複雜系統之簡介(梁鈞泰) - 中研院物理研究所- Academia Sinica

Let us do some statistics:
Although each walk looks random, after
many such walks, their averages are no
longer random:
∆x
X = x 1 + x 2 + x 3 +…+ x n
= n =0
X 2 = (x 1 + x 2 + x 3 +…+ x n ) (x 1 + x 2 + x 3 +…+ x n )
= x 1
2
+ x 2
2
+ x 3
2
+…+ x n
2
+ x 1 x 2 + x 1 x 3 +…+ x 1 x n
+ x 2 x 1 + x 2 x 3 +…+ x 2 x n +…
+ x n x 1 + x n x 2 +…+ x n x n-1
= + + +…+ + 0
= n = (∆x) 2 n
= [ (∆x) 2 /∆t ] (n ∆t)
= C t
where C≣ (∆x) 2 /∆t is called the diffusion constant