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.
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