t+Ar 2 
E| } {|-BX2(t)— F(X1(0))|Ar+ | fAidddr 
(2) d 
Dy = lim 
Ar—>0 At 
t+Ar 
= lim E}{-BX2—F(X))|’Ar+ 2{/- 8X2 - F(X))} | frat 
At—>0 oie Pe. 
SS & Ne 9 os ae 
t+Ar t+Ar 
+ lim E it | | Ft p)ft2)at dt 
At—>0O At 
t t 
Wo 
Ss 12.29 
5 ( ) 
Now, the process is stationary, so with Eq. 12.19, the Fokker—Planck equation becomes 
Wo ap a a 
eee aes (np) +5 [(BxX2+ F&XyIp] (12.30) 
This is called the stationary case for Kramer’s equation and has been solved by several 
scholars. Following Caughey—Wu, the solution of Eq. 12.30 is 
x) 
48 |x 
D(X1, X2) = p(x, x) =C exp -- ee | Feat k (12.31) 
0 
Here C is the coefficient for normalization and € is a dummy variable for x;. Equation 
12.31 is in the form of 
E 
p(x, x) =C exp, —48 -—}, (12.32) 
Wo 
where E is the total energy per unit mass of this system 
333 
