# A question about the Boltzmann calculation in the reply to the Zermelo's recurrence objection

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.

• 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... – Yvan Velenik Feb 2 at 12:34