3.3 Building the Fokker-Plank’s Matrices
29
Fig. 3.3 Periodic Potential: (a) V (x) =
Vo
2π
− sin
2πx
L
+
1
4 sin
4πx
L
. (b) V (x) =
Vo
2π
sin
2πx
L
−
1
4 sin
4πx
L
3.3.1 Periodic Potential Slightly Tilted
The sign + corresponds tilted to the right, Fig. 3.3b, correspondingly the sign − the
potential is tilted to the left, Fig. 3.3a.
V (x) = ±
V o
2π
sin
2πx
L
−
1
4
sin
4πx
L
(3.17)
dV (x)
dx
= ±
V o
L
cos
2πx
L
−
1
2
cos
4πx
L
(3.18)
d 2 V (x)
dx 2 = ±V 0
2π
L 2
− sin
2πx
L
+ sin
4πx
L
(3.19)
The corresponding dimensionless equations are
V ( ˜
x) = ±
1
2π
sin (2π ˜
x) −
1
4
sin (4π ˜
x)
(3.20)
d
V ( ˜
x)
d ˜
x
= ±
cos (2π ˜
x) −
1
2
cos (4π ˜
x)
(3.21)
d 2
V ( ˜
x)
d ˜
x 2 = ±2π
− sin (2π ˜
x) + sin (4π ˜
x)
(3.22)
The corresponding factor
V 0 = V 0 /K B T in the former dimensionless equations is
omitted because is considered explicitly in the Euler’s equations. From now on, we