Typical ways to introduce a random component are 1. The amplitude of these displacements must not be too large, in order to avoid overlap of atomic cores. A few percent of the lattice spacing is usually more than adequate. When doing this, the system will have a small total linear momentum, corresponding to a translational motion of the whole system. Since this is somewhat inconvenient to have, it is common practice to subtract this component from the velocity of each particle in order to operate in a zero total momentum condition.
Initial randomization is usually the only place where chance enters a molecular dynamics simulation.
The subsequent time evolution is completely deterministic.