Negentropy

In

... if I had been catering for them [physicists] alone I should have let the discussion turn on free energy instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to energy for making the average reader alive to the contrast between the two things.

Information theory

In

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

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.

Correlation between statistical negentropy and Gibbs' free energy

There is a physical quantity closely linked to

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).

Brillouin's negentropy principle of information

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

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

Notes

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