ABSTRACT
In the case of an ensemble average, the idea is to collect statistics over an extensive collection of system replicas. Then, the measures of microscopic (mechanical) properties are ensemble-averaged, i.e., calculated over the collection of replicas. The significance of stationary probability distributions can be traced back to Boltzmann and Maxwell. Boltzmann builds up on the concept of thermodynamic analogy introduced by Helmholtz to show the role of stationary probability distributions. Lastly, molecules may be considered completely rigid bodies. The most straightforward approach then consists in using a Taylor expansion and propagating the equations of motion according to this expansion. The Euler algorithm is a self-starting algorithm since one only needs the positions, velocities, and the forces at the beginning of the trajectory to provide accurate values for entire trajectory. Explicit reversible integrators can also be developed using operator factorization techniques.
