DEFINITIONS FOR A FUZZY ENTROPY MODEL (Man at play)

Introductory notes:

• Daily bread of the mathematicians, the concept of infinite is embarrassing for the physicists. More, the rigor of the mathematical signs " = " or " ≡ " doesn't seem adapted to physical reality. In this field, let us to propose to replace them by the fine approximate signs " ~ " or " ≈ ", which would avoid ambiguities. In the case of the infinitesimal calculus, a monument of the mathematical physics, some critics had objected that secondary terms were regarded as negligible; G. Leibniz had to concede that infinitesimal was a useful fiction.
• A lot has been written about symmetry with regard to the time of the equations of mechanics; this symmetry shows us simply that the model is approximate and that in this field also the map is not the territory; the solar system itself is only apparently steady. In mechanics total energy represents the sum of potential and kinetic energies. You must take account of both to launch a satellite. The variation of potential energy between two points A and B does not take account of the used way, considered reversible. The kinetic energy yes.

History:

The model, too much mathematical, of R. Clausius's Entropy is only an approximation of the Fuzzy Entropy, representation of an inaccessible fraction of Reality.

Rigorously:  "to calculate the variation of entropy accompanying a transformation carried out irreversibly, you must imagine a reversible process that should have the same initial and final states that this transformation ". (Paul Arnaud, cours de chimie physique) It would be even necessary to prove that at least one reversible way exists.

The difference in work between the reversible expansion and the irreversible expansion of a perfect gas may be significant, but where to buy perfect gas? Reversible and quasi-static processes exist on a small scale; they exist on our macroscopic level only in thought experiments.

L. Boltzmann described the same physical phenomenon differently; by creating statistical mechanics, a true science of uncertainty, he invented a "fuzziness" with a great future ahead of it. Had J. M. Keynes assimilate this change of paradigm, when it introduced this uncertainty into the economic theories? (ref. 75)

Like for the variations of potential energy in mechanics, these variation formulations of the traditional Entropy do not take account of the followed way. They do not represent reality satisfactorily; various possibilities exist to modify the temperature of a system, by heating, friction, Joule effect ... They are not applicable to the cases of dissipative systems, such the life.

State of the building site:

The study of the dissipative systems, in particular by the School of Brussels, showed the need for developing "dynamic" or kinetic Entropy, but without time symmetry, contrary to traditional dynamics.

Paul Glansdorff and Alkiviadis Grecos, in Encyclopedia Universalis - 2002, observe that research is in hand in two directions: 

• a development of the theorem H, in the statistical vein cleared by Boltzmann,
• a more deterministic orientation by functional equations (functions of functions).

By the increasing and inescapable use of the energy resources this field is one of the burning issues of the day. A human being, transforming biologically approximately 100W, can convert approximately 100 000W of noble energy into heat by simply turning a car ignition key.

Here is that should challenge crowds of scientists. However, many brilliant minds have first a passion for the mysterious borders of our world, from Higg's boson to pulsars. At the point to wonder whether, in their heads, Evolution does not impose another threshold, (see § Philosophical Presuppositions) a better evaluation of the Entropy consequences being able to lead to a restriction of its production; a restriction against-nature for our species.

If you don't admit that Evolution imposes an undervaluation of certain parameters in our judgments, how to understand such slow advances in thermodynamics, a crucial science for our common fate? How to understand that we hardly exceeded thermostatic and that the necessary mathematical tools are long in coming? On Earth, not at the Universe confines, a liquid, heated up in a simple pan, poses a problem when you get vortices of Bénard. Even if physicists manage to model how it's unfolding, see the see the recent commendable  Maximum Entropy Production (MEP) of R. C. Dewar, the subjacent logic of the process is not clarified. A precedent: the fall of a body described by G. Galileo, clarified later by a “law” of mechanics. 

According to a honourable member of the Royal Society, I should have noted the name, it would be already too late. In spite of this unfavourable forecast, let us try to overcome the handicap of this scientific field, manifestly little prized; let us anticipate the model to come and define tools usable, may be the concepts of Field of Entropy and Catalysis of Entropy that I suggest. Moreover, how to correctly describe a star or a black hole without a convincing model of Kinetic Entropy?

A rigorous definition of Entropic Catalysts will show us the extent of the task if we must, one day, control the uses of energy. (see the page: Production of Fuzzy Entropy zS by the Man)

Definitions:

Thermodynamic System:  (in short: system) discernible set, called sometimes particle, separable or not from its environment called sometimes Universe, usable for a real experiment or a thought experiment. General term including elementary particles.

A phenomenon of Entropic origin can engage when:

• a Fuzzy Entropic Potential offers the favorable niche appropriateness by its parameters geographical,  geological, climatic, physical or physicochemical,
• local thermodynamic changes of temperature, pressure and Fuzzy Entropy occur, due to these physicochemical variations,
• by chance an area is created where the Fuzzy Entropy differs from that of its environment,
• a real or virtual border is formed around this area that we may call system, for example, in meteorology, a low-pressure system in a general atmospheric circulation. (Jupiter's Great Red Spot)

Fuzzy Entropy zS: Conjecture to define strongly to assure the foundations of the model; in the vein of the Fuzzy Logic or the Fuzzy Space Time of the quantum physics; then used in the place of the Entropy S, usually non-recognised in “out of equilibrium thermodynamics”.

Today and while waiting better, we can write without annoying anybody:

dzS ~ Q_irreversible / T              with  T cst

with an uncertainty margin as little as we can, not as we want; we are speaking of physics, not of mathematics.

L. Bolzmann formula, where uncertainty is intrinsic, may also be written:

zS ~ k ln

zS would be too an extensive size (log neperian).

zS Total Fuzzy Entropy may be regarded as the sum of two Entropies:

• a constant Realized Fuzzy Entropy, for example the Fuzzy Entropy of a change of "state" for a frozen eel or frog (rana sylvestrica), brought back to a temperature of homeostatic balance,
• a Kinetic Fuzzy Entropy, when their metabolism start again.

Fuzzy Entropic Field: physical-mathematical being, defined by applying the notion of continuum to Fuzzy Entropy, with the possible use of the tools of the analysis. 

Field in the broad sense: a whole of values, which may be of probability, in a space.

Fuzzy Entropy Potential: to define, in Cartesian coordinates, like an integral of surface, .(see Newtonian Potential in Dictionnaire des Mathématiques of A. Bouvier and M. George – Presses Universitaires de France, or Wikipedia)

Currently, we can only conjecture that a Fuzzy Entropy Potential authorizes variations in temperature, from where may emerge systems of low relative Fuzzy Entropy.

It would tend towards a maximum.

Rem: the thermodynamic potential, or free enthalpy, does lead towards a minimum, under very restrictive conditions.

Well of Fuzzy Entropy: a Field of Fuzzy Entropy may be represented by a surface where Systems would dig wells, like gravitational masses digging wells in a surface classically representing the field of gravity.

Dimension of Fuzzy Entropy zS: would have the dimension of energy if T (K) is considered dimensionless; in MKSA System, where temperature is basic size, Fuzzy Entropy would be measured in Joule . K^-1.

Realized Fuzzy Entropy RzS: to define. A priori corresponding to the traditional notation S.

Corresponds to a plateau in the energy conversion, to the emergence of a system, living or not.

In the case of living system, corresponds to the emergence of tools, goods or services.

Kinetic Fuzzy Entropy KzS: to define, effective production of Fuzzy Entropy; called “source” by some authors. (ref. 7)

zS Jump: from a Fuzzy Entropy's Potential plateau to another one above by Kinetic Fuzzy Entropy.

Endogenous KzSn:internal KzS produced by transformation in a system. (ex: by basal metabolism in an amoeba)

Exogenous KzSx: external KzS produced by a system (ex: Earth's atmosphere under the impact of a meteorite)

Catalysis: in chemistry, starting or changing of speed of reaction (and increasing the selectivity), generated by some bodies which are left unchanged at the end of the process. Misused here to compare Fuzzy Entropies internal and external during an event.

Catalyst of KzS: system whose endogenous KzS production involves an exogenous KzS production. Ex: a rolling stone causing an avalanche in mountain, a processor driving a milling machine, a maintenance engineer opening a dam floodgate.

All would occur like if systems were locally and temporarily shaped. Then, they would behave like tools to produce elsewhere Fuzzy Entropy, superior sometimes of several orders of magnitude. (ref 59)  More than an analogy with gravity, and the role of the drop or sand grain in the mathematical study of relaxation or catastrophe systems?

A relationship with the “principle of least action”? (Koenig - Maupertuis)

Before that we call life, did this Catalyst of KzS appear in the ordered groups of molecule called membranes?

Catalysts Chain of KzS: A Primary Catalyst may set in motion a Secondary or Derived Catalyst, often with an increased production of Fuzzy Entropy:  

• Temporal Chain: a same Catalyst successively maintains various energy transformations. Ex. A truck driver who takes his car, then a lift, to go back home,
• Spatial Chain: structure of several KzS's catalysts nested, with multiplier effect. The unit would then behave like a Super-Catalyst. Not-exhaustive examples:
• ATP molecules, muscle fibres, hand, the pilot to simplify, throttle levers, jet engines, … , Airbus A340 on takeoff,
• an animal or human hierarchy.

Collective Catalysts of KzS: set of Catalysts whose meeting produces a higher Collective KzS than the sum of individual KzS :

• by simple gathering of individual Catalysts. Ex: predators that meet to attack big preys; in the same way, herds of presys that protect themselves by their number,
• by the task sharing within a community and use of particular competences of its members. Ex: the bees, (ref. 50) which contribute to the GDP by the pollination; the first assembly line work developed at the 16th century in the shipyards of Venice, which could launch two hundred boats in two months; assembly line work systematized at the slaughter-houses of Chicago and adapted to automotive engineering by Ford.

Network of Chains of KzS: to see the graph theory.

Financial Capital: palpable (sometimes), bijective substitute for Realized Fuzzy Entropy.

Cash flow: bijective substitute for Kinetic Fuzzy Entropy.

KzS Efficiency Factor Sf: term used in chemistry. Picked up here in applying the name to the ratio exogenous / endogenous of Kinetic Fuzzy Entropy of a system; for a living system "endogenous" would mean its basal metabolism:

Sf = KzSx _exogenous / KzSn _endogenous

With the notation of Prigogine:

dKzS ~ d_iKzS + Sf d_iKzS

dKzS ~ d_iKzS (1 + Sf)

To answer an objection, the Factor of Efficiency of one individual Catalyst can be indeed enormous, in the case of an operator who pressed the fire button of a giant rocket for example; but at the same moment, billion of individuals had a very weak Sf.

Complexification of a Thermodynamic System: Like definite higher, a thermodynamic system can be  relatively simple, almost mechanical in the case of a disturbance for example, where the component count is weak. In the case of a cyclone too, internal energy conversions are higher than external conversions, even if its drag on the ground causes a human and material disaster.

A system may be more complex if physicochemical phenomena come into play: 

• use of the royal road of carbon, then  oxygen,
• energy input, starting the coupling of physicochemical reactions,
• coupling that "must connect the release of energy to the organization process  ..." (ref. 78) for example, later, by nucleotide ATP,
• formation of a real wall called membrane, which includes the internal components, separates them from the universe and filters the interactions with the environment,
• increasing Entropy Potential difference between the system and the universe, with two consequences:
• the chemical reactions become more and more complex and the internal request for energy is increasing,
• the increasing weakness of the relative Fuzzy Entropy of the system is compensated, then over-compensated by the external increase in Fuzzy Entropy of the universe.

In this remarkable enclosure, becoming an individual of a species, a continuity of events is unfolding, which will astonish us still a long time; among them:

• the research of the energy sources,
• the adaptation to internal constraints and locally external, due in particular to confrontation or (and) co-operation with other individuals or species,
• the cellular division, the mitosis,
• the apoptose,
• the symbiosis,
• the mixing of genes by the sexual reproduction,
• the adaptation and mutations, in answer to the stress in an environment sometimes suddenly more difficult,
• the emergence of always more powerful biochemical catalysts, the ribozymes then the enzymes, increasing the genic repertory of a species vis-à-vis the environmental changes and also the chances of mutations; they accelerate mixing by:
• the fast maturation of the individuals of a given species (taking into account their size),
• the precociousness of their fertile period, which often begins before the maturity,
• the limited duration of the fertile period; assertion consolidated by the comparatively long reproduction duration of some species of social insects, of par with a very slow evolution.
• the links between some species, like the mutualistic (ref. 91), host-parasite or prey-predatory couples and their coevolutions .

The Life: its definition, which you were waiting impatiently, comes from the preceding ones naturally.

Thermodynamic singularities of low relative Fuzzy Entropy emerge in a usual way on Earth; those that interest us particularly appear:

• in a liquid or solid environment,
• under pressure conditions ranging  between ~0atm. (~6500m) and ~1000atm (~10 000m)
• under temperature conditions ranging between ~0°C and  ~150°C (?)
• in niches which allow the Fuzzy Entropy:
• to equip these singularities with complex molecules, randomly synthesized by the evolution,
• to submit these singularities to the filter of the Darwinian selection,
• if they succeed in surmounting the internal and external obstacles, to push them to self-reproduce and get almost identical individuals.

What exclude, quoted previously and inter alias, the rolling stone (the true one), the meteorite that lights a forest fire, and in the immediate future the microprocessor.

On the other hand, these limitations should frame all numerous species of unicellular animals like amoebas that invented the locomotion, a strong producer of Fuzzy Entropy. Will these limitations frame the bacteria and also the viruses, which still make debate because at the border of the alive world?

We conjecture that a singularity of low relative Fuzzy Entropy, limited by the preceding terrestrial conditions, is alive when its Factor of Efficiency Sf is higher than ~2. (See the 3^2nd curve of the power ratios final résumé)

Let us await a counterexample, probably long in coming.

N. B:

• Sf may sometimes be temporally suspended: virus, bacteria, spores, seeds, deep-freezing without irreversible damage of eggs, sperm, parts of alive beings or whole alive beings,
• researchers use Darwinian processes in the study of proteins; a new stage of the Evolution with exciting prospects (ref. 85) 

The Life Curve: Let us conjecture that a point represents any alive species close to the curve illustrated in the 6^2nd table of the final résumé, in Napierian log of Sf weighted by the existence duration of the species.

Will it be possible to carry on this curve the numerous species of bacteria and virus?

Remarks:

• Traditional thermodynamics is interested in the macroscopic parameters of the matter state, pressure, volume, temperature ...  With the trivial Factor of Efficiency Sf, number without dimension, we seek to disregard the preceding parameters, as traditional thermodynamics disregarded microscopic level. We don't have to question us on the state of thermodynamic balance or out-balance of a studied particle. The coarse quantified studies of the second part of the site are only outlines, even if they appear already instructive. Their results will be improved when managing to model the fluctuating parameters of the different components of a system far from equilibrium.

Let us forecast that we will have one day a model of Total Entropy, which will satisfy a significant majority of researchers.

• The two financial conjectures would justify the relation between Fuzzy Entropy and money, of terrorism and anti-terrorism. (see § "philosophical presuppositions - Present Prospects")
• If the preceding definitions must be clarified, revised or spiked, studies must continue, in the general interest.

Resume of tasks:

To give a mathematical satisfactory definition of:

Realized Fuzzy Entropy,

Kinetic Fuzzy Entropy,

Fuzzy Entropy Potential,

Total Fuzzy Entropy,

Efficiency Factor Sf.

To define hypothetic Fuzzy Entropy Fields.

To calculate a hypothetic Constant of Fuzzy Entropy.