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Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Exercise 2 – Energy & Entropy (Microcanonical Ensemble)
1.
Paramagnetism
Find the equilibrium value at temperature 
of the fractional magnetization
M / Nm  2 s / N of the system of N spins, each of magnetic moment m in a magnetic
field B. The spin excess is s. Take the entropy as the logarithm of the multiplicity
g ( N , s)  g ( N , 0) exp  2s 2 / N  :
 ( s )  ln g ( N , 0)  2s 2 / N
for
s  N . Hint: Show that in this approximation  (U )   0 
U2
2m 2 B 2 N
with
 0  ln g ( N , 0) .
Further, show that
2.
1


U
, where U denotes <U>, the thermal average energy.
m B2 N
2
Quantum harmonic oscillator
a. Find the entropy of a set of N oscillators of frequency  as a function of the total
quantum number n. Use the multiplicity function g ( N , n) 
( N  n  1)!
(see Kittel,
n !( N  1)!
1.55) and use the Stirling approximation for large N: ln N !  N ln N  N
b. Let U denote the total energy n of the oscillators. Express the entropy as
 (U , N ) . Show that the total energy at temperature  is U 
N 
.
exp( /  )  1
This is the Planck result. We will derive it later using more powerful methods that do
not require the direct calculation of the multiplicity function
Ben-Gurion University of the Negev
Department of Physics
Thermodynamics & Statistical Mechanics 1
3.
‫גוריון בנגב‬-‫אוניברסיטת בן‬
‫המחלקה לפיסיקה‬
1 ‫תרמודינמיקה ומכניקה סטטיסטית‬
Additivity of entropy for two spin systems
Given two systems of N1 ≈ N2 = 1022 spins with multiplicity functions g1(N1,s1) and g2(N2,s-s1),
the product g1g2 as a function of s1 is relatively sharply peaked at s1 = ŝ1. For s1 = ŝ1 + 1012,
the product g1g2 is reduced by 10-174 from its peak value. Use the Gaussian approximation of
the multiplicity function.
a. Compute g1g2 / (g1g2)max for s1 = ŝ1 + 1011 and s = 0
b. For s = 1020, by what factor must you multiply (g1g2)max to make it equal to
 g ( N , s ) g ( N , s  s ) ? Give the factor to the nearest order of magnitude
1
1
1
2
2
1
s1
c. How large is the fractional error in the entropy when you ignore this factor?
4.
Lattice defects model
The model consists of N molecules organized in a lattice. n molecules out of the N may move
into one of M available interstitial sites. In the drawing the interstitial sites are the points in
which the lines intersect.
Every molecule that moves into one of the interstitial sites contributes  to the energy, so
that for n molecules that moved the energy is n .
a. What is the multiplicity function of the energy (calculate the number of possibilities
that n molecules have to change their locations)
b. Calculate the entropy and equilibrium temperature of a system with n defects
c. Calculate the most probable energy for the extreme cases    (   0 ) and
 0 ( )