Answers to Physics 176 OneMinute Questionnaires
Lecture date: February 10, 2011
Are Einstein solids related to BoseEinstein condensates?
No, they are not. Einstein’s model of a solid addresses the vibrational prop
erties of a crystal, for example how vibrations of atoms in crystals contribute
to the heat capacity. A BoseEinstein condensate is a quantum state of mat
ter that occurs only at extremely low temperatures. The condensate arises
when a collection of identical massive bosons (a certain class of fundamental
particles that have integer spin, such as a gas of rubidium atoms) can be
described by a single “coherent” macroscopic wave function. We will discuss
briefly the properties of BoseEinstein condensates and why they occur later
this semester.
What is the purpose and motivation behind calculating
Ω
total
?
The goal of Section 2.3 of Schroeder and of the related discussion in class was
to explain why the multiplicity of some macroscopic object spontaneously
increases toward a maximum value, and why the transfer of energy (heat)
between subsystems is an irreversible probabilistic phenomenon.
The first step in the discussion was to calculate the multiplicity Ω
total
of a
thermally isolated Einstein solid that consists of two interacting macroscopic
subsystems A and B. Our discussion led to several insights. One was to label
a macrostate of the total solid by the amount of energy
q
A
in subsystem A,
which then determines the amount of energy
q
B
=
q

q
A
in subsystem B
since energy is conserved in the isolated total system.
A second important insight was that, generally, two subsystems interact
weakly with one another (they affect each other only for atoms near the
surface where they contact one another). This implies that the behavior of
subsystem A is statistically uncorrelated with the behavior of subsystem B
(at least on times short compared to a relaxation time) which in turn implies
that the total multiplicity is approximately equal to the product of the
multiplicities of the subsystems:
Ω
total
≈
Ω
A
×
Ω
B
.
(1)
This insight allows us to calculate Ω
total
for any specified amount of
energy
q
A
in subsystem A which led to a third insight as discussed in Sec
tion 2.3 of Schroeder: the total multiplicity Ω
total
(
q
A
) generally has a large
narrow peak as a function of
q
A
(especially if number of oscillators in each
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
subsystem is large).
This means that some macrostates have many more
accessible microstates than other macrostates and so, if all accessible mi
crostates are equally likely, certain macrostates are much more likely to be
observed than others. This explains why the multiplicity of a closed system
will “spontaneously” increase toward a maximum value, corresponding to
the macrostate with the most accessible microstates.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Behringer
 Physics, Energy, Entropy, Schroeder, total multiplicity, energy qA

Click to edit the document details