Umkehreinwand

Loschmidt s Umkehreinwand (1876). Denote by r, and IY two phases of the molecular motion for which 

The gas model is a conservative mechanical system. Therefore if it can perform the motion which has gone through the phases in the order of Eq. (11), it can also perform the motion in the order 

Hence for every motion of the model in which H decreases from Hi to H , there exists a motion in which, in a precisely opposite way, H increases from Hn to Z/i.G0 

During the motion from phase T to phase (rO, therefore, contrary to the formulation above of the //-theorem, the H-function has again assumed higher values. 

The monotonic decrease of the //-function (and as a consequence the irreversible approach of any distribution to the Maxwell-Boltzmann distribution) follows from the fact that in the calculations of the //-theorem the Stosszahlansate is used for every successive At, without exception. Therefore the Umkehreinwand and the Wieder-kehreinwand are mainly directed to this application of the Stosszahlansatz.61 

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This deeper formulation of the question was shelved in Boltzmann s investigations by the soon emerging formulation of the 17-theorem (1872). Boltzmann took up this problem again only to counter Loschmidt s Umkehreinwand (Section 7b) and to obtain a modified formulation of the 17-theorem. He tried to show that, if we consider a motion of unlimited duration, then the Maxwell-Boltzmann distribution very strongly dominates in time over all other distributions, and hence the tendency to approach this particular distribution is quite understandable. 

This point is essential for the understanding of Boltzmann s position with respect to the Umkehreinwand and the Wiederkehreinwand.,8 ... 

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... 

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