The hypothesis of molecular chaos

The hypothesis of molecular chaos. Let us now return to the group of motions we discussed in Section 14c. 

At initial time tA this group of motions originates from all those phases which correspond to a given distribution of state Za, i.e., from that exact set to which statements (X) apply. It follows from (X) therefore that in the first time interval, i.e., from tA, to tA+At, the overwhelming majority of the motions under consideration will satisfy the Stosszahlansatz. 

At a later time Ib the paths of the group under consideration will have reached various r-points correspond- 

We therefore need a supplementary statement, which can be given as follows 116 

Only if we also accept statement (XI) can we justify the assertion that the overwhelming majority of motions under consideration satisfy the Sto88zahlan8atz not only in the first time interval from la to 4+At, but also in the subsequent ones. 

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The BE was found intuitively. Here, the hypothesis of molecular chaos, that is, the assumption that any pair of particles enters the collision process uncorrelated (statistical independence) is the most important one. Only this assumption allows the formulation of a closed equation for the single-particle distribution function. 

In this sense Jeans160 has made statement (1) above more precise. Statement (2), on the other hand, which, in our eyes, represents what Boltzmann actually meant by the hypothesis of molecular chaos, 161 is still awaiting a corresponding formulation. The following considerations are based mainly on the work of Jeans and attempt to establish a connection with the criticisms which Bur-bury162 has repeatedly made of the Stosszahlansatz. 

Although the formulation of the hypothesis of molecular chaos still contains many gaps, it certainly shows clearly that the Umkehreinwand and the Wiederkehrein-wand affect only the original formulation of the Stosszahl-amatz in fact, they prove its untenability. The improved statistical version of the Stosszahlamaiz, however, takes into account all those requirements arising from these objections.168... 

This last assumption about equal frequencies is often referred to as the hypothesis of molecular chaos. However, the same phrase is also applied to another, considerably deeper statement, which will be developed in Section 18c. The confusion of the two meanings has an important role in our discussion of the /-theorem. Hence it seemed mandatory to reserve the expression the hypothesis of molecular chaos in our discussion exclusively for the concept introduced in Section 18c. Cf. note 161. 

M. Planck, Acht Vorleaungen Uber theoret. Phyaik. (Leipzig, 1909), 3rd lecture. The "special physical hypothesis introduced by Planck to exclude the spontaneous occurrence of observable decreases in entropy (he calls it the hypothesis of "elementary disorder") consists of the following statement The number of collisions which take place in a real gas never deviates appreciably from the Stoaazahlanaatz (cf. Section 18). The hypothesis denoted in Section 18c as the "hypothesis of molecular chaos" would, on the other hand, permit such deviations. 

Further statistical development of the Stosszahlansatz Hypothesis of molecular chaos... 

We may wish, for example, to know the probability of finding a gas molecule at a definite spot in the box within which we suppose the gas to have been enclosed. If no external forces act on the molecules, we shall be unable to give any reason why a particle of gas should be at one place in the box rather than at another. Similarly, in this case there is no assignable reason why a particle of the gas should move in one direction rather than in another. We therefore introduce the following hypothesis, the principle of molecular chaos For the molecules of gas in a closed box, in the absence of external forces, all positions in the box and all directions of velocity are equally probable. 

The doctrine of molecular chaos, leading to the interpretation of entropy as probability, is in a somewhat different case again. It is based, though not upon direct experiment, upon the primary hypothesis of aU chemistry, that of the existence of molecules, and upon the assumption, common to most of physics, that these particles are in motion. It is related very closely to such facts of common observation as diffusion and evaporation, and it takes its place among the major theories about the nature of things. In scope and significance it is of a different order from rather colourless assertions about the geometry of lines and surfaces constructed with the variables of state. 

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