#3 RHO_0 and <b**2> for partial PDFs incorrect ?
It seem rho_0 is calculated incorreclty for partial
PDFs. Also it seems <b**2> might be incorrect for
partial PDFs AND systems with multiple phases.</b**2>
Email from Jae-Ho Chung reporting this:
The number of pairs (on long-range average, not local)
is same as the number of partices, therefore, the
pair density is the same as number density, which is
rho_o. It is like 1 X rho_o, because each particle
equally sees rho_o around itself. HOWEVER, for partial
case, number density and pair density are somewhat
different, and it is the pair density which has to go
into the baseline expression. If you have A, B, .... n
different particles, ho_o = rho_A + rho_B + ... +
rho_n, each of which is NUMBER density. For a
self-distance correlation, say between A and A, the
particle A, which has concentration rho_A/rho_o, will
see the self-density rho_A around itself. Therefore,
(rho_A/rho_o)rho_A is the proper expression for the
partial PAIR density. On the other hand, for a non-self
correlations, say between A and B, the particle A,
which has the concentration rho_A/rho_o, will see
particles B, which has the number density rho_B, and
vise versa. Therefore, the partial pair density for A-B
is (rho_A/rho_o)rho_B + (rho_B/rho_o)rho_A =
2rho_Arho_B/rho_o. As a sanity
check, the summation of partial pair densities has to
be equal to the total pair densities. So,
(rho_A/rho_o)rho_A + (rho_B/rho_o)rho_B + ... +
(rho_n/rho_o)rho_n + 2rho_Arho_B/rho_o + ... +
2rho_Arho_n/rho_o
- 2rho_Brho_n/rho_o = (rho_A + rho_B + ... +
rho_n)2/rho_o = rho_o2/rho_o = rho_o. I think this is
correct. It will be straightforward to apply this into
the calculation regardless of the number of different
particles.To summarize, it is -4pi(rho_i)2/rho_o for
self correlations, and -4pi2rho_irho_j/rho_o for
non-self correlations. For a case including many
different particles, for instance differential pdf, I
guess it is best to calculate all separately and sum
them up. Hope this helps.
