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Ininformation theory andstatistics,negentropy is used as a measure of distance to normality. It is also known asnegative entropy orsyntropy.

The concept and phrase "negative entropy" was introduced byErwin Schrödinger in his 1944 bookWhat is Life?.^[1] Later, theFrenchphysicistLéon Brillouin shortened the phrase tonéguentropie (transl. 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.^{[citation needed]}

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.

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Ininformation theory andstatistics, negentropy is used as a measure of distance to normality.^[4][5][6] Out of allprobability distributions with a givenmean andvariance, the Gaussian ornormal distribution is the one with the highestentropy.^{[clarification needed][citation needed]} 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.^{[citation needed]}

Negentropy is defined as

${\displaystyle J(Y)=h(Y_G)-h(Y),}$

where${\displaystyle h(Y_G)=\frac{1}{2}\log \left(2\pi \mathrm{e}\cdot {\sigma }^2\right)}$ is thedifferential entropy of a normal distribution${\displaystyle Y_G\sim N(\mu ,{\sigma }^2)}$ with the same mean${\displaystyle \mu }$ and variance${\displaystyle {\sigma }^2}$ as${\displaystyle Y}$, and${\displaystyle h(Y)}$ is the differential entropy of${\displaystyle Y}$, with${\displaystyle p_Y}$ as itsprobability density function:

${\displaystyle h(Y)=-\int p_Y(u)\log p_Y(u)\mathrm{d}u}$

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

The negentropy of a distribution is equal to theKullback–Leibler divergence between${\displaystyle Y}$ and a Gaussian distribution with the same mean and variance as${\displaystyle Y}$ (seeDifferential entropy § Maximization in the normal distribution for a proof):$${\displaystyle J(Y)=D_{KL}(Y\Vert Y_G)}$$In particular, it is always nonnegative (unlike differential entropy, which can be negative).

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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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