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发帖数: 53 | 1 【 以下文字转载自 Mathematics 讨论区 】
发信人: joep (joep), 信区: Mathematics
标 题: Re: statistics question
发信站: BBS 未名空间站 (Sat Jan 4 11:57:55 2014, 美东)
This sounds like a physics-inspired problem, so let's try a physicist's
approach
1. define a 2-body interaction energy function E(r) -- to achieve the
desired 2-body distance pdf P(d), set $E(r)=-(\ln P(d))/\beta$. See 3 below.
2. define that there is no direct 3-body, 4-body etc. interaction. Hence the
system of N particles total energy is $H=\sum_{i\neq j} E_{ij}(r_{ij})$
3. define the physical system's distribution to be Boltzman distribution,
namely the probability the system is in state is: $exp^{-\beta H}$. Note
this is the JOINT pdf of the 2N coordinates.
4. Once you know the joint-pdf of a multi-random-variable system, there
should be well-defined ways to simulate them from a uniform distribution? (
for a single variable, one can use cdf-inversion method, etc. etc. For multi
-variable...look it up a bit...)
5. Once you have simulated the above joint-distribution, you can numerically
compute the 2-body distribution Q(d) for any 2 spheres.
6. Now, because of INdirect 3-body, 4-body interactions coming from $E(r_{12
})+E({r_23})+...$, you will find Q(d) found in 5 is not identical to the
target P(d). However, Q(d) should be similar to P(d).
7. Now repeat 1 to 6, treating it as an iterative procedure, where you add
corrections to E(r) in 1 so that Q(d) finally converges to P(d).
Technicality: how to add corrections to E(r)? You might want expand it using
a set of orthonormal basis functions (may want to restrict the N spheres to
lie with a box of size L by L; and then use sine(x) , cosine(x) as the
basis function). Then the problem of iteration in functional space becomes
iteration in the expansion coefficients. | s*****u
发帖数: 164 | |
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