Technische Universität Berlin
Institut für Mathematik
Mathematical Physics II, Prof. Yuri Suris, Dr. Matteo Petrera
SS 17
http://www3.math.tu-berlin.de/geometrie/Lehre/SS17/MP2/
Exercise Sheet 2
(Kinetic theory of gases)
Due date: 11.05.17
• To get the Übungsschein (necessary condition for the oral exam) you need to collect 60% of
the total sum of points. Each exercise sheet has 20 points.
• Please work in fixed groups of 2 students.
• Please justify each step of your computations. Results without any explanation are not
accepted. Please write in a readable way. Unreadable handwriting will not be corrected. Feel
free to write your answers either in English or in German.
• Please turn in your homework directly to me (Matteo Petrera) at the beginning of the Tutorial.
No homework will be accepted after the deadline has passed.
(6 pts)
Exercise 1
Consider a closed and isolated system of N 1 distinguishable but independent identical particles. Each particle can exist in one of two states with energy
difference " > 0. Given that m particles are in the excited state, the total energy
of the system is m " with a degeneracy of
Z (N; m) := (N
• Give a combinatorial interpretation of
N!
:
m)!m!
Z (N; m).
• Define the entropy of the system by
S := log Z (N; m);
where is the Boltzmann constant. Determine S in the Stirling approximation.
From now on work in this approximation.
• The absolute temperature of the system is defined by
@S
T :=
@E
!
1
;
where E is the total energy. Compute the inverse temperature := ( T ) 1 .
• Find the density of excited states m=N as a function of . Use your result to
express the entropy as a function of .
Turn over
• Determine the entropy in the limit T
! 0.
(8 pts)
Exercise 2
Consider a free ideal gas at equilibrium described by the Maxwell distribution:
%0 (p) := n
2m
!3=2
2
e kpk =(2 m) :
Here := ( T ) 1 , m is the mass of the particles and n
of particles per unit volume.
(1)
:= N=V is the number
• Define the Boltzmann functional by
H :=
Z
R3
and the entropy by
%0 (p) log %0 (p) dp;
S := V H:
Compute explicitly H and prove that
!
3
V 4 m E 3=2
+ N;
S = N log
N
3N
2
where E
:= (3NT )=2 is the internal energy.
• Compute the average velocity of the particles,
hkvki%
0
:= hkpki%0 =m.
Let > 0 be the diameter of each particle. Consider pairs of colliding particles
with momenta p1 and p2 . Choosing a reference frame translating with one of the
particle, the frequency of collisions per unit volume is defined by the positive
number
Z
:=
2
m
R6
kp
1
p2 k %0 (p1 ) %0 (p2 ) dp1 dp2 :
Since every collision involves only two particles, the total number of collisions
to which a particle is subject per unit time can be found by dividing 2 by the
density n of particles.
• Compute explicitly .
• Prove that the mean-free path of each particle, defined by
:=
n
hkvki%0 ;
2
=
p1
is given by
Note that does not depend on T !
1
2 2 2 n
:
Turn over
(Hint: The following integral is useful:
Z +1
0
xn e
ax2
1
dx = a (n+1)=2
2
n + 1
2
a > 0; n 2 N ;
;
R
where is the Euler -function defined by (x) := 0+1 tx 1 e t dt; x > 0: The
Euler -function satisfies the identities
p (x + 1) = x (x); (n + 1) = n!:
Special values are: (n + 1=2) = (2n)! =(4n n!))
(6 pts)
Exercise 3
Consider an ideal gas in a box [0; L]3 at equilibrium at temperature T and subject
to an external conservative force whose potential energy is
U (q ) := U0 cos
2 ` q1
!
L
U0 > 0; ` 2 N:
;
Here q1 2 [0; L] is the first component of the position vector q . The gas is
described by a Boltzmann-Maxwell distribution of the form
%(q; p) := %0 (p)
e
U (q )
I0 ( U0 )
;
where %0 is the Maxwell distribution (1) and the function I0 is a modified Bessel function (see the hint below) which guarantees the correct normalization of
%(q; p).
• Determine an approximated formula for %(q; p) in the limit U0
• Prove that the total internal energy,
E := N
is given by
3
2
*
kpk
E = N T
2
2m
+ U (q )
N U0
+
%
1.
;
I1 ( U0 )
;
I0 ( U0 )
where the functions I0 and I1 are modified Bessel functions (see the hint below). Note that E is the total internal energy of a free ideal gas if the external
force is switched off.
(Hint: The following integral representation and series expansion of the
modified Bessel functions of the first kind are useful:
In (z ) =
with z
2 C; n 2 N)
1Z
0
ez cos cos(n ) d =
z n X (z 2 =4)j
;
2 j 0 j !(n + j )!