  The diffusion of a particle in a crowded environment typically proceeds
through three regimes: for very short times the particle diffuses freely until
it collides with an obstacle for the first time, while for very long times
diffusion the motion is Fickian with a diffusion coefficient $D$ that depends
on the concentration and type of obstacles present in the system. For
intermediate times, the mean-square displacement of the particle often
increases approximately as $~t^\alpha$, with $\alpha<1$, typical of what is
generally called anomalous diffusion. However, it is not clear how one can
identify or choose a time or displacement interval that would give a reliable
estimate of $\alpha$. In this paper, we use two exact numerical approaches to
obtain diffusion data for a simple Lattice Monte Carlo model in both time
limits. This allows us to propose an objective definition of the transient
regime and a unique value for $\alpha$. Furthermore, our methodology directly
gives us the length scale over which the transient regime switches to the
steady-state regime. We test our proposed approach using several types of
obstacle systems, and we introduce the novel concept of excess diffusion
lengths. Finally, we show that the values of the parameters describing the
anomalous transient regime depend on the Monte Carlo moves used to describe the
dynamics of the particle, and we propose a new algorithm that correctly models
the short time diffusion of a particle on a lattice.
