Negentropy

Negentropy is an obsolete synonym of free entropy.^[1] The concept and phrase "negative entropy" were introduced by Erwin Schrödinger in his 1943 popular-science book What is life?^[2] Later, Léon Brillouin shortened the phrase to negentropy.^[3]^[4] In a note to What 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 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.

The main problem with the term "negentropy" is that the value it represents is nonnegative:

To explain the difficulty with negentropy as information we notice that what is here physically important is not −S, but the so-called entropy defect, i.e., D = k log n − S, which is always nonnegative (we have in mind the case (1-5)), and may represent "information at our disposal" or "free entropy," similarly as free energy F = U − TS, (where U is internal energy and T is absolute temperature) which is also nonnegative.

—Ingarden, Roman S.; Kossakowski, A.; Ohya, Masanori ♦ Information dynamics and open systems: classical and quantum approach p. 17

Due to its unjustified negativity, the term "negentropy" is often misunderstood as entropy deficit resulting from entropy export. However, a system, which has exported all of its entropy (e.g., liquid helium at −271°C), has no room for information^[5] and is unable to do work.

General concept

Entropy is a measure of a system's internal energy (regardless of its availability).^[6] The total entropy of a system is subdivided into the bound entropy and the free entropy. The bound entropy is a measure of a system's anergy (the unavailable part of the system's internal energy).^[7] The free entropy (also known as negentropy) is a measure of a system's exergy (the available part of the system's internal energy):^[8]

The bound entropy is determined by the microscopic degrees of freedom of the network and is not available as useable information. In analogy with free energy, which is the energy available to do work, we have free entropy, which is the entropy available to produce useful information. Thus, the free entropy is precisely the neg-entropy discussed by Szilárd, Wiener, Brillouin and so many others.

—West, Bruce J.; Scafetta, Nicola ♦ Disrupted Networks: From Physics to Climate Change Studies of Nonlinear Phenomena in Life Science, Vol. 13. World Scientific, 2010, p. 150

A system's free entropy is inversely proportional to the system's volume (a compressed gas is capable of doing work, which cannot be said about an expanded gas).^[9] When a gas is being compressed, its total entropy remains unchanged,^[10] its bound entropy decreases, while its free entropy increases. Thus, negentropy (free entropy) is volume-specific entropy surplus above background.

Negentropy (condensed entropy) is synonymous with information,^[11] because condensation of dissipated energy is accompanied by its self-organization. For example, when a portion of a gas becomes subjected to a sufficiently strong compression, it automatically forms a hot crystal^[12] capable of doing work.

Gravity as the source of negentropy

In relativistic quantum theory, a system cannot be localized to a precision better than its Compton wavelength,^[13] expressed as λ = h c/E = c/f.

At f = 1 Hz, a quantum's energy (E = f h) is equal to h (the breakeven point between wave-likeness and particle-likeness), while the quantum's Compton wavelength (the radius of nonlocality, instantaneous propagation) is 1 light-second (300 thousand kilometres). According to the theory of relativity, superluminal propagation is propagation into the past. Therefore:

A relativistic wave, whose frequency is above 1 Hz, begins its second period in the future; the third period, still further in the future, and so on. Such a wave is an electromagnetic wave; in its quantum—the photon—the particle-like magnetic phase dominates over the wave-like electric phase. Since the electromagnetic field propagates into the future, its thermodynamic entropy increases with time.
A relativistic wave, whose frequency is below 1 Hz, begins its second period in the past; the third period, still further in the past, and so on. Such a wave is a gravitoelectromagnetic wave; in its quantum—the graviton—the wave-like electric phase dominates over the particle-like magnetic phase.^[14] Since the gravitoelectromagnetic field propagates into the past, its thermodynamic entropy decreases with time.^[15]

Information theory

In information theory and statistics, negentropy is used as a measure of distance to normality.^[16]^[17]^[18] Consider a signal with a certain distribution. If the signal is Gaussian, the signal is said to have a normal distribution. 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

J(p_{x})=S(\phi _{x})-S(p_{x})\,

where $S(\phi _{x})$ stands for the differential entropy of the Gaussian density with the same mean and variance as $p_{x}$ and $S(p_{x})$ is the differential entropy of $p_{x}$ :

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.^[19]^[20] Negentropy can be understood intuitively as the information that can be saved when representing $p_{x}$ in an efficient way; if $p_{x}$ were a random variable (with Gaussian distribution) with the same mean and variance, would need the maximum length of data to be represented, even in the most efficient way. Since $p_{x}$ is less random, then something about it is known beforehand, it contains less unknown information, and needs less length of data to be represented in an efficient way.

Correlation between statistical negentropy and Gibbs' free energy

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. The said quantity is the amount of entropy that may be increased without changing an internal energy or increasing its volume.^[21] 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 ^[22]^[23]^[24] (both quantities differs just with a figure sign) and then Planck for the isothermal-isobaric process ^[25] 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,^[26] applied among the others in molecular biology^[27] and thermodynamic non-equilibrium processes.^[28]

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

where:

J

- negentropy (Gibbs "capacity for entropy")

\Phi

– Massieu potential

Z

- partition function

k

- Boltzmann constant

Organization theory

In 1988, on the basis of Shannon's definition of statistical entropy, Mario Ludovico^[29] gave a formal definition of sintropy, as a measurement of the degree of organization internal to any system formed by interacting components. According to that definition, sintropy is a quantity complementary to entropy. The sum of the two quantities defines a constant value, specific of the system of which that constant value identifies the transformation potential. By use of such definitions, the theory develops equations apt to describe/simulate any possible evolution of the system, either toward higher/lower levels of "internal organization" (i.e., sintropy) or toward the system's collapse.^[30]

In risk management, negentropy is the force that seeks to achieve effective organizational behavior and lead to a steady predictable state.^[31]

Notes

^ West, Bruce J.; Scafetta, Nicola ♦ Disrupted Networks: From Physics to Climate Change Studies of Nonlinear Phenomena in Life Science, Vol. 13. World Scientific, 2010, p. 150
^ Schrödinger, Erwin What is Life - the Physical Aspect of the Living Cell, Cambridge University Press, 1944
^ Brillouin, Leon: (1953) "Negentropy Principle of Information", J. of Applied Physics, v. 24(9), pp. 1152-1163
^ Léon Brillouin, La science et la théorie de l'information, Masson, 1959
^ Brillouin, Léon ♦ Science and information theory Courier Dover Publications, 2004, p. 8 ♦ "Every type of constraint, every additional condition imposed on the possible freedom of choice immediately results in a decrease of information."
^ Dubois , F. ♦ Concavity of thermostatic entropy and convexity of Lax's mathematical entropy Aerospatiale, Strategic and Space Systems Division, 1990 ♦ "The thermodynamic properties of a gas with mass M, internal energy E enclosed in volume V are completely determined by these three parameters. In effect, the thermostatic entropy S is a function of the triplet (M, V, E): S=∑(M, V, E)."
^ Honerkamp, J. ♦ Statistical physics Springer, 2002, p. 298 ♦ "The maximum fraction of an energy form which (in a reversible process) can be transformed into work is called exergy. The remaining part is called anergy, and this corresponds to the waste heat."
^ Corning, Peter A. ♦ Holistic Darwinism: synergy, cybernetics, and the bioeconomics of evolution University of Chicago Press, 2005, p. 329 ♦ "… negentropy is really another term for an increase in available energy."
^ Newman, Jay ♦ Physics of the life sciences Springer, 2008, p. 336 ♦ "To be useful, internal energy has to be concentrated. The more dilute or disorganized the internal energy, the less useful it is ..."
^ Nesbitt, Brian ♦ Handbook of Valves and Actuators Butterworth-Heinemann, 2007, p. 65 ♦ "Entropy is very unusual when compared to other gas properties; entropy only changes when heat transfer occurs. Entropy is not dependent upon temperature, pressure or volume."
^ Ingarden, Roman S.; Kossakowski, A.; Ohya, Masanori ♦ Information dynamics and open systems: classical and quantum approach Springer, 1997, p. 17
^ Ginzburg, Vitaliĭ Lazarevich ♦ The physics of a lifetime: reflections on the problems and personalities of 20th century physics Springer, 2001, p. 27
^ Ji, Xiangdong (2004). "Viewing the proton through "color" filters". In Boffi, S.; Ciofi degli Atti, C.; Giannini, M. M. (ed.). Perspectives in hadronic physics: 4th international conference held at ICTP, Trieste, Italy, 12-16 May 2003. p. 24. {{cite book}}: |access-date= requires |url= (help); External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)CS1 maint: multiple names: editors list (link)
^ "McGraw-Hill Dictionary of Scientific and Technical Terms". Retrieved 2 Dec 2010. (The gravitoelectric field dominates over the gravitomagnetic field.)
^ Corning, Peter A. ♦ Holistic Darwinism: synergy, cybernetics, and the bioeconomics of evolution University of Chicago Press, 2005, p. 340
^ Aapo Hyvärinen, Survey on Independent Component Analysis, node32: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science
^ Aapo Hyvärinen and Erkki Oja, Independent Component Analysis: A Tutorial, node14: Negentropy, Helsinki University of Technology Laboratory of Computer and Information Science
^ Ruye Wang, Independent Component Analysis, node4: Measures of Non-Gaussianity
^ P. Comon, Independent Component Analysis - a new concept?, Signal Processing, 36 287-314, 1994.
^ Didier G. Leibovici and Christian Beckmann, An introduction to Multiway Methods for Multi-Subject fMRI experiment, FMRIB Technical Report 2001, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain (FMRIB), Department of Clinical Neurology, University of Oxford, John Radcliffe Hospital, Headley Way, Headington, Oxford, UK.
^ Willard Gibbs, A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, Transactions of the Connecticut Academy, 382-404 (1873)
^ Massieu, M. F. (1869a). Sur les fonctions caractéristiques des divers fluides. C. R. Acad. Sci. LXIX:858-862.
^ Massieu, M. F. (1869b). Addition au precedent memoire sur les fonctions caractéristiques. C. R. Acad. Sci. LXIX:1057-1061.
^ Massieu, M. F. (1869), Compt. Rend. 69 (858): 1057.
^ Planck, M. (1945). Treatise on Thermodynamics. Dover, New York.
^ Antoni Planes, Eduard Vives, Entropic Formulation of Statistical Mechanics, Entropic variables and Massieu-Planck functions 2000-10-24 Universitat de Barcelona
^ John A. Scheilman, Temperature, Stability, and the Hydrophobic Interaction, Biophysical Journal 73 (December 1997), 2960-2964, Institute of Molecular Biology, University of Oregon, Eugene, Oregon 97403 USA
^ Z. Hens and X. de Hemptinne, Non-equilibrium Thermodynamics approach to Transport Processes in Gas Mixtures, Department of Chemistry, Catholic University of Leuven, Celestijnenlaan 200 F, B-3001 Heverlee, Belgium
^ Mario Ludovico, L'evoluzione sintropica dei sistemi urbani - Elementi per una teoria dei sistemi auto-finalizzati (Sintropy in the Evolution of Urban Systems - Elements for a Theory of Self-Organized Systems), Bulzoni, Roma 1988-1991
^ Here is a summary of the sintropy theory.
^ Pedagogical Risk and Governmentality: Shantytowns in Argentina in the 21st Century (see p. 4).