Übung 12.06.2015

Übungen zur LV Stochastische Prozesse in der Physik
Sebastian Rosmej / Reinhard Mahnke
16. Juni 2015
Übungsaufgaben Serie 5
(bitte zur Übung am 19.06.2015 die Ergebnisse mitbringen, auch zum
Vorrechnen und diskutieren an der Tafel)
Fokker-Planck Gleichung
Aufgabe 5:
Study of Fokker–Planck dynamics p(x, t) with known linear drift f (x) = −γ x
given by
∂p(x, t)
∂
σ 2 ∂ 2 p(x, t)
= − [f (x)p(x, t)] +
∂t
∂x
2 ∂x2
;
p(x, t = 0) = δ(x − x0 )
(1)
with natural boundary conditions.
Summary: What to do?
The task is to solve the one–dimensional Fokker–Planck equation
∂p(x, t)
∂
+
j(x, t) = 0
∂t
∂x
(2)
with flux j(x, t) including given drift f (x) = −dV (x)/dx and constant diffusion coefficient D
j(x, t) = −
dV (x)
∂p(x, t)
p(x, t) − D
dx
∂x
(3)
getting the probability density p(x, t) taking into account initial condition
p(x, t = 0) = δ(x − x0 ) and natural boundary conditions limx→±∞ j(x, t) = 0.
The result is
∞
ψ0 (x) X −λn t
p(x, t) =
e
ψn (x0 )ψn (x)
ψ0 (x0 ) n=0
1
(4)
where the eigenfunctions ψn (x) and eigenvalues λn are determined from the
eigenvalue equation
d2
−D 2 + VS (x) ψn (x) = λn ψn (x)
dx
(5)
with Schrödinger potential
"
1 d2 V (x)
1
VS (x) = −
−
2
2 dx
D
1 dV (x)
2 dx
2 #
(6)
The lowest eigenvalue is always zero (λ0 = 0) and the corresponding eigenfunction is related to the stationary solution via
exp (−V (x)/D)
pst (x) = ψ0 (x)2 = R +∞
dx exp (−V (x)/D)
−∞
(7)
Task:
Calculate Rthe probability density p(x, t) for the given harmonic potential
V (x) = − f (x)dx with f (x) = −γ x based on the solution of the eigenvalue
problem known from quantum mechanical harmonic oscillator.
2