In animate thermodynamics, negative entropy is a mathematical synonym for order, in an entropic sense. The term comes from Austrian physicist Erwin Schrödinger's famous 1944 booklet What is Life?, wherein he tried to explain the second law to a lay audience, stating that negative entropy is the amount of order that an organism "sucks from its environment" as its lives or "avoids decay to thermodynamical equilibrium or of maximum entropy". [1]

Mathematics
The idea of the verbal expression 'negative entropy' being synonymous to 'order' stems from a combination of the following three expressions:

$\frac{1}{X} = X^{-1} \,\!$ Rule for inverse functions
$\log(a^b) = b \log(a) \,\!$ Rule for logarithms
$S = k \log W \!$ Entropy expression from statistical mechanics.

$\frac{1}{X} = X^{-1} \,\!$	Rule for inverse functions
$\log(a^b) = b \log(a) \,\!$	Rule for logarithms
$S = k \log W \!$	Entropy expression from statistical mechanics.

In short, Schrodinger equates the multiplicity W of the Boltzmann entropy equation with disorder, pure and simple, which he reasons applied to all systems; then equates the inverse of multiplicity with order, as in:

$W^{-1} = Order \,\!$

then carries the negative sign over to the left side of the statistical entropy expression, using the rule for logarithms, to argue that negative S equals order.

Derivation
In his 1944 book What is Life?, Schrödinger reasoned that it is not energy that living beings feed on that keeps them at bay from decay but “negative entropy”. In rephrasing this statement, he says “the essential thing in metabolism is that the organism succeeds in freeing itself from all the entropy it cannot help producing while alive.” In making these ball-park statements, Schrödinger calls on the statistical concept of order and disorder, connections that were revealed, as he says, by the investigations of Boltzmann and Gibbs in statistical physics. On this basis, he situates the following definition:

where k is the Boltzmann constant and D, he says, is a “quantitative measure of the atomistic disorder of the body in question”. Here, to note, Schrodinger fails to mention that this expression is generally valid only for ideal gases. In any event, Schrödinger reasons that this statistical expression applies to living organisms. Moreover, to make is verbal argument mathematical, he states that “if D is a measure of disorder, its reciprocal, 1/D, can be regarded as a direct measure of order.” In addition, “since the logarithm of 1/D is just the minus of the logarithm of D, we can write can write Boltzmann’s equation thus:

Hence, as Schrödinger states:

“The awkward expression negative entropy can be replaced by a better one: entropy, taken with the negative sign [ – entropy], is itself a measure of order.”

Thus, he concludes “the device by which an organism maintains itself stationary at a fairly high level of orderliness”, a state he equates with a low level of entropy, consists in “sucking orderliness from its environment”.

Negentropy
In 1953, through the guise of information theory, Schrödinger's negative entropy usage was shortened into the term "negentropy" by French physicist Léon Brillouin. [2]

Difficulties
After his lecture, wherein he discussed negative entropy, Schrodinger famously had to add a note to Chapter 6, where explains that:

“My remarks on negative entropy have met with doubt and opposition from physicist colleagues.”

He goes on to explain that had he been lecturing to them, he would have turned the discussion to free energy, but judged the concept too intricate for the lay audience. In a 1946 review of Schrödinger’s What is Life?, author H.J. Muller stated, supposedly, that biologists have, in the previous decades, commonly defined negative entropy as "potential energy". [3] Muller, it seems, is referring here to Schrodinger's note on free energy. When concept of negative entropy is taken literally and measured in actual organisms, the concept looses its meaning, and a move to free energy discussions prevails. [4]

See also
● Positive entropy

References
1. Schrödinger, Erwin. (1944). What is Life? (ch. 6 “Order, Disorder, and Entropy). pgs. 67-75 Cambridge: Cambridge University Press.
2. Brillouin, Léon. (1953). "Negentropy Principle of Information", J. of Applied Physics, v. 24:9, pp. 1152-1163.
3. Muller, H. J. (1946). “A physicist stands amazed at genetics.” (PDF). J. Hered. 37:90-92.
4. Stockard U. von and Liu, J. S. (1999). “Does Microbial Life Always feed on Negative Entropy? Thermodynamic Analysis of Microbial Growth.” Biochimica et Biophysica Acta (BBA) – Bioenergetics, Vol. 1412, Issue 3, 4 Aug., Pgs. 191-211.

Further reading
● Iniquez, Cruz. (2008). Negative Entropy: A Brief Incursion into the Uncharted Universe of Decreasing Entropy (abs) (pdf). Infinity Publishing.