md2011-01-note 6 - It is worth noting that we assumed here...

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I It is worth noting that we assumed here that after a long enough time ρ i ( d ) is independent of the initial state ( r N ( 0 ) , p N ( 0 ) ) I If this is true, we can improve on our statistics by running several MD simulations in parallel (f.ex. in a PC cluster) and average over the individual results: ρ i ( d ) = initial conditions lim t 1 t R t 0 d t 0 ρ i ( r ; r N ( 0 ) , p N ( 0 ) , t 0 ) number of initial conditions . (6) I As a limiting case, we consider averaging over all possible initial conditions for constant N , V and E . I In that case, we can introduce an integral: initial conditions f ( r N ( 0 ) , p N ( 0 )) number of initial conditions R E d r N d p N f ( r N ( 0 ) , p N ( 0 )) Ω ( N , V , E ) , (7) where f is an arbitraty function of the initial coordinates and Ω ( N , V , E ) = R E d r N d p N . Notes
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I This now corresponds to the classical “ensemble average”, which we denote by h . . . i to distinquish it from a time average (as above). I Changing the order of the averages, we can write: ρ i ( r ) = lim t 1 t Z d t 0 ρ i ( r ; r N ( 0 ) , r N ( 0 ) , t 0 ) NVE . (8) I
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