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I. Introduction

The fact that there are things uncontrollable by Man must have been one of the first ideas in the evolution of our social conscience. Scientifically though, it was the atomistic thought of matter structure that revealed for the first time a way to theorize the action upon uncontrollable matter. Cauchy was the one who saw, perhaps for the first time, the richness of mechanical equilibrium of forces when it comes to discuss it with respect to the ponderable matter. He realized that the static equilibrium is only apparent at a certain scale, and that underneath there are uncontrollable motions of molecules that add up to give what we observe at that scale. When speaking of motions we only use the mechanical model of what we imagine going on in the unseen part of the Universe. For, since the times of Cauchy, an idea has been developed according to which what is going on there is a deformation in general, which can be modeled as a sum of motions only in particular cases. This way the uncontrollable is scientifically reduced to one of its particular instances.

    The instance we are talking about is nowhere more evident than in Thermodynamics at the point of its connection with Statistical Mechanics. Here the story starts with the second law. Along the quest for the second law of Thermodynamics it has been revealed that there are things uncontrollable when it comes to describing the ponderable matter, regardless of its mechanical state, and that was the essential characteristic separating Thermodynamics from other scientific enterprises to date. Clausius was apparently the first to notice, with the clear intention to quantify the phenomenon, that from a certain amount of work put into a piece of material, only a part is transformed into heat, the rest being concealed by the material in internal processes, as internal energy (Clausius, 1852). Hints about this internal energy came even earlier from observations of the constancy of the temperature during the change of the state of matter, for instance from solid to liquid and further on to gas. In these changes of the state of aggregation heat that is pumped into the matter and the second law is all about the circumstances when heat and work are entering the processes related to matter.

In our opinion it is just a matter of convenience that, in building the grounds of Thermodynamics Clausius exercised special care to illustrate the issues with those bodies that are gases in their natural state, because in the processes related to gases the internal energy is recoverable in a cyclic transformation: there is thus no need for energy to change the state of aggregation, so that uncontrollability is thereby minimized. It seems to us however, that contemporary Thermodynamics still bears heavily the imprint of its inception point, the ideal gas. And in the kinetic theory of gases there are not too many things uncontrollable, at least at the experimental level. Then, as J. J. Thomson puts it, the second law of Thermodynamics tackles the problem of recovering of a part of internal (or intrinsic) energy, in the form of mechanical work (Thomson, 1887). The distinguished scholar goes on to show to what extent the principles of the Classical Mechanics could still be applied without relying upon the then newly born axiom of the second principle of Thermodynamics. However the fact is that the Thermodynamics came out as it did, i.e. as a phenomenological science, and attempts like J.J. Thomson’s were greatly discouraged in the aftermath. Every now and then, nonetheless, there appear some notable ones. Louis De Broglie’s ”Thérmodynamique de La Particule Isolée” is such an example (De Broglie, 1964). Explicitly or implicitly, De Broglie’s idea, like Thomson’s for that matter, centers about the action functional, therefore around the Hamiltonian formalism. In our opinion this fact does not serve best the purpose of the enterprise.

It seems to us that the Hamiltonian formalism is not an adequate approach of the subject, for it does not allow uncontrollable energy to come into play, but only at a statistical level. And, if it is to go beyond the achievements of the Classical Thermodynamics, the uncontrollability is the essential ingredient from the outset. This fact has been repeatedly stated, especially in connection with the old Cauchy’s approach of the Mechanics of Materials; we are not bringing up anything new by this statement. In this last field the problem of uncontrollability has been circumvented by the idea of continuum. However, Continuum Mechanics – the science in charge with the description of the behavior of materials – preserves the spirit of the Classical Mechanics, mostly regarding the concept of controllability. To be specific, a subjacent idea seems to exist that every motion, including pure continuum displacements, is controllable, even if the Man is not capable of actually implementing this control. The belief is that, given enough time, the Technology is capable of developing means of control at any level. This entertains the idea of ever increasing power of technological scrutiny of the Man, and there is no secret how disastrous the consequences are on occasions.

Here it is perhaps the moment to get into our notion of controllable. We understand it in the sense of restrainable. Sometimes in Physics, or even Engineering, it seems sufficient to just measure a quantity in order to declare it under control. Not here: we think control in the sense that we are able to keep the values of a certain physical quantity within certain specified limits. For instance, the Uncertainty Principle is, first and foremost, a principle of control exercised through complementary variables; it does not refer only to measurements, in spite of the fact that it is always discussed from that point of view. And while we are on this example it is worth noticing that Classical Mechanics, by accepting the fact that the quantities can be unconditionally measured, entertained in the background the idea of complete controllability. Naturally, the Continuum Mechanics inherited this idea. And yet this is the only field that seems to be in dire need of recognizance of the difference between controllable and measurable. The point is that if a quantity is measurable it is not necessarily controllable.

The present essay is about a case of positively admitting our impossibility of control at all structural levels. It is a kind of show … showing that rather than being poor in consequences, this principle of impossibility, when applied with the right line of reasoning, reveals what the classical descriptions miss about their concepts, and then something more.

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