Time distribution of many small clocks in thermal motion

We shall make some assumptions. My knowledge of relativity theory is basic while that of quantum mechanics is negligible, so someone more knowledgeable about these topics can comment on how realistic the following assumptions are.

First, we shall take molecules of an ideal gas to be our clocks. Some periodic internal process within the molecule is supposed to act like a clock. We shall assume that whenever exchange of energy between molecules occurs due to a collision it manifests entirely as kinetic energy of the molecules involved. In what follows, whenever we speak of time we shall mean the time of an observer w.r.t. whom the mean motion of molecules is zero.

Let $g(s)$ be the probability density function such that $g(s)\delta s$ gives the probability that any given molecule travels for a distance lying in the interval $[s,s+\delta s]$ between consecutive collisions. Let $f(v)$ be the probability density function for molecular speed $v$. We shall assume that $f$ and $g$ are statistically independent. This means that knowledge that a molecule has speed $v$ does not alter the probability values for flight distance $s$ between collisions, and vice versa. Then the probability for a molecule having speed $v$ and flight distance $s$ (between consecutive collisions) is simply $f(v)g(s)$.

For a given speed $v$, a molecule in flight for a distance $s$ measures a proper time $\tau=(s/v)\sqrt{1-v^2}$, in $c=1$ units. For a given $v$, probability that the measured proper time of molecule is $\leq \tau$ is equal to the probability that $(s/v)\sqrt{1-v^2}\leq\tau$ i.e. $s\leq \tau v/\sqrt{1-v^2}$. Accounting for all possible values of $v$, the c.d.f. for $\tau$ is obtained: \begin{align} P(\tau)=\int_0^1dv~f(v)G(\tau v/\sqrt{1-v^2}) \end{align} where $G$ is the c.d.f. corresponding to the p.d.f. $g$. The p.d.f. for $\tau$ is: \begin{align} p(\tau)=\frac{dP}{d\tau}=\int_0^1dv~\frac{v}{\sqrt{1-v^2}}f(v)g(\tau v/\sqrt{1-v^2}) \end{align}

Using $p(\tau)$ we may calculate mean and variance of proper time $\tau$ if it exists: $\mu_\tau,\sigma^2_\tau$. This is for one collision. For $n$ collisions the total measured proper time of a molecule is $T_n=\tau_1+\tau_2+...+\tau_n$. If the variance $\sigma^2_\tau$ is finite, then assuming that $\tau_i$ are independent variables, by virtue of central limit theorem we have (for large $n$, which happens over a large enough observation time): \begin{align} z_n & \equiv \frac{T_n-n\mu_\tau}{\sqrt{n}\sigma_\tau} \\ \phi(z_n) & =\frac{1}{\sqrt{2\pi}}e^{-z_n^2/2} \end{align}

This shows that it is practically certain (for large $n$) that all the molecules will have measured the same proper time, equal to $n\mu_\tau$.

P.S. I could not find an expression for $g(s)$ in the links for kinetic theory of gases. Anybody knows?