Description
1 Spin system microstates
Consider an isolated system of spins in a magnetic field. A spin up particle has energy −µB
and a spin down particle µB. If there are five particles in the ensemble consider the following
microstates:
State A: ↑↑↑↓↓
State B: ↓↑↑↓↓
State C: ↓↓↑↑↑
State D: ↓↓↓↑↑
According to the fundamental postulates of statistical physics which of these states are guaranteed to occur with the same probability as each other? For which will the probabilities
probably be different? Explain your answer
2 Toy system microstates
A “toy” system consists of two interacting subsystems, A and B. Subsystem A has two
particles (labeled a1 and a2) and subsystem B has three particles (with individual particles
labeled similarly). Particles in either system can only have discrete energies. The state of
each particle is labeled by the number of energy units that it has. Macrostates for the pair
of systems are describe by the particle numbers and total energies for the systems. The
following are various microstates of the system.
Subsystem A Subsystem B
Particle a1 a2 b1 b1 b3
Microstate 1 4 1 1 1 1
Microstate 2 2 2 1 2 1
Microstate 3 3 2 1 1 2
Microstate 4 3 2 1 0 2
Microstate 5 0 5 1 1 1
a) Which microstates represent the same macrostate?
b) For which of these microstates is it possible to state with certainty that they will occur
with the same probability?
1
3 Stirling’s approximation
Consider the standard form of Stirling’s log approximation
ln n! ≈ n ln n − n +
1
2 ln (2πn).
A weaker version is
ln n! ≈ n ln n − n.
Check both versions of Stirling’s log approximation for n = 15 and n = 150. Comment on
their accuracy.
4 Interacting Einstein solids: analytical approximations
a) Use Stirling’s approximation to show that for an Einstein solid
Ω(N,q) ≈
!
1 + q
N − 1
“N−1 !
1 +
N − 1
q
“q
#
N + q − 1
2π(N − 1)q
≈
$
1 + q
N
%N !
1 +
N
q
“q
#
N + q
2πNq
whenever q % 1 and N % 1.
b) Assume that q % N. Using the Taylor series approximation,
$
1 +
x
n
%n
≈ ex
whenever n % 1, show that
Ω(N,q) ≈
$eq
N
%N 1
√
2πN .
c) Suppose that two Einstein solids, labeled A and B, interact. Show that the multiplicity
for the combined system is
Ω = κ q
NA
A q
NB
B
where κ is a term that does not depend on qA or qB.
d) Determine an expression for the entropy for the combined system.
e) If the system is isolated, then q = qA + qB is fixed. Use this to rewrite the entropy in
terms of qA only. Show that the entropy attains a maximum when
qA
qB
= NA
NB
.
Explain in words what this means for the way in which energy units are shared between
the two subsystems when they are in thermal equilibrium.
2
5 Interacting spin systems
Consider two spin systems, A and B, that interact. For each the magnetic field is such that
µB = 1. Let EA be the energy of A, NA be the number of particles in A with similar variables
representing B. Suppose that NA and NB are fixed and the total energy E = EA +EB is also
fixed.
a) Let NA+ be the number of particles in subsystem A with spin up. Show that
EA = NA − 2NA+.
b) Noting that the total energy for the system is fixed, explain how you would describe a
macrostate of the system. Which variables are necessary to describe the macrostate?
Are any of these variables redundant?
c) Show that the multiplicity of the macrostate is
Ω =
! NA
NA−EA
2
“! NB
NB−EA+E
2
”
.
Now suppose that NA = 4 and NB = 12 and initially EA = 4 and EB = 4. The systems are
allowed to interact.
d) List all possible macrostates for the composite system and the probabilities with which
each occurs.
e) Determine the equilibrium macrostate. Determine the energy per particle in each subsystem in this macrostate. In which direction did the energy flow as the systems reached
equilibrium?
f) Determine the mean energy for each subsystem.
6 Counting states: electron in a box
a) Consider an electron in a one-dimensional box with length L. Determine Γ(E) if E =
110 h2
8mL2 where m is the mass of the electron.
b) Consider an electron in a three-dimensional cubic box whose sides have length L. Determine Γ(E) if E = 15 h2
8mL2 where m is the mass of the electron.
7 Gould and Tobochnik, Statistical and Thermal Physics, 4.12, page 198. The OSP program
name is Particle in a Box. The points represent the actual number of states; the line plots the
expression of Eq. (4.39); this is the asymptotic relation. The data can be obtained via Views
→ Data Table.
8 Volumes of hyperspheres
The volume of a sphere of radius R in n dimensions is given by:
Vn(R) = 2πn/2
nΓ(n/2) Rn.
Verify that this is correct for n = 1, 2, and 3.
3
2020 HW 16 Q1
If the states have the same erergy ther the probabilities with which
they Occew ore the same For any microslate
こ Energy N+ (-MB) +N- (MB)
Thus for state A
EA = 3(-MB) +2MB =0 EA=-MB
Similarly EB=MB
Ec-MB
Eo= uB
Thus A,C will accur with the some probability as stach other
B,D 1
but A,B are probadly diffterent.
2020 HW 16 Q2
a) The EA EB combination must be the same
microstate EA EB E=EA+ER
S 3 8
Z
4 4
8
3 5 4
१
4 5 3 8
5
5 3 8
So microstates (1, 4,5 repesent the some macrostate.
The other two belong to distinct macrostates.
b) Only microstates with the same total energу. Тиль
T,2,4,5
occur withe the same probability
2020 HW16 Q3
Exact Stirling
U
% dif6
In n!
27.899 27.894
nun – + & In 2Tin
0.02% 25.621
nlan-n
150 605.020 605,010 0.00% 601.595
8.2%
0.6%
A
accwale to witlin 0.1% less accwate
reasonable for
n lorge
ercugli
a) N! = NNeN √ZTN
R(N,4)= (N1)= (Nag-1
2
(N+q-1) N+g-e-(N+9-1) √2T (N+q-1)”
qe- √2Tq (N-I)N.1@-(N-I) ([2TT(N-1)
N+a-11N-1
a NN+a-1
21g/N-1)
2N-1 N-119 +
q
NN+9-1
2T19 (N-1)
N
a
N
19
N
N (+) e
So R(NA) (NeN
с) R= Лал
NA eqA Then RA= NA VZITNA
1e9B NB
3
NB VZTNG
19 N+g
N + q 2 q
2020 HW 16 94
VZTN
.ealN
N VZTN
Sa
NA NA/e Ne NB VA 2T INANB NA NB
e NA/e NB
こ
NA NB
2TI VNANP, NA (NB qA 9s
let this be K.
NA Ng So R= KANA qS
d) S= knл = kink fink + NA lnqA+ Ne hge]
e) S = kink + KNA IngA + kN8 na-9A)
2020 HW 16 Q4.
se
=
KNA KNB 0
ƏQA gA 9-9A
NA NB NA N3 =0 =D =D GA NA こ
9A 9-95 gA 9B 9B NB
The ratio of erergies in.A vs B is the same as the ratio of oscillator
nuubers
2020 HW 16 Q5
a) EA= -MB NA+ +UB NA-
= – NA+ +NAThen NA-= NA-NA+ =D EA= NA-2NA+
b) We need EA,Es or NAI, Ns+. But since E= EAtE& is
fixed we could just use.
EITHER EA (or NA+)
OR EB (or Ng+)
c) Multiplicity of A is
NA
A ( NA
Multiplicity of B is
NB Ro Ne+
Multiphicity is
2 = 16 NA NB
NA+ NB+
Now NA+= NA-EA
Z
NB-EB NB+ = こ
Z
gives
NA NB S2= NA-EA NG-E+EA
2
NB-(E-EA)
Z
2020 HW16 Q5
We list these by NA+ ond Nos Note that
E= EA+EB =D 8 = NA-2NA+ + NB-2NB+
= 8 = 4+12-2(NA+ +NB+)
=D -8=-2 (NA+ +N B+)
NA-2NA+
=4-2NA+
NA+ N&+ EA
=7 4= NA+ +N&+
E = NB-2Ns 12-2NB+ RA Ле Prab
0 4 4 4 495 495 0.272
1 3 Z 6 4
220 880 0.484
2 2 0 8 6 66 396 0.218
3 1 -2 10 4
12 48 0.026
4 0 -4 12 0.000S
Σ= 1820
e) NA+= =D EA= 2
– erergy per pahcle= 2/4=0.5
No+ =3 EB=6 -0 erugy = 1=0.5
1) EA = ΣEΛprob(EA) = 2
ЁB= Σ Es prob (Es) = 6
2020 HW16 9ь
a) The stales cre labeled by n=1,2,3… with enegy En= h28ML2
So n= 1,2,3,4,, 10 all have ES 100 8M72
Thus r(E) = 10
The states cve labeled by nx,Ag te-1,2, with Rnergy
h² E= BmL² (nx²tny²+n)
Stales are
Ux ny E n multiples of
3
2 6
2 6
U 2 6
1 | 3
3
3
27
8M7
4 18
Axnynz E
I 4
18 122 9 =17
4
18 D 221 9
212 9
2 3
3 2
2 3
14
2 3
3 2
3 2
2 2 2 12
2
2 3 17
2 3 2
t1
3 2 2 17
Let N = acheal nunber
N₂= asymptoric nunber
Then we woat IN2-N ×100 1.
N
G+T Ch4
Prob 4.12
Data indicates this occus when R = 117 since heve
N, = 10859
N2= 10751
Phys 362
2020 HW 16 98
2TT /2
V=1 V = R’
1 (12)
(12) = T2 =D V = 2R
Thes is length of segmont -R R
0=2 V2 = 271 R² = TTR² = aea circle ☑
2 (1)
=1
n=3 V3 = 23/2 R3
3 Π(34)
= V3 = TR3 =D Volume spkere
