Negentropy 361356 221675396 2008-06-25T15:55:59Z 213.41.233.135 /* Information theory */ minor fix in definitions of S(\phi_x) and S(p_x) '''Negative [[entropy]]''' or '''negentropy''' or '''syntropy''' of a living system is the entropy that it exports to maintain its own entropy low (see [[entropy and life]]). The concept and phrase were introduced by [[Erwin Schrödinger]] in his [[1943]] 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 people|Italian]] [[mathematics|mathematician]] [[Luigi Fantappiè]], who tried to construct a unified theory of [[biology]] and [[physics]]. (This attempt has not gained renown or borne great fruit.) [[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.}} ==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> Consider a [[Wiktionary:signal|signal]] with a certain [[Distribution (mathematics)|distribution]]. If the signal is [[Gaussian]], the signal is said to have a [[normal distribution]]. Negentropy is always nonnegative, is invariant by any linear invertible change of coordinates, and vanishes [[iff]] the signal is Gaussian. Negentropy is defined as :<math>J(p_x) = S(\phi_x) - S(p_x)\,</math> where <math>S(\phi_x)</math> stands for 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> 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, 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 can be understood intuitively as the [[Information theory|information]] that can be saved when representing <math>p_x</math> in an efficient way; if <math>p_x</math> were a random variable (with Gaussian distribution) with the same mean and variance, would need the maximum length of data to be represented, even in the most efficient way. Since <math>p_x</math> is less random, then something about it is known beforehand, it contains less unknown information, and needs less length of data to be represented in an efficient way. ==Correlation between statistical negentropy and Gibbs' free energy== [[Image:Gibbs-plot.jpg|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.]] 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 [[Josiah Willard Gibbs|Willard Gibbs]] created a diagram illustrating concept of free energy corresponding to [[free enthalpy]]. On the diagram one can see the quantity called [[capacity for entropy]]. The said quantity is 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 adopted in statistics and theory information, definition of negentropy. Similar physical quantity introduced in 1869 [[M. F. Massieu|Massieu]] for [[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> Recently, Massieu-Planck [[thermodynamic potential]], known also as ''[[free entropy]]'', plays a great role in 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-equilibriumi 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> ::'''<math>J = S_\max - S = -\Phi = -k \ln Z\,</math>''' ::where: ::<math>J</math> - negentropy (Gibbs "capacity for entropy") ::<math>\Phi</math> – [[Free entropy|Massieu potential]] ::<math>Z</math> - [[Partition function (statistical mechanics)|partition function]] ::<math>k</math> - [[Boltzmann constant]] ==Organization theory== 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> ==Notes== {{reflist}} ==See also== * [[Ectropy]] * [[Exformation]] * [[Extropy]] * [[Free entropy]] * [[Entropy in thermodynamics and information theory]] * [[Entropy and life]] {{thermodynamics-stub}} [[Category:Thermodynamic entropy]] [[Category:Entropy and information]] [[Category:Statistical deviation and dispersion]] [[de:Negentropie]] [[es:Negentropía]] [[fr:Néguentropie]] [[ja:ネゲントロピー]] [[no:Negentropi]] [[pl:Negentropia]] [[simple:Negentropy]] [[it:negentropia]]