t******y
发帖数: 239 | 1 n个小球随机撒进一个长度为L的一维盒子里，小球之间的最近邻距离的分布应当如何？
小球体积不计。
直观想象，这个分布应该在[0，L]区间，并且在0和L处为零；最大值在L/n处。求具体的
数学表达应该从何处着手？
预先多多谢过。 | s***e
发帖数: 911 | 2 你把这系统看成一维质点的分布问题. 假设前后质点间距离是b1,b2,...,bn(n-1个粒子),
并且给定最近邻居粒子间相互作用是:
U(b)=Infinity for b<=0, and =0 else.
那么这系统的分布是:
\rho(b1,..,bn)正比于Exp[-(U(b1)+...+U(bn)]*\Delta(L-(b1+b2+...+bn)),
最后一项保证这些粒子局限在[0,L]内.
现在你可以求任何量了,当然都涉及到最b1,b2,..,bn从零到无穷的积分.
你作这积分,需要把delta函数作富里页变换, 这样对b_i的积分们就可以被分开作了.
【在 t******y 的大作中提到】
: n个小球随机撒进一个长度为L的一维盒子里，小球之间的最近邻距离的分布应当如何？
: 小球体积不计。
: 直观想象，这个分布应该在[0，L]区间，并且在0和L处为零；最大值在L/n处。求具体的
: 数学表达应该从何处着手？
: 预先多多谢过。
| t******y
发帖数: 239 | 3
According to my discussion, if n=2 and L=1, the curve is linear:
p(x)=2-2x, indicating that the exponential relation may not be
intrinsic. If n=3, it becomes too complex for my brain...
I was supprised to learn from the Math board (I also posted there)
that this problem is indeed difficult. It is related to so called
"ordered statics" and has not been solved yet...
This problem should be self-similar, which means (n+1) ball case
should be related to the n-ball case. Since I know p(x) for n=2,
I | m****i
发帖数: 159 | 4 just a guess:
(L-2x)^(n-2)
then normalize it.
的
【在 t******y 的大作中提到】
: n个小球随机撒进一个长度为L的一维盒子里，小球之间的最近邻距离的分布应当如何？
: 小球体积不计。
: 直观想象，这个分布应该在[0，L]区间，并且在0和L处为零；最大值在L/n处。求具体的
: 数学表达应该从何处着手？
: 预先多多谢过。
| s***e
发帖数: 911 | 5 我考虑了一下直接计算几率的可能性. 从n=2开始:
\rho(x1,x2)=\rho(x2)*\rho(x1|x2),
其中\rho(x2)=1/L, \rho(x1|x2)=1/x2 for 0
distribution is normalized, and =1/(4L), =1/(2L), =1/(4L).
Everything seems reasonable so far.
The distribution of d=(x2-x1) is calculated as:
\rho(d)=Integrate[\rho(x1,x2)*\Delta(d-(x2-x1)),{x1,0,L},{x2,0,L}]
I did a rough calculation on this. Again you need to to the transformation
for the delta function. But it's very simple. You can use this calcl
【在 t******y 的大作中提到】
:
: According to my discussion, if n=2 and L=1, the curve is linear:
: p(x)=2-2x, indicating that the exponential relation may not be
: intrinsic. If n=3, it becomes too complex for my brain...
: I was supprised to learn from the Math board (I also posted there)
: that this problem is indeed difficult. It is related to so called
: "ordered statics" and has not been solved yet...
: This problem should be self-similar, which means (n+1) ball case
: should be related to the n-ball case. Since I know p(x) for n=2,
: I
| c*****l
发帖数: 135 | |
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