1. No category

Technische Universität Berlin
Institut für Mathematik
, Prof. Yuri Suris, Dr. Matteo Petrera
Mathematical Physics II
http://www3.math.tu-berlin.de/geometrie/Lehre/SS15/MP2/
SS 15
Exercise Sheet 2
(Kinetic Theory of Gases)
Due date:
04.05.15
60% of
the total sum of points. Each exercise sheet has 20 points.
Please work in xed 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
ˆ To get the Übungsschein (necessary condition for the oral exam) you need to collect
ˆ
ˆ
free to write your answers either in English or in German.
ˆ 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
dierence " > 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
N!
:
m)!m!
1. Give a combinatorial interpretation of Z (N; m).
2. Dene 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.
3. The absolute temperature of the system is dened by
@S
T :=
@E
!
1
;
where E is the total energy. Compute the inverse temperature := ( T ) 1 .
4. 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
5. 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) :
(1)
Here := ( T ) 1 , m is the mass of the particles and n := N=V is the number
of particles per unit volume.
1. Dene the Boltzmann functional by
H :=
and the entropy by
Z
R3
%0 (p) log %0 (p) dp;
S := V H:
Compute explicitly H and prove that
!
V 4 m E 3=2
3
S = N log
+ N;
N
3N
2
where E := (3NT )=2 is the internal energy.
2. Compute the average velocity of the particles, hkv ki%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 dened by the positive
number
2 Z
:=
kp p2k %0(p1) %0(p2) dp1 dp2:
m R6 1
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.
3. Compute explicitly .
4. Prove that the mean-free path of each particle, dened by
n
:= hkv ki%0 ;
2
is given by
1 1
= p
:
2 2 2 n
Note that does not depend on T !
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 dened by (x) := 0+1 tx 1 e t dt; x > 0: The
Euler -function satises 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 rst 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 modied Bessel function (see the hint below) which guarantees the correct normalization of
%(q; p).
1. Determine an approximated formula for %(q; p) in the limit U0 1.
2. Prove that the total internal energy,
E := N
is given by
3
2
*
kpk
2
2m
+ U (q )
+
%
;
I1 ( U0 )
;
I0 ( U0 )
where the functions I0 and I1 are modied 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 o.
E = N T
N U0
(Hint: The following integral representation and series expansion of the
modied Bessel functions of the rst 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 )!