Properties as Time Averages of Trajectories

If we start a system at some reasonable (i.e., low-energy) phase point, its energy-conserving evolution over time (i.e., its trajectory) seems likely to sample relevant regions of phase space. 

Certainly, this is the picture most of us have in our heads when it comes to the behavior of a real system. In that case, a reasonable way to compute a property average simply involves computing the value of the property periodically at times r, and assuming 

The ergodic hypothesis assumes Eq. (3.9) to be valid and independent of choice of to- It has been proven for a hard-sphere gas that Eqs. (3.5) and (3.9) are indeed equivalent (Ford 1973). No such proof is available for more realistic systems, but a large body of empirical evidence suggests that the ergodic hypothesis is valid in most molecular simulations. 

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