  The bright, nearby binary $\alpha$ Centauri provides an excellent laboratory
for testing stellar evolution models, as it is one of the few stellar systems
for which we have high-precision classical (mass, radius, luminosity) and
asteroseismic ($p$-mode) observations. Stellar models are created and fit to
the classical and seismic observations of both stars by allowing for the free
variation of convective mixing length parameter $\alpha_{\text{MLT}}$. This
system is modeled using five different sets of assumptions about the physics
governing the stellar models. There are 31 pairs of tracks (out of ${\sim}
150,000$ generated) which fit the classical, binary, and seismic observational
constraints of the system within $3\,\sigma$. Models with each tested choice of
input physics are found to be viable, but the optimal mixing lengths for Cen A
and Cen B remain the same regardless of the physical prescription. The optimal
mixing lengths are $\alpha_{\text{MLT,A}} /\alpha_{\odot}= 0.932$ and
$\alpha_{\text{MLT,B}}/\alpha_{\odot} = 1.095$. That Cen A and Cen B require
sub- and super-solar mixing lengths, respectively, to fit the observations is a
trend consistent with recent findings, such as in Kervella et al. (2017), Joyce
and Chaboyer (2018), and Viani et al. (2018). The optimal models find an age
for $\alpha$ Centauri of $5.3 \pm 0.3$ Gyr.
