Problem Set #2 for 5.72 (spring 2012)
I. A diffusive oscillator satisfies the Fokker-Planck equation
𝜕𝑝
𝜕2
𝜕
= 𝐷 2 𝑝 + 𝛾 (𝑥𝑝),
𝜕𝑡
𝜕𝑥
𝜕𝑥
with 𝛾 = 𝐷𝑚𝜔2 𝛽. 1) Find the equilibrium distribution.
2) Show 𝑥̅ = 𝑥0 𝑒 −𝛾𝑡 . 3) Assume that 𝑝 takes the form of
(𝑥−𝑥0 𝑒 −𝛾𝑡 )2
1
−
2𝛼(𝑡)
𝑝(𝑥0 , 𝑥, 𝑡) =
𝑒
.
2𝜋𝛼(𝑡)
Find 𝛼(𝑡) explicitly. 4) Confirm that solution (2) satisfies equation (1).
Cao
(1)
(2)
II. Consider one-dimensional diffusion under the action of a constant force 𝐹.
1) Write the Fokker-Planck equation for the diffusive particle and show 𝜁𝑣̅ = 𝐹.
2) Show that the solution to the F-P equation is
(𝑥 − 𝑥0 − 𝐷𝛽𝐹𝑡)2
1
exp −
.
𝑝(𝑥0 , 𝑥, 𝑡) =
4𝐷𝑡
√4𝜋𝐷𝑡
3) Show that the solution satisfies the stationary condition
𝑝𝑒𝑞 (𝑥0 )𝑝(𝑥0 , 𝑥, 𝑡)𝑑𝑥0 = 𝑝𝑒𝑞 (𝑥),
where 𝑝𝑒𝑞 (𝑥) = 𝑒 𝛽𝐹𝑥 is the equilibrium distribution.
III. *Define the Fourier transform as 𝑝(𝑘, 𝑡) = ∫ 𝑒 −𝑖𝑘𝑥 𝑝(𝑥, 𝑥0 , 𝑡)𝑑𝑥.
1) Show that the F-P equation for the diffusive oscillator can be written as
𝜕𝑝
𝜕
= −𝐷𝑘 2 𝑝 − 𝛾𝑘
𝑝.
𝜕𝑡
𝜕𝑘
2) Solve for 𝑝(𝑘, 𝑡) with the initial condition 𝑝(𝑘, 𝑡 = 0) = 𝑒 −𝑖𝑘𝑥0 . 3) Give 𝑝(𝑥0 , 𝑥, 𝑡) explicitly.
IV. Diffusive oscillator with a sink at 𝑥𝑠 = 0.
1) Solve for 𝑝(𝑥0 , 𝑥, 𝑡) subject to the initial condition 𝑝(𝑥0 , 𝑥, 0) = 𝛿(𝑥0 − 𝑥) and
the absorbing boundary condition 𝑝(𝑥0 , 𝑥𝑠 , 𝑡) = 0. 2) Find the mean first passage time from 𝑥0 to 𝑥𝑠 . MIT OpenCourseWare
http://ocw.mit.edu
5.72 Statistical Mechanics
Spring 2012
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