component analysis]].<ref>P. Comon, Independent Component Analysis – a new concept?, ''Signal Processing'', '''36''' 287–314, 1994.</ref><ref>Didier G. Leibovici and Christian Beckmann, [http://www.fmrib.ox.ac.uk/analysis/techrep/tr01dl1/tr01dl1/tr01dl1.html An introduction to Multiway Methods for Multi-Subject fMRI experiment], FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.</ref>
+The negentropy of a distribution is equal to the [[Kullback–Leibler divergence]] between <math>p_x</math> and a Gaussian distribution with the same mean and variance as <math>p_x</math> (see ''{{section link|Differential entropy#Maximization in the normal distribution}}'' for a proof). In particular, it is always nonnegative.
 ==Correlation between statistical negentropy and Gibbs' free energy==
-[[File:Wykres Gibbsa.svg|275px|thumb|right|[[Willard Gibbs]]’ 1873 '''available energy''' ([[Thermodynamic free 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 free 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.]]
+[[File:Wykres Gibbsa.svg|275px|thumb|right|[[Willard Gibbs]]’ 1873 '''available energy''' ([[Thermodynamic free 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.]]
-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]]. The said 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, [http://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] {{Webarchive|url=https://web.archive.org/web/20081011011717/http://www.ecm.ub.es/condensed/eduard/papers/massieu/node2.html |date=2008-10-11 }}, 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/abs/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>
-::'''<math>J = S_\max - S = -\Phi = -k \ln Z\,</math>'''
+:: <math>J = S_\max - S = -\Phi = -k \ln Z\,</math>
 ::where:
-::<math>J</math> - negentropy (Gibbs "capacity for entropy")
+::<math>S</math> is [[entropy]]
+::<math>J</math> is negentropy (Gibbs "capacity for entropy")
-::<math>\Phi</math> – [[Free entropy|Massieu potential]]
+::<math>\Phi</math> is the [[Free entropy|Massieu potential]]
-::<math>Z</math> - [[Partition function (statistical mechanics)|partition function]]
+::<math>Z</math> is the [[Partition function (statistical mechanics)|partition function]]
-::<math>k</math> - [[Boltzmann constant]]
+::<math>k</math> the [[Boltzmann constant]]
+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).
-==Risk management==
-In [[risk management]], negentropy is the force that seeks to achieve effective organizational behavior and lead to a steady predictable state.<ref>[http://www.kent.ac.uk/scarr/events/Grinberg-%20(2).pdf Pedagogical Risk and Governmentality: Shantytowns in Argentina in the 21st Century] (see p. 4).</ref>
 ==Brillouin's negentropy principle of information==
-In 1953, [[Léon Brillouin]] derived a general equation<ref>Leon Brillouin, The negentropy principle of information, ''J. Applied Physics'' '''24''', 1152-1163 1953</ref> stating that the changing of an information bit value requires at least kT ln(2) energy. This is the same energy as the work [[Leó Szilárd]]'s engine produces in the idealistic case. In his book,<ref>Leon Brillouin, ''Science and Information theory'', Dover, 1956</ref> he further explored this problem concluding that any cause of this bit value change (measurement, decision about a yes/no question, erasure, display, etc.) will require the same amount of energy.
+In 1953, [[Léon Brillouin]] derived a general equation<ref>Leon Brillouin, The negentropy principle of information, ''J. Applied Physics'' '''24''', 1152–1163 1953</ref> stating that the changing of an information bit value requires at least <math>kT\ln 2</math> energy. This is the same energy as the work [[Leó Szilárd]]'s engine produces in the idealistic case. In his book,<ref>Leon Brillouin, ''Science and Information theory'', Dover, 1956</ref> he further explored this problem concluding that any cause of this bit value change (measurement, decision about a yes/no question, erasure, display, etc.) will require the same amount of energy.
 ==See also==
 * [[Exergy]]
-* [[Extropy]]
 * [[Free entropy]]
 * [[Entropy in thermodynamics and information theory]]
 ==Notes==
-{{reflist}}
+{{Reflist|20em}}
+{{Wiktionary}}
-[[Category:Thermodynamic entropy]]
 [[Category:Entropy and information]]
 [[Category:Statistical deviation and dispersion]]
+[[Category:Thermodynamic entropy]]