The negentropy, also negative entropy or syntropy or entaxy, 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" were introduced by Erwin Schrödinger in his 1943 popular-science book What is Life? Later, Léon Brillouin shortened the phrase to negentropy, to express it in a more "positive" way: a living system imports negentropy and stores it. 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. Thus, negentropy has units when defined based on specific entropy per unit mass, and 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.
In information theory and statistics, negentropy is used as a measure of distance to normality. Out of all distributions with a given 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 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 S is the differential entropy of the Gaussian density with the same mean and variance as p_x and S is the differential entropy of p_x:
Negentropy is used in statistics and signal processing. It is related to network entropy, which is used in Independent Component Analysis.
Correlation between statistical negentropy and Gibbs' free energy
There is a physical quantity closely linked to free energy, 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. The said 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 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.