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V. Historical Facts to Judge By

As we have seen above it is not at all unnatural to consider the existence of some classes of forces between the molecules of an ideal gas, compatible with the classical definition of the temperature, as we know it from Clausius and Kelvin’s times. It is also natural to conceive that the equation of state of an ideal gas contains a term completely independent of the temperature. This would mean that the temperature is exclusively related to the kinetic energy of the molecules of a gas and, moreover, the temperature is an expression of this kind of energy in any instance it is defined. This last conclusion seems to be the one indicated by the historical facts related to the problem at hand.

     Historically, when the definition of temperature has been extended to some other matter formations, it was considered that it must be correlated not with the kinetic energy, but with the whole mechanical energy of the particles thought to compose the matter. Thus, even if the particles of the ensemble which represents a portion of matter are not free to move like the molecules of an ideal gas, the temperature is still defined, and is taken to represent the total mechanical energy, kinetic and potential, of the constitutive particles of the piece of matter in question.

     The first ensemble which underwent a significantly detailed study under this idea of definition of the temperature was the ensemble of oscillators representing the blackbody radiation. Here the application of the definition of the temperature was facilitated by the so-called theorem of equal energy partition (equipartition) on the degrees of freedom of the constitutive components of light – the oscillators (Lavenda, 1992). A first conflict due to the discrepancy between theory and experiment popped up immediately. In order to straighten up the things without however abandoning the idea that the temperature is related to the whole mechanical energy of the oscillators, the statistics had to be improved (Planck, 1900). This statistics remained essentially that expressed by the distributions of exponential type, which are a general characteristic of Physics when it comes to statistical problems (Lavenda, 1992, passim), except that they were limited to a subclass having a special relationship between the energy estimator and the energy variance. This special relationship generated the first quantization in Planck’s form, a well known and largely publicized success.

     Inspired by this success, the men of science went further on and applied the philosophy to some other oscillator ensembles. This time the notorious example was the one of an ensemble of oscillators representing a solid material. Everything went just fine up until the moment when the Physics got in the realm of low temperatures. There it got stuck as it stumbled again upon the discrepancy between theory and experiment. One of the ways out of this problem was the proposal to accept as estimator of the energy of constitutive ensemble of the solid matter, a part which is independent of temperature. The idea took shape in the period 1915 – 1920, through works related to the specific heat of solids at low absolute temperatures (Compton, 1915). The argument was that in a solid, unlike in the ideal gas that inspired the statistical model helping in defining the temperature, some degrees of freedom become “agglomerated” so that they don’t contribute to the energy defining the temperature. In other words there is a part of mechanical energy of the component particles of a solid which has no contribution in the definition of temperature.

     While this conclusion appears as an ad hoc assumption, the truth is that it has a natural reason in the very classical definition of the temperature. As we have seen in the previous section, the classical definition of temperature can be taken as an indication of noncentral forces in the interaction between the molecules of an ideal gas. Moreover, if the interaction between molecules is represented by forces having a central component inversely proportional with the distance between molecules, then we have naturally in the estimator of energy of molecular ensemble a part independent of temperature. In view of this development, the theory of “agglomeration” appears as an assumption forced upon us by the natural course of reason: the forces of interaction are there, a theory must necessarily account for them. Otherwise the theory is bound to fail in situations where there are static pressures which cannot be kinetically explained. And the case of a solid plainly qualifies for such a situation. Thus, the failure of the quantum theory in the case of low temperatures can be viewed as a natural backlash of the very same forces casually overlooked in the initial definition of the temperature. However, the things didn’t go historically so smoothly. As it turns out the quantum theory had indeed something to say, above and beyond the classical theory.

     The quantum theory, which inspired the concept, could solve the dilemma within energetical approach, but only with the extra hypothesis of existence of a zero-point energy for the elementary oscillators of the ensemble representing the solid body. This energy must be proportional with the frequency of the oscillator (Milonni, Shih, 1991). At the beginning such an assumption appeared as a mere mathematical trick, especially designed to make the quantum and classical theories match one another. Suggestions have been advanced that an electromagnetic radiation might be responsible for this trick, thus giving it a physical boost (Einstein, Stern, 1913). However, because these suggestions couldn’t avoid some subtle statistical problems related, again, to the very definition of the temperature, the issue remained at the stage it was initiated about a half of century, when Timothy Boyer approached it. The Boyer’s moment of knowledge is a special one from many points of view, but for what we are interested right now it can be summarized to the following points.

     The main argument of the classical kinetic theory of gases, aside from calculating the pressure from surface stress considerations as done before, is a dynamical one, involving the law of conservation of the momentum. Boyer has shown that, from a phenomenological point of view, the momentum is not conserved in the processes of collision of the molecules with the walls of container, but only in the case of existence of an electromagnetic radiation entering each process of individual collision (Boyer, 1969). Indeed, one of the most important results of the Classical Electrodynamics is that any acceleration of a charged particle involves an emission of electromagnetic radiation. In the collisions we are talking about, enormous accelerations are bound to appear, and we have no reason to believe that the particles of an ideal gas are not electrically charged. Even if initially they are electrically neutral, the electric charge can be produced in the very short process of collision.

     This is actually the phenomenological argument to accept the existence of the zero-point radiation. In turn this radiation is physically responsible for the reason of existence of a part independent of temperature in the classical definition of temperature. Therefore in a place in which the forces should have been placed by their natural right so to speak, they have been first rejected forever. They cannot come back but only through the intermediary of invention of some other beings that may not have more right to existence than the forces that have been eliminated in the first place. As it happens however, the zero-point electromagnetic radiation has indeed rights to existence from a phenomenological point of view. But the Boyer’s reasoning ties up the electromagnetic radiation, or at least the part of it independent of temperature, to the forces which, a century before Boyer have been banished from existence.

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