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# Henri Poincaré, Quanta hypothesis: Thermodynamics and probability

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Let us go back to the kinetic theory of gases; the gases are formed of molecules which circulate in all directions with great velocities; their trajectories would be rectilinear if from time to time they did not collide each other, or if they did not collide the walls of the vase. The chances of these collisions end by establishing a certain mean distribution of velocities, whether we consider their direction or their size; this average distribution tends to recover on its own as soon as it is disturbed; so that, in spite of the inextricable complication of the movements, the observer who can only see the averages perceives only very simple laws which are the effect of the game of probabilities and large numbers. He observes the statistical equilibrium. It is for example that the speeds will be equally distributed in all directions, because if they stopped a moment of being, if they tended to take a common direction, the collisions in a very short time would have made them to lose.

Calculation leads to another consequence; the average force that each molecule will take on average is proportional to the number of its degrees of freedom; let me explain ; a body can take a number of very small, different movements; for example, a material point can move along the three axes, it has three degrees of freedom; a sphere can undergo a translation parallel to each of the three axes, or a rotation around these three axes, it has six degrees of freedom. Now, a molecule is not a simple material point, it is susceptible to deformation, it will therefore have several degrees of freedom; for example, one molecule of argon will have 3, one oxygen molecule will have 5. So, according to the law that we enunciate and which is called the equipartition law, if in the statistical equilibrium an argon molecule possesses at a certain temperature the living force 3, an oxygen molecule must possess the living force 5; in other words, the constant-volume molecular specific heats of argon and oxygen will have to be between them as 3 to 5.

And this law, properly interpreted, is not only true for gases; it results from the very form that has always been attributed to the equations of Dynamics and which is the Hamilton form. If the general laws of Dynamics are applicable to liquids and solids, these bodies must obey the equipartition law, mutatis mutandis.

The principle of Carnot, or the second principle of Thermodynamics, teaches us that the world is moving towards a final state from which it can no longer deviate; he thus teaches us that statistical equilibrium is possible; if it were not, we could always find some artifice allowing to realize what one called the perpetual movement of second kind, allowing for example to heat a steam engine with ice, taking advantage of what this ice, however cold it may be, is not at absolute zero, and therefore contains a certain quantity of heat. If the laws of statistical equilibrium were not the same when we put in contact the bodies A and B, or the bodies B and C, or finally the bodies C and A, it would be easy, by bringing together sometimes two of these bodies, sometimes two others, of constantly changing the conditions of this equilibrium; these bodies would never know the definitive repose, and there would be no true statistical equilibrium; Carnot’s principle would be false.

By what singular coincidence are the conditions of this equilibrium always the same, whatever the bodies put together? the foregoing considerations make us understand it because the general laws of Dynamics, expressed by Hamilton’s differential equations, apply to all bodies.

These conceptions had hitherto always been confirmed by experience, and the verifications are now numerous enough that they can not be attributed to chance. It will be necessary, therefore, if new experiments bring out exceptions, not to abandon the theory, but to modify it, to enlarge it so as to allow it to embrace new facts.

It is not that certain objections have been presented to all minds from the first day. Molecules, atoms themselves, are not material points; if they have dimensions, is it permissible to equate them with absolutely rigid bodies; or whatever simple would be the molecule of argon, it can not be a mathematical point, it will be a sphere; why can this sphere not turn, and if it turns, it will be 6 degrees of freedom instead of 3. (It would be pointless to say that the ratio of specific heats would not be changed if one gave 6 degrees of freedom to argon and 10 to oxygen. It is well 3 degrees of freedom and not 6 that requires the kinetic theory of gases based on the virial theorem.) Unless it is not supposed that collisions, capable of modifying the translation of the molecule, are absolutely without influence on its rotation; that they cannot subject this molecule to the slightest deformation, etc. Moreover, each line of the spectrum corresponds to a degree of freedom. Needless to say, the oxygen spectrum has more than 5 lines. Why do certain degrees of freedom seem to play no role; why are they, so to speak, stiff as long as no mysterious circumstances intervene?