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The negentropy has different meanings in information theory and theoretical biology. In a biological context, the negentropy (also negative entropy, syntropy, extropy, ectropy or entaxy[1]) 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. In other words, negentropy is reverse entropy. It means things becoming more orderly. By 'order' is meant organisation, structure and function: the opposite of randomness or chaos. The concept and phrase "negative entropy" was introduced by Erwin Schrödinger in his 1944 popular-science book What is Life?[2] Later, Léon Brillouin shortened the phrase to negentropy,[3][4] to express it in a more "positive" way: a living system imports negentropy and stores it.[5] 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.

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.[6] Thus, negentropy has SI units of (J kg−1 K−1) when defined based on specific entropy per unit mass, and (K−1) 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 theoryEdit

In information theory and statistics, negentropy is used as a measure of distance to normality.[7][8][9] 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


where   is the differential entropy of the Gaussian density with the same mean and variance as   and   is the differential entropy of  :


Negentropy is used in statistics and signal processing. It is related to network entropy, which is used in independent component analysis.[10][11]

Correlation between statistical negentropy and Gibbs' free energyEdit

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.

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.[12] 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[13][14][15] (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process.[16] 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,[17] applied among the others in molecular biology[18] and thermodynamic non-equilibrium processes.[19]

  is entropy
  is negentropy (Gibbs "capacity for entropy")
  is the Massieu potential
  is the partition function
  the Boltzmann constant

Brillouin's negentropy principle of informationEdit

In 1953, Léon Brillouin derived a general equation[20] 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,[21] 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 alsoEdit


  1. ^ Wiener, Norbert
  2. ^ Schrödinger, Erwin, What is Life – the Physical Aspect of the Living Cell, Cambridge University Press, 1944
  3. ^ Brillouin, Leon: (1953) "Negentropy Principle of Information", J. of Applied Physics, v. 24(9), pp. 1152–1163
  4. ^ Léon Brillouin, La science et la théorie de l'information, Masson, 1959
  5. ^ Mae-Wan Ho, What is (Schrödinger's) Negentropy?, Bioelectrodynamics Laboratory, Open university Walton Hall, Milton Keynes
  6. ^ 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
  7. ^ Aapo Hyvärinen, Survey on Independent Component Analysis, node32: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science
  8. ^ Aapo Hyvärinen and Erkki Oja, Independent Component Analysis: A Tutorial, node14: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science
  9. ^ Ruye Wang, Independent Component Analysis, node4: Measures of Non-Gaussianity
  10. ^ P. Comon, Independent Component Analysis – a new concept?, Signal Processing, 36 287–314, 1994.
  11. ^ 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.
  12. ^ 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)
  13. ^ Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. C. R. Acad. Sci. LXIX:858–862.
  14. ^ Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. C. R. Acad. Sci. LXIX:1057–1061.
  15. ^ Massieu, M. F. (1869), Compt. Rend. 69 (858): 1057.
  16. ^ Planck, M. (1945). Treatise on Thermodynamics. Dover, New York.
  17. ^ Antoni Planes, Eduard Vives, Entropic Formulation of Statistical Mechanics, Entropic variables and Massieu–Planck functions 2000-10-24 Universitat de Barcelona
  18. ^ 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
  19. ^ 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
  20. ^ Leon Brillouin, The negentropy principle of information, J. Applied Physics 24, 1152–1163 1953
  21. ^ Leon Brillouin, Science and Information theory, Dover, 1956