= Solution
A local <stellar relaxation time> depends on the <number density> $n$ of scatterers. Calling $N$ a total number, as the PDF does, leaves the displayed expression dimensionally wrong: $\sigma^3/(G^2m^2N)$ has units of time per volume. The intended local expression must use $n$, or $N/V$ for a homogeneous sample of <volume> $V$. We shall also make the normalization of the <stellar relaxation time> explicit, since an unspecified representative one-dimensional <velocity dispersion> does not fix an exact numerical coefficient.
In a <small-angle gravitational encounter>, approximate the relative trajectory by a straight line of <speed> $u$ and <impact parameter> $b$. The perpendicular acceleration of one <star> due to the other is $Gmb/(b^2+u^2t^2)^{3/2}$. Its impulse is therefore
$$
|\Delta v_\perp|=Gmb\int_{-\infty}^\infty\frac{dt}{(b^2+u^2t^2)^{3/2}}=\frac{2Gm}{bu}.
$$
A <star> encounters $2\pi b\,db\,nu$ scatterers per unit <time> in this interval of <impact parameters>. Independent deflections have random directions, so their <variances> add as in a <random walk>. Hence the accumulated squared <velocity> change has rate
$$
D_v\equiv\frac{d\langle|\Delta\mathbf v|^2\rangle}{dt}=\int_{b_{\min}}^{b_{\max}}\left(\frac{2Gm}{bu}\right)^2 2\pi bnu\,db=\frac{8\pi G^2m^2n}{u}\ln\Lambda,\qquad\Lambda=\frac{b_{\max}}{b_{\min}}.
$$
The <Coulomb logarithm in stellar dynamics> arises because each logarithmic interval of <impact parameter> contributes equally. The lower cutoff is approximately the strong-deflection scale, $b_{\min}\simeq G(2m)/u^2$ for equal masses, or the physical collision scale if larger. The upper cutoff is a scale on which the local density and straight-line encounter approximation cease to apply, such as the local disk thickness. Thus $\Lambda$ is dimensionless; it is not a number density or an arbitrary dimensional argument of a logarithm.
To display the normalization issue, use the representative relative <speed> $u=\sqrt2\sigma$ and define $T(\kappa)$ by $D_vT=\kappa\sigma^2$. This gives
$$
T(\kappa)=\frac{\kappa}{4\pi\sqrt2}\frac{\sigma^3}{nG^2m^2\ln\Lambda}.
$$
Choosing $\kappa=3/4$ recovers precisely the printed coefficient after repairing $N$ to $n$:
$$
\boxed{T_R^{\rm printed\ convention}=\frac{3}{16\pi\sqrt2}\frac{\sigma^3}{nG^2m^2\ln\Lambda}.}
$$
This is an explicitly defined convention, not a uniquely derivable normalization from the supplied data. If instead relaxation means a mean-square change equal to the full initial random-speed variance, $3\sigma^2$, the same encounter approximation gives $T_R=4T_R^{\rm printed\ convention}$. Averaging the encounter <speed> distribution instead of using one representative $u$ changes the coefficient again. The robust result is the scaling $T_R\propto\sigma^3/(nG^2m^2\ln\Lambda)$.
For a transparent solar-neighbourhood estimate, take $n=0.1\,{\rm pc}^{-3}$, $m=M_\odot$, $\sigma=30\,{\rm km\,s}^{-1}$ and $b_{\max}=300\,{\rm pc}$. With $G=4.3009\times10^{-3}\,{\rm pc}\,M_\odot^{-1}({\rm km\,s}^{-1})^2$, one obtains $b_{\min}=4.78\times10^{-6}\,{\rm pc}$, $\ln\Lambda=17.96$, and
$$
T_R^{\rm printed\ convention}\simeq3.35\times10^{13}\,{\rm yr},\qquad T_R^{\rm full\ variance}\simeq1.34\times10^{14}\,{\rm yr}.
$$
These illustrative parameters give an order $10^{13}$--$10^{14}$ year <stellar relaxation time>, much longer than a galactic lifetime of order $10^{10}$ years. The solar neighbourhood is consequently a <collisionless stellar system> with respect to encounters between individual <stars>. Scattering by massive clouds and collective structure is a different process; the estimate does not assert that every mechanism of disk heating is negligible.
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