Gas, Kinetic Theory of
### I. General Considerations
Since the 17th century, the term *gas* has designated a particular state of aggregation of matter in which its constituent particles (now called molecules) not only lack any mutual cohesion or semi-cohesion—such as that found in solids and liquids—but instead exhibit a marked tendency toward unrestricted expansion. Mechanics, having studied the laws of motion of solids, also examined those of liquids and gases based on their collective properties; similarly, physics, particularly regarding their thermal behavior, pursued this line of inquiry.
However, with the establishment of the atomic and molecular hypothesis in the first half of the 19th century, two major problems arose: the dynamic study of individual molecules and atoms, and the interpretation of the collective properties of bodies in their various states of aggregation, particularly gases, in relation to their molecular constitution. The first problem, now termed microscopic, could not be addressed at the time and remains one of the primary objectives of modern physics; the second, however, was significantly advanced, and in the case of gases, it not only yielded the theoretical results anticipated but also others that far exceed the narrow scope of its original formulation. For this reason, it now constitutes one of the most fascinating and instructive chapters of physics.
### II. Constitution of Gases
As early as 1738, D. Bernoulli considered gases as composed of an immense number of material corpuscles, heavy, widely separated relative to their dimensions, and each possessing a very high velocity. Based on this, he demonstrated that such a system must obey Boyle-Mariotte’s law. Today, Bernoulli’s corpuscles are replaced by molecules, whose vast number was determined by Loschmidt using Avogadro’s law, and it is assumed that each molecule is subject to the laws of mechanics. Clearly, a first series of gas properties must follow as a statistical consequence of this assumption. For simplicity, monoatomic molecules are considered, especially since it is known that, when limiting these considerations to the statistical aspect, transition to more complex cases presents no insurmountable difficulties.
Molecules are generally far apart but occasionally approach, collide, and rebound. However, at very small distances, they repel one another, so their collisions likely occur only between their so-called spheres of action (repulsive) when elastic. From this perspective, Joule (1848), Krönig (1856), and Clausius (1857) began to give more concrete form to Bernoulli’s ideas, followed by Maxwell, who in 1859 took the decisive step that may be considered the foundation not only of the kinetic theory of gases but also of statistical mechanics in the broadest sense.
### III. The Maxwellian Distribution of Velocities
Experience shows that a gas enclosed in a container and isolated from external influences, regardless of its initial distribution, rapidly assumes a stationary state. Pre-Maxwellian researchers believed that to interpret this stationary state, it sufficed to assume that the molecules possessed a certain average velocity, however constituted (even excluding the special case where all possessed that velocity as their own). Maxwell, however, recognized that to characterize this stationary state, it was not enough to assign an average velocity; it was also necessary that the number of molecules possessing each of the countless velocities present remain constant. This does not mean that every molecule must retain its velocity unchanged, but rather that the molecules losing a given velocity in a short time due to collisions are as numerous as those gaining it for the same reason.
Two important questions thus arose: 1) What was the particular distribution of velocities corresponding to the stationary state of the gas? 2) What was the primary reason why this velocity distribution was always rapidly achieved?
Maxwell, aided by intuitive analogies with V. Herschell’s law of errors, believed he had demonstrated that in the stationary state, the number of molecules with velocity components u, v, w (along Cartesian axes) is given by an expression of the form a e^(-b(u² + v² + w²)), where “a” and “b” are positive constants dependent on the physical state of the gas.
However, Maxwell himself recognized the flaws in his 1860 demonstration and in 1866 proposed a second, based on the collision law of elastic bodies. He deduced the statistical condition under which the number of molecules leaving a given velocity equals the number acquiring it during each infinitesimal time interval, and rediscovered the previous law, which from then on was called Maxwell’s distribution law.
In 1870, however, L. Boltzmann thoroughly examined Maxwell’s demonstration and first proved that the law is not only sufficient but necessary for the stationary state; then he sought—and believed he demonstrated—that, as experience confirms, if the stationary state of a gas is disturbed in any way, the gas, left to itself, spontaneously returns to the stationary state. Maxwell’s law would thus represent the limiting distribution toward which every other velocity distribution tends.
### IV. Loschmidt’s Objection and Boltzmann’s Probabilistic Interpretation
Loschmidt observed (1876) that the irreversible transition from any distribution to the Maxwellian one could not follow from the reversible laws of dynamics. Consequently, Boltzmann concluded that this transition was not strictly necessary at all times but only extremely probable. Thus, the stationarity of Maxwell’s law would not be a rigorous necessity but only a requirement of extreme probability. Among the infinite possible distributions, those of Maxwellian type represent the overwhelming majority; hence, even if the gas randomly passes through all possible states, it will practically always appear in the Maxwellian distribution.
Now that the necessity of resorting to probabilistic considerations was established, Maxwell’s law can be derived more directly from them. The conclusion, however, remains the same.
At this point, it is highly interesting to consider the extremely high probabilities involved, which are practically equivalent to the greatest certainties. Attempts have been made to illustrate this with intuitive examples, such as supposing a monkey strikes a typewriter and all letters and spaces are continuously recorded. What is the probability that the monkey will type a page of the *Divine Comedy*? Theoretically, this case cannot be excluded, though it is practically impossible. Yet the probability that a gas contained in a one-liter vessel will spontaneously assume a non-Maxwellian distribution for even one second is of the same order!
### V. Physical Implications
The realization that a physical law, hitherto considered certain and always verified, is instead an extremely probable but not certain law has led to revisions of various other physical notions and laws. Even classical physics now admits that many fundamental laws are only probable. *See the entry on THERMODYNAMICS.*