Let's say I have two molecular dynamics (MD) simulators, A and B, which implement the following interface: $$ \left\{\vec{x}_i(t = 0), T_i, t \right\} \rightarrow \left\{\vec{x}_i(t), \vec{F}_i(t) \right\} $$ where $\vec{x}_i$ is the position of atom/ion $i$, $T_i$ is the type of atom/ion $i$ (e.g. hydrogen or carbon), and $\vec{F}_i$ is the force on atom/ion $i$. Given a set of times, they can produce discrete trajectories: $$ \mathcal{T} = \left\{ \left(t, \vec{x}_i(t), \vec{F}_i(t)\right) : t \in (dt) \mathbb{N} \right\} $$ Unless A and B are identical, there are going to be differences in the trajectories. Given a pair of trajectories, $(\mathcal{T}_A, \mathcal{T}_B)$, how should I decide if A and B are in agreement (for that particular trajectory)? or what would I have to do to convince you that $B$ was reproducing the behavior of $A$?
To give some context, B is an altered version of A, using a slightly different Hamiltonian, so they have identical interfaces. I understand that it may be useful to sample the neighborhood of initial conditions around $\vec{x}(t=0)$, but producing a trajectory is computationally intensive, so using a small number of trajectories is preferable. The regime is ab initio molecular dynamics using plane-wave DFT. I'm particularly interested in the motion of adatoms on surfaces.