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Ex. Set # 11
1. A balance for measuring weight consists of a sensitive spring which hangs from a
fixed point. The spring constant is is K, i.e. the force opposing a length change x is
Kx. The balance is at a temperature T and gravity accelaration is g. A small mass
m hangs at the end of the spring.
a. Write the partition function and evaluate the average <x>. Use the virial theorem
x
H
  k B T to evaluate the fluctuation ( x   x ) 2  . What is the minimal mass
x
m which can be meaningfully measured?
b. Write a Langevin equation for x(t) with friction  and a random force A(t).
Assuming
 A(t ) A(0)  C (t ) show
that
 A( ) A( ')  2 C (   ')
and
~
~
evaluate the spectrum | x ( ) | 2  where x ( ) is the Fourier transform of
~
~
x (t )  x(t )   x. Evaluate [ x (t )] 2  and from (a) find the coefficient C. [you
may use  d /[( 2  K / m) 2   2 2 ]  m / K ].
c. Consider response to a force which couple to the velocity, i.e. the Langevin
equation acquires a term  F / t . Evaluate the dissipation function Im v ( ) ,
the power spectrum of the velocity  v ( ) and show that the fluctuation dissipation
theorem holds.
2. a) An electrical circuit has components with capacitance C and an inductance L.
Assume equilibrium at temperature T and find the fluctuations of the charge on the
capacitor <Q2> and of the current in the circuit <I2>.
b) The circuit has now also a resistance R and a voltage source V0 cos t with
frequency . Write a Langevin equation for this circuit and identify the response
function  Q ( )  Q  / V0 . Use this to write the energy dissipation rate.
c) Use the fluctuation dissipation relation to identify the fluctuations  Q 2 ( )  of the
circuit in (b). Evaluate  Q 2 (t )  and compare with (a).
Hint:
 (
2
d / 2
1

2 2
2 2
 0 )   
20
3. The discreteness of the electron charge e implies that the current is not uniform in
time and is a source of noise. Consider a vacuum tube in which electrons are
emitted from the negative electrode and flow to the positive electrode; the
probability of emitting any one electron is independent of when other electrons are
emitted. Suppose that the current meter has a response time . The average current
is <I> so that the number n of electrons during a measurement period is on average
<n> = <I>/e.
a) Show that the fluctuations in n are <n2> = <n>. (Hint: Divide  into
microscopic time intervals so that in each interval ni=0 or ni=1.)
b) Consider the meter response to be in the range 0<||<2/. Show that the
fluctuations in a frequency interval d are d<I2> = e<I>d/2. At what
frequencies does this noise dominate over the Johnson noise in the circuit?
4. Consider a damped harmonic oscillator with mass M whose coordinate x(t) satisfies
x   x   2 x  A(t )
where the power spectrum of A( t ), in the quantum case, is
A ( ) 

M
coth

2 k BT
a. Deduce the power spectrum of the velocity v ( ).
b. Show that for   0 v ( )   (  ) and that
1
1

< Mv 2 
 coth
.
2
4
2 k BT
Explain why is this the
expected result.
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c. Bonus: Evaluate < Mv 2  with   0.
2