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I'm reading the Boltzmann "Reply to Zermelo's Remarks on the Theory of Heat" (Entgegnung auf die wärmetheoretischen Betrachtungen des Hrn. E. Zermelo, Annalen der Physik 57, pp. 773-84 (1896); English trans, Stephen Brush, Kinetic Theory, vol.2, p.218), which I found on the following website: https://www.informationphilosopher.com/solutions/scientists/boltzmann/zermelo.html. In the appendix of the article it is described a numerical calculation of the recurrence time for the molecules of air confined in a container of volume $1 \ \text{cm$^3$}$ in the hypothesis that the average distance between the centres of two neighbouring molecules is about $10^{-6} \ \text{cm}$. Boltzmann constructs around a single molecule a cube of edge $10^{-7} \ \text{cm}$. Here I summarize the other hypothesis:
i)the number of molecules in the container is $n = 10^{18}$
ii)Initially the velocity of a single molecule is $500 \ \text{m/s}$;
iii)Each molecule experiences $4 \cdot 10^{9}$ collisions per second and so the total number of collision per second is $b = \frac{4 \cdot 10^9 \cdot 10^{18}}{2} = 2 \cdot 10^{27}$;
iv) The velocity of a molecule is assumed to be the same as its initial state if its velocity components return to values that differ by no more than 1 metre (per second ?) from their original values.

Now I don't understand why Boltzmann assumes that the first molecule can have any speed $v$ with $0\leq v \leq a = 500 \cdot 10^9 \ \text{m/s}$ (if so it would exceed the speed of light! Is there an error in the translation?), the second any between zero and $\sqrt{a^2 - v_1^2}$ where $v_1$ is the velocity of the first molecule, the third $\sqrt{a^2 - v_1^2 - v_2^2}$ and so iteratively $v_n = \sqrt{a^2 - v_1^2 - \dots - v_{n-1}^2}$ and calculates the total number of possible combination of velocity to be:
$N = (4\pi)^{n-1} \int_{0}^{a} v_1^2 dv_1 \int_0^{\sqrt{a^2 - v_1^2}} v_2^2 dv_2 \dots \int_0^{\sqrt{a^2-v_1^2 \dots -v_{n-2}^2}}v_{n-1}^2 dv_{n-1} = \frac{\pi^{(3n-3)/2} \cdot a^{3(n-1)}}{2\cdot3\cdot 4\cdot \dots 3\frac{n-1}{2}}$ where $n$ is the total number of molecules. I think that this quantity is intended to be $N$ times the unitary volume in the space of velocity of dimensions ($n - 1$). Now he argues that this total number of combinations $N$ lasts an average time of $\frac{N}{b}$. I hope I haven't misunderstood the hypothesis.

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    $\begingroup$ I guess that the value of $a$ comes from assuming that all the kinetic energy of the trillion molecules, each having initially a velocity of $500$m/s, gets concentrated in the first molecule, which would indeed lead to $v_1 = \sqrt{10^{18}\cdot 500^2} = 500\cdot 10^9$m/s (note: I have used trillion $=10^{18}$, rather than $10^{12}$ as it is much more likely that Boltzmann used the long scale). As to your remark on the speed of light, I'd simply note that his paper dates from 1896, so nearly a decade before the special theory of relativity... $\endgroup$ – Yvan Velenik Feb 2 at 12:34

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