Negentropy

In information theory and statistics, negentropy is used as a measure of distance to normality.[4][5][6] 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

${\displaystyle J(p_{x})=S(\varphi _{x})-S(p_{x})\,}$

where ${\displaystyle S(\varphi _{x})}$ is the differential entropy of the Gaussian density with the same mean and variance 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 in statistics and signal processing. It is related to network entropy, which is used in independent component analysis.[7][8]

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

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.[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 by Massieu for the isothermal process[10][11][12] (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process.[13] 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,[14] applied among the others in molecular biology[15] and thermodynamic non-equilibrium processes.[16]

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

where:

${\displaystyle S}$ is entropy

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

${\displaystyle \Phi }$ is the Massieu potential

${\displaystyle Z}$ is the partition function

${\displaystyle k}$ the Boltzmann constant

In particular, mathematically the negentropy (the negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (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 work Leó 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