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Home Work Set No. 1 (Professor Sow-Hsin Chen) Spring Term 2005. Due March 7, 2005
1. This problem concerns calculations of analytical expressions for the self-intermediate scattering
function (ISF) of the test particle in a liquid which captures both the short-time and the long-time
behavior of the function correctly.
In the classical limit the self-ISF in the Gaussian approximation is given by
1 1
− Q2 ()∆X (t) 2 − Q2W (t)
iQ[]X (t)−X (0) iQ∆X (t) 2 2
Fs (Q,t) = e = e = e = e (1)
where the width function W(t) is defined by
2 t t t
W(t) =∆()X(t) = ∫ dt1 ∫ dt 2 Vx (t1)Vx (t 2) = 2d∫ t'()t− t' Vx (0)Vx (t') . (2)
0 0 0
(a) Show that Eq.(2) is valid when the velocity auto-correlation function Vx (t1)Vx (t2 ) depends
only on the time difference t’= t1- t2, such as in a steady state.
(b) Define the normalized velocity auto-correlation function
Vx (0)Vx (t) 2 2 k T
ψ(t) = , where V = V = B (3)
2 x 0
Vx M
and its Fourier transform (the density of states function)
∞∞
1 iωt 1
ψ(ω) = ∫ dte ψ(t) = ∫ dt cosωtψ(t). (4)
2π−∞π 0
Show that (by using Eq.(2) and Eq.(4)) the width function can be written as
∞
2 1− cosωt
W(t) = 4V0 ∫ dω 2 ψ(ω) . (5)
0 ω
(c) Show that if the velocity of the test particle satisfies the Langevin equation, then the normalized
velocity auto-correlation function is given by
t
−
ψ(t) = e τ. (6)
1
2
Calculate the width function W(t) and give its short-time and long-time limits. Relate the
relaxation time τ to the friction constant of the test particle in the liquid. Give also the density of
states function in this case.
(d) By examining long time behavior of the width function, derive the general relationships between
the diffusion constant D of the test particle and the normalized velocity auto-correlation function
ψ(t). Give also the relation between D and the density of states function ψ(ω).