The two-dimensional Fokker—Planck equation was derived as 
ap a a bP ap 
eS Mee Gp Bie Oy EE (12.70) 
a am az’ Daze 
where Z; =x, Z,; =x, and pis the transitional probability density function. 
The stationary solution of p is obtained by (dp/dt) = 0. As g(x, x) in Eq. 12.69, in 
its general form 
2(x, x) = yF(x, x) + G(x), (12.71) 
is used. 
Then the joint probability density function of swaying motion p(x, x) is the 
solution of Eq. 12.70, when (dp/dt) = 0, 
pO =i as|-Eo ae U(x) — dx (2772) 
yD, 
where 
U(x) = | G(A)dh (12.73) 
0 
is the potential energy for restoring forces, C is a normalization constant, 
and 
Q(y) = [2a (12.74) 
0 
Here 
1 
= J2v— 12.75 
Biv) TIA) | F(x) ¥2(v — U(x) dx ( ) 
R 
C(v) = as | vv—U(x) dx (12.76) 
R 
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