Problem set # 3 for 5.72 (spring 2012)
Cao
I. A Gaussian variable with ⟨𝑥⟩ = 0 and ⟨𝑥 2 ⟩ = 𝑔 is described by the probability
function
𝑥2
1
− 2𝑔
𝑝(𝑥) =
𝑒
.
�2𝜋𝑔
∞
1) Calculate 𝐼 (𝑧) = ∫−∞ 𝑝(𝑥)𝑒 𝑧𝑥 𝑑𝑥.
2) Show that ⟨𝑥 2𝑛 ⟩ =
(2𝑛)!
2𝑛 𝑛!
𝑔𝑛 .
II. A Brownian particle in the potential of a constant force 𝐹 follows
𝑚𝑣̇ + 𝜁𝑣 = 𝐹 + 𝑓,
where 𝑓 is white noise with ⟨𝑓⟩ = 0 and ⟨𝑓(𝑡)𝑓(0)⟩ = 𝑔𝛿(𝑡).
1) Given the initial velocity 𝑣0 , show that
𝐹
⟨𝑣(𝑡)⟩ = 𝑣0 𝑒 −𝛾𝑡 + (1 − 𝑒 −𝛾𝑡 ),
𝜁
𝜁
where 𝛾 = .
𝑚
2) From the stationary condition ⟨𝑣 2 ⟩ − ⟨𝑣⟩2 =
𝐹 2
3) Show that 𝐶(𝑡) = ⟨𝑣(𝑡)𝑣(0)⟩ = � � +
𝜁
𝑘𝐵 𝑇
𝑚
𝑘𝐵 𝑇
𝑚
, prove 𝑔 = 2𝑘𝐵 𝑇𝜁 .
𝑒 −𝛾𝑡 .
4) Calculate the mean square displacement ⟨|𝑥(𝑡) − 𝑥(0)|2 ⟩.
III. *Chandraskhar’s theorem (see McQuarrie). White noise 𝑓 is a Gaussian variable with
𝑡
⟨𝑓⟩ = 0 and ⟨𝑓(𝑡1 )𝑓(𝑡2 )⟩ = 𝑔𝛿(𝑡1 − 𝑡2 ). If 𝐴 = ∫0 𝑎(𝜏)𝑓(𝜏)𝑑𝜏, the probability
distribution of 𝐴 is
𝑡
𝐴2
exp �−
𝑃(𝐴) =
�,
2𝛼(𝑡)
�2𝜋𝛼(𝑡)
where 𝛼(𝑡) = 𝑔 ∫0 𝑎2 (𝜏)𝑑𝜏.
1
(1)
1) Prove the theorem, i.e., Eq. (1)
2) For a Brownian particle, given the initial velocity 𝑣0 , find the probability
distribution of 𝑣 at time 𝑡, 𝑝(𝑣0 , 𝑣, 𝑡).
3) Show that the probability distribution of the displacement at time 𝑡, subject to the
initial condition 𝑥(0) = 𝑥0 and 𝑣(0) = 𝑣0 , is
2
𝑣0
−𝛾𝑡 )�
(1
�𝑥
−
𝑥
−
−
𝑒
0
1
𝛾
𝑝(𝑥0 , 𝑣0 ; 𝑥, 𝑡) =
exp �−
�,
2𝛼(𝑡)
�2𝜋𝛼(𝑡)
𝐷
with 𝛼(𝑡) = (2𝛾𝑡 − 3 − 𝑒 −2𝛾𝑡 + 4𝑒 −𝛾𝑡 ).
𝛾
1
IV. Zwanzig (J. Stat. Phys. 9, p 215, 1973) proposed the system-bath Hamiltonian
𝑝2
1
1
𝑐𝑖
𝐻=
+ 𝑉(𝑞) + � 𝑚𝑖 𝑥̇ 𝑖2 + 𝑚𝑖 𝜔𝑖2 �𝑥𝑖 −
𝑞�
2𝑀
2
2
𝑚𝑖 𝜔𝑖2
𝑖
where {𝑚𝑖 , 𝜔𝑖 , 𝑥𝑖 } define a set of bath harmonic modes.
1) Show that the reduced equation of motion for 𝑞 is
𝑡
𝑀𝑞̈ + � 𝜁(𝑡 − 𝜏)𝑞̇ (𝜏)𝑑𝜏 + ∇𝑉 (𝑞 ) = 𝑓(𝑡).
0
2) Write the explicit form for 𝑓(𝑡) as a function of 𝑥𝑖 (0) and 𝑥̇ 𝑖 (0).
3) Use the equilibrium conditions for 𝑥𝑖 (0) and 𝑥̇ 𝑖 (0) to prove
𝑐𝑖2
𝜁(𝑡) = 𝛽⟨𝑓(𝑡)𝑓(0)⟩ = �
cos(𝜔𝑖 𝑡).
𝑚𝑖 𝜔𝑖2
𝑖
2
2
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5.72 Statistical Mechanics
Spring 2012
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