Ininformation theory andstatistics,negentropy is used as a measure of distance to normality. The concept and phrase "negative entropy" was introduced byErwin Schrödinger in his 1944 popular-science bookWhat is Life?^[1] Later,FrenchphysicistLéon Brillouin shortened the phrase tonéguentropie (negentropy).^[2][3] In 1974,Albert Szent-Györgyi proposed replacing the termnegentropy withsyntropy. That term may have originated in the 1940s with the Italian mathematicianLuigi Fantappiè, who tried to construct a unified theory ofbiology andphysics.Buckminster Fuller tried to popularize this usage, butnegentropy remains common.

In a note toWhat 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 onfree energy instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near toenergy for making the average reader alive to the contrast between the two things.

This sectionis missing information about the mathematical treatment of negentropy in information theory. Please expand the section to include this information. Further details may exist on thetalk page.(December 2024)

Ininformation theory andstatistics, negentropy is used as a measure of distance to normality.^[4][5][6] Out of alldistributions with a given mean and variance, the normal orGaussian distribution is the one with the highestentropy. 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 vanishesif and only if the signal is Gaussian.

Negentropy is defined as

${\displaystyle J(p_x)=S({\phi }_x)-S(p_x)}$

where${\displaystyle S({\phi }_x)}$ is thedifferential entropy of the Gaussian density with the samemean andvariance as${\displaystyle p_x}$ and${\displaystyle S(p_x)}$ is the differential entropy of${\displaystyle p_x}$:

${\displaystyle S(p_x)=-\int p_x(u)\log p_x(u)du}$

Negentropy is used instatistics andsignal processing. It is related to networkentropy, which is used inindependent component analysis.^[7][8]

The negentropy of a distribution is equal to theKullback–Leibler divergence between${\displaystyle p_x}$ and a Gaussian distribution with the same mean and variance as${\displaystyle p_x}$ (seeDifferential entropy § Maximization in the normal distribution for a proof). In particular, it is always nonnegative.

There is a physical quantity closely linked tofree 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 tofree enthalpy. On the diagram one can see the quantity calledcapacity 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 byMassieu for theisothermal process^[10][11][12] (both quantities differs just with a figure sign) and by thenPlanck for theisothermal-isobaric process.^[13] More recently, the Massieu–Planckthermodynamic potential, known also asfree entropy, has been shown to play a great role in the so-called entropic formulation ofstatistical mechanics,^[14] applied among the others in molecular biology^[15] and thermodynamic non-equilibrium processes.^[16]

${\displaystyle J=S_{max}-S=-\mathrm{\Phi }=-k\ln Z}$

where:

${\displaystyle S}$ isentropy

${\displaystyle J}$ is negentropy (Gibbs "capacity for entropy")

${\displaystyle \mathrm{\Phi }}$ is theMassieu potential

${\displaystyle Z}$ is thepartition function

${\displaystyle k}$ theBoltzmann constant

In particular, mathematically the negentropy (the negative entropy function, in physics interpreted as free entropy) is theconvex conjugate ofLogSumExp (in physics interpreted as the free energy).

In 1953,Léon Brillouin derived a general equation^[17] stating that the changing of an information bit value requires at least${\displaystyle kT\ln 2}$ energy. This is the same energy as the workLeó 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.

• Exergy
• Free entropy
• Entropy in thermodynamics and information theory

Look upnegentropy in Wiktionary, the free dictionary.

• Entropy and information
• Statistical deviation and dispersion
• Thermodynamic entropy

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