TD2 Statistical Physics (M1)
Classical Harmonic Oscillator
Exercise 1: Slightly coupled oscillators
Let’s consider an assembly of N one-dimensional harmonic oscillators, with mass m and frequency .
The classical Hamiltonian can be written:
N
 p 2 m 2 2 
Hq i , p i     i 
q i 
2
i 1  2m

The goal proposed in this exercise is to study the assembly of N oscillators in the Microcanonical
ensemble with a total energy E. This corresponds to the situation of “slightly coupled oscillators”,
since each oscillator has its own frequency, but they are supposed to exchange energy by some way
to satisfy the conservation of the total energy E, but not of the energy of each oscillator separately.
Remember that the number of classical states within a total energy less than E is given by
E, N  
1
d
hN 
Where the integral is restricted to the phase space domain defined by Hq i , p i   E .
1)
Prove that
E, N  
N
1 2
  V2 N
hN  
 E
where VD R  is the volume of a sphere with radius R in D-dimension. Compute E, N .
2)
Deduce from 1) the expression of the microcanonical entropy SE, N  and of the
corresponding temperature TE, N .
Exercise 2: Test of the Ergodicity.
I.
Let’s first consider a single one-dimensional oscillator, with mass m, frequency , and energy
E. Its Hamiltonian can be written as:
p 2 m 2 2
Hx, p  

x
2m
2
We want to see if this isolated system can be described within the ergodicity hypothesis, or not.
I.1) Qualitative discussion.
Compare the trajectory of the oscillator in the phase space (x,p), as obtained from the equations of
motion, with the surface of energy E defined by Hx, p  E .
I.2) Quantitative discussion.
a) Compute the temporal averages x t and x 2
t
b) In order to test the ergodicity hypothesis, we have to compute the same averages, but now on a
statistical ensemble given by the probability density
 x, p  . Hx, p  E
corresponding to the trajectory in the phase space ( is a normalization constant).
This distribution just considers that every state with the energy E has the same probability
(microcanonical ensemble). And dx.dp. x, p is the number of microstates around the coordinates
(x,p) of the phase space. The statistical average x S is thus given by
x
S
  dx.dp.x. x , p 
For the calculations, we suggest to use the following change of variables: x, p   , J  with
x , J  
2J
. sin  and p , J   2mJ . cos 
m
with 0    2 and 0  J  
 Show that this change of variables yields to
dx.dp. Hx, p  E  d.dJ. J  E
that is  is equiprobable.
 Prove that    / 2
 Compute the statistical averages x S and x 2
S
. Is the ergodicity hypothesis valid?
II.
Determine the probability to find x between xo and xo+x for an oscillator with an energy
between Eo and Eo+E using:
II.1) the calculation of the area of the corresponding accessible domain in phase space and the
corresponding number of microstates.
II.2) the time spent by the particle between xo and xo+x and the period of motion.
III.
Let’s consider now an assembly of 2 oscillators with the total energy E. What conclusion can
you get from the comparison between the temporal trajectories of this system in the phase space,
and the surface of energy E?