The equivalence between the canonical and microcanonical ensembles
when applied to large systems
V. Gurarie
Citation: Am. J. Phys. 75, 747 (2007); doi: 10.1119/1.2739571
View online: http://dx.doi.org/10.1119/1.2739571
View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v75/i8
Published by the American Association of Physics Teachers
Related Articles
An Introduction to Statistical Mechanics and Thermodynamics.
Am. J. Phys. 81, 798 (2013)
Using computation to teach the properties of the van der Waals fluid
Am. J. Phys. 81, 776 (2013)
Stefan–Boltzmann law for the tungsten filament of a light bulb: Revisiting the experiment
Am. J. Phys. 81, 512 (2013)
University student and K-12 teacher reasoning about the basic tenets of kinetic-molecular theory, Part I: Volume
of an ideal gas
Am. J. Phys. 81, 303 (2013)
Predicting the future from the past: An old problem from a modern perspective
Am. J. Phys. 80, 1001 (2012)
Additional information on Am. J. Phys.
Journal Homepage: http://ajp.aapt.org/
Journal Information: http://ajp.aapt.org/about/about_the_journal
Top downloads: http://ajp.aapt.org/most_downloaded
Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html
Downloaded 09 Oct 2013 to 130.239.76.10. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
The equivalence between the canonical and microcanonical ensembles
when applied to large systems
V. Gurariea兲
Department of Physics, University of Colorado, Boulder, Colorado 80309
共Received 4 October 2006; accepted 20 April 2007兲
A straightforward technique is suggested that demonstrates that a microcanonical ensemble and
canonical ensemble behave in exactly the same way in the thermodynamic limit. The canonical
distribution is derived for a closed system, without the need to introduce a large reservoir that
exchanges energy with the system. The derivation also clarifies the issue of the energy interval
which arises when introducing the microcanonical ensemble. © 2007 American Association of Physics
Teachers.
关DOI: 10.1119/1.2739571兴
I. INTRODUCTION
When teaching statistical mechanics at either the undergraduate or graduate level, an educator invariably encounters
two rather significant gaps in the logic of the subject. The
larger of the two gaps concerns the introduction of the microcanonical ensemble and the definition of entropy. The microcanonical ensemble refers to a thermally insulated system
whose energy E does not change in time. Introductory textbooks on statistical physics usually begin by defining the
entropy as the logarithm of the statistical weight ⌫共E兲,1
S共E兲 = kB ln ⌫共E兲,
共1兲
where ⌫共E兲 is the number of microstates 共energy eigenstates兲
that correspond to the energy E and kB is Boltzmann’s constant. Once the entropy is defined as in Eq. 共1兲, the temperature can be found according to
1 ⳵ S共E兲
=
.
T
⳵E
共2兲
Subsequently we can invert Eq. 共2兲 to find the energy E as a
function of temperature T. The heat capacity CV = ⳵E / ⳵T,
Helmholtz free-energy F = E − TS, and other relevant thermodynamic quantities can then be easily calculated.
Equation 共2兲 can be used to obtain the thermodynamic
properties of, for example, a collection of harmonic oscillators. The energy of M identical harmonic oscillators of frequency ␻ is equal to
冉 冊
M
M
1
M
E = 兺 ប␻ ni +
= ប␻ + ប␻ 兺 ni ,
2
2
i=1
i=1
共3兲
where ni are the non-negative integers which characterize the
microstates of the oscillators. The problem of finding ⌫共E兲
amounts to finding all such distinct sets of non-negative integers ni, i = 1 , . . . , M, such that Eq. 共3兲 is satisfied. The result
can be found in closed form, as long as the energy takes one
of the values defined by
E = ប␻
M
+ ប␻K,
2
共4兲
where K is an arbitrary non-negative integer. If E does not
take one of the values given in Eq. 共4兲, then Eq. 共3兲 cannot
be satisfied. Although this condition selects special values of
energy E such that ⌫共E兲 is nonzero, in the limit of a very
large number of oscillators E ប␻ and E can effectively be
747
Am. J. Phys. 75 共8兲, August 2007
http://aapt.org/ajp
considered a continuous variable ignoring the fact that
S共E兲 = kB ln ⌫共E兲 is not well-defined outside the discrete set
of values given in Eq. 共4兲. The entropy and heat capacity of
a system of harmonic oscillators can thus be found, as discussed in many textbooks.2
However, in most realistic examples the energy levels En
of the system are not degenerate, or En ⫽ Em if n ⫽ m. Thus
⌫共E兲, defined as the number of energy levels whose energy is
exactly E, at most equals unity if E = En for some n, and
otherwise ⌫共E兲 = 0. The example most commonly discussed
following that of a collection of harmonic oscillators is an
ideal gas. The energy of an ideal gas consisting of N point
particles of mass m in a box of linear size L can be written in
the form
3N
ប 2␲ 2
E=
兺 n2 ,
2mL2 i=1 i
共5兲
where ni are a set of 3N positive integers characterizing the
state. For a generic value of energy E, there is at most one set
of these integers such that Eq. 共5兲 is satisfied.1 Taken literally, we would conclude that the entropy of the ideal gas is
equal to either 0 or minus infinity, which obviously does not
make sense.
Most textbooks declare at this point that instead of counting the number of energy levels whose energy is exactly E,
we should count the number of energy levels whose energy
lies in the interval between E and E + ⌬E. More precisely, it
is suggested the function ⍀共E兲 be introduced, which gives
the total number of energy levels such that En ⬍ E. Then, we
define
⌫共E兲 ⬅ ⍀共E + ⌬E兲 − ⍀共E兲 ⬇
⳵ ⍀共E兲
⌬E.
⳵E
共6兲
Here, ⌬E is argued to be some interval of energy whose
precise value is not important. By calculating the entropy
according to Eq. 共1兲, we find
冉
S = kB ln
冊
⳵ ⍀共E兲
+ kB ln共⌬E兲.
⳵E
共7兲
It is then shown that the first term on the right-hand side of
Eq. 共7兲, which can be calculated for an ideal gas, grows with
the size of the system, while the second term, by assumption,
remains the same. Thus, if the system is large, the following
relation is valid:
© 2007 American Association of Physics Teachers
Downloaded 09 Oct 2013 to 130.239.76.10. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
747
冉
S = kB ln
冊
⳵ ⍀共E兲
,
⳵E
共8兲
which is proposed to be used for practical applications of Eq.
共1兲 and which is now independent of ⌬E 共which we did not
know anyway兲. Equation 共8兲 is now poorly defined, because
it involves a logarithm of a dimensionful quantity, but some
relatively obscure system-dependent manipulations can fix
even that problem.1,2
The route that we have outlined is the one usually taken
when introducing the notion of entropy and temperature. Yet
it leaves many questions unanswered. First of all, why did
we, after successfully employing ⌫共E兲 to calculate the thermodynamics of a collection of harmonic oscillators, suddenly change gears and introduce a completely new way of
calculating the entropy in Eq. 共8兲? The answer “because it
works” cannot be termed satisfactory. Second, what is the
meaning of ⌬E? The answer, “it does not matter in the thermodynamic 共large system size兲 limit” is not satisfactory either. After all, entropy should be a uniquely defined quantity,
and defining it up to 共albeit small兲 arbitrary additive terms
kB ln ⌬E leaves the feeling that something is missing. For
example, in later applications of the notion of entropy,
Nernst’s theorem 共the third law of thermodynamics兲 is discussed, which states in one of its formulations that the entropy goes to zero when the temperature approaches absolute
zero. But, how can we talk about entropy being zero if it is
defined up to arbitrary, although small, terms?
One possible answer put forward by more advanced books
on statistical physics,3 emphasizes the fact that no statistical
system in the real world is truly isolated. It always interacts
with the environment, with the result being that its energy is
fluctuating, not only because of the random statistical energy
exchange, but also because any quantum-mechanical system
which is not truly insulated will experience a broadening of
its energy levels. Thus, ⌬E at least represents this broadening. Because the typical size of this broadening for a macroscopic body can be shown to be much bigger than the typical
energy level spacing, it includes many energy levels. Yet this
explanation raises more questions than it answers. Does the
answer for the entropy depend on the size of the broadening?
Does the answer depend on the mechanism that led to the
broadening? Why should we consider the levels to be broadened by ⌬E, but still count unbroadened levels of an idealized insulated system, such as given in Eq. 共5兲, for the purpose of calculating ⍀共E兲?
Some advanced textbooks3 fix the problem of the ambiguity of ⌬E by stating that ⌬E = 1 / ␳, where ␳ is the “probability of the most likely energy level.” However, at this point
we have not yet even defined the probability. Moreover, in
the future development of the subject the probability is defined as ␳ ⬃ eS/kB, which amounts to circular reasoning.
The second gap in introductory statistical physics naturally
occurs during the derivation of the canonical distribution.
The standard derivation of the canonical distribution involves coupling the system of interest to a large reservoir.
Thus, its energy is no longer constant, but fluctuates due to
exchange of energy with the reservoir. The free energy of the
system can be calculated from the partition function Z −
= 兺ne−En/kBT,
F = − kBT ln Z.
In turn, E can be calculated using
748
Am. J. Phys., Vol. 75, No. 8, August 2007
共9兲
E=F−T
⳵F
.
⳵T
共10兲
Remarkably, the thermodynamic properties of a system can
be calculated using the canonical approach 关Eq. 共9兲兴 or using
the microcanonical approach 关Eq. 共8兲兴. If the system is large,
it can be shown, example by example, that the results for the
thermodynamic quantities do not depend on which of the two
approaches we take. For example, typically we derive the
Bose-Einstein distribution first by applying the microcanonical ensemble to a harmonic oscillator and later by calculating
the oscillator’s partition function. Both approaches result in
the same Bose-Einstein distribution. This equivalence occurs
despite the fact that the microcanonical and canonical approach are physically distinct. One assumes that the system
is completely isolated, while the other assumes that the system freely exchanges energy with its environment. Yet the
thermodynamics of the system under these different conditions are identical.
The textbooks usually argue that if a system is large, the
fluctuations of its energy, even in the canonical approach, are
small, and so the canonical and microcanonical techniques
must both lead to the same thermodynamics in this limit.
This argument definitely has merit. However, the fact that the
results are the same must have a simple mathematical explanation rooted in the definitions Eq. 共8兲 and Eq. 共9兲, in contrast to the hand-waving arguments presented in this paragraph.
In this paper we demonstrate that there exists a straightforward method, apparently not discussed in existing textbooks, that connects the microcanonical and canonical ensembles and shows mathematically that the thermodynamical
quantities which follow from them are identical in the thermodynamic limit. In the process it establishes the precise
meaning of the energy interval ⌬E. The particular result for
⌬E given in the following in Eq. 共36兲 does not seem to have
been featured in the literature before. This technique appears
to resolve both issues discussed in Sec. I.
II. DERIVATION OF THE GIBBS DISTRIBUTION
Consider a system with energy levels En. We define the
statistical weight in the following way, which generalizes
Eqs. 共1兲 and 共8兲:
⌫共E兲 = ⌬E 兺 ␦共E − En兲,
共11兲
n
where ␦ is the Dirac delta function. ⌬E has dimensions of
energy and has to be introduced because the Dirac delta
function has dimensions of its inverse argument, and ⌫共E兲 is
a dimensionless quantity equal to the number of microstates.
Thus, ⌬E must be some quantity with dimensions of energy.
It is straightforward to see that ⌬E is, as before, the energy
interval over which the energy levels are counted. The sum
over the delta functions in Eq. 共11兲 is none other than the
density of states g共E兲 introduced in many textbooks,1,2
g共E兲 = 兺 ␦共E − En兲.
共12兲
n
The density of states is the quantity such that
V. Gurarie
Downloaded 09 Oct 2013 to 130.239.76.10. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
748
冕
E2
共13兲
dEg共E兲
E1
is the number of energy levels En such that E1 ⬍ En ⬍ E2. A
direct substitution of Eq. 共12兲 into Eq. 共13兲 confirms that Eq.
共12兲 is the correct expression for the density of states. Then,
⌫共E兲 = ⌬Eg共E兲 ⬇
冕
d⑀g共⑀兲,
共14兲
is the number of energy levels in the energy interval ⌬E; the
smaller the value ⌬E, the better the last approximate equality
holds. Thus, Eq. 共11兲 is the natural definition of ⌫共E兲, is
consistent with its usual definitions, and is compatible with
both the situation where the energy levels are degenerate,
such is the case for harmonic oscillators, and when the energy levels are not degenerate, as for an ideal gas.
The entropy is now defined as
e
= ⌬E 兺 ␦共E − En兲.
共15兲
n
Defined in this way, the entropy is a function of energy E and
the energy interval ⌬E. We will see that for a sufficiently
large system the entropy dependence on ⌬E can be completely neglected, consistent with what the textbooks say regarding the energy interval ⌬E. We will see that it will be
possible to fix ⌬E by requiring that this definition of entropy
be equivalent to the one given in the canonical ensemble.
We now take the following expression for the Dirac delta
function well known from Fourier analysis:
␦共x兲 =
冕
⬁
−⬁
dp ixp
e ,
2␲
共16兲
and apply it to Eq. 共15兲. We find
eS/kB = ⌬E 兺
n
冕
i⬁
−i⬁
d␤ ␤共E−E 兲
n .
e
2␲i
I⬇
冋兺 册
e −␤En .
冕
dxe f共x0兲−共1/2兲f ⬙共x0兲x = e f共x0兲
2
冑
2␲
.
f ⬙共x0兲
共17兲
共18兲
共22兲
In principle, we can expand even further, thus producing
better and better approximations to the integral I. However,
we already have a good approximation as long as x does not
deviate too much from x0. To give a more precise determination of how good the approximation employed in Eq. 共22兲
is, we need to calculate corrections by expanding f共x兲 in a
Taylor series up to fourth order. We will not present this
derivation here, but the end result is that Eq. 共22兲 is a good
approximation to the integral I if
冏
冏
f ⵳共x0兲
1.
共f ⬙共x0兲兲2
共23兲
In some applications the saddle point x0 may turn out to be
complex, in which case the contour of integration has to be
deformed first to pass through that point. If f共x兲 has several
maxima, the largest of them gives the leading contribution to
I.
With these considerations in mind, we apply the saddlepoint technique to the integral in Eq. 共19兲. The saddle point
must satisfy
E − F共␤兲 − ␤
⳵ F共␤兲
= 0.
⳵␤
共24兲
Once the solution of Eq. 共24兲, denoted by ␤s, is known, the
entropy can be computed by comparison with Eq. 共21兲, or
eS/kB ⬇ e␤sE−␤sF共␤s兲 .
The integral over ␤ goes along the imaginary axis of the
complex plane of ␤.
Next we introduce the notation
F共␤兲 = − ␤−1 ln
共21兲
To obtain a better approximation to I, we expand about x0 in
a Taylor series, keep the first two terms, and find
E+⌬E
E
S/kB
I ⬇ e f共x0兲 .
共25兲
In the spirit of the first saddle-point approximation of Eq.
共21兲, we neglect everything that is not in the exponential in
Eq. 共19兲, including ⌬E. We will later perform a more detailed calculation to explore the fate of the terms we neglected 关see Eq. 共34兲, which is an improved version of Eq.
共25兲 just as Eq. 共22兲 is an improved version of Eq. 共21兲兴. For
now let us explore the consequences of Eq. 共25兲. It follows
from Eq. 共25兲 that
n
S/kB = ␤sE − ␤sF共␤s兲.
With its help, Eq. 共17兲 becomes
eS/kB = ⌬E
冕
i⬁
−i⬁
d␤ ␤E−␤F共␤兲
e
.
2␲i
共19兲
We would like to compute the integral in Eq. 共19兲 by the
saddle-point technique. The saddle-point or steepest descent
technique4 is generally used to calculate integrals of the form
I=
冕
dxe f共x兲 ,
共20兲
where the function f共x兲 has a sharp maximum for x = x0 with
f ⬘共x0兲 = 0. As a first approximation, I is then given by
749
Am. J. Phys., Vol. 75, No. 8, August 2007
共26兲
Equation 共24兲 has exactly one solution. To see that, we
rewrite the saddle-point equation, 共24兲, using the definition
Eq. 共18兲, as
E=
兺 nE ne −␤En
.
兺 ne −␤En
共27兲
The expression on the right-hand side 共which is the average
energy in the canonical ensemble兲 is equal to the groundstate 共or lowest兲 energy E0 if ␤ is taken to infinity. As ␤
decreases, the right-hand side grows monotonically. Thus,
for every E that lies between the smallest and the highest
energy level of the system 共or infinity, if there is no highest
energy of the system兲, there is an unique solution ␤s.
V. Gurarie
Downloaded 09 Oct 2013 to 130.239.76.10. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
749
Fig. 1. The plane of complex ␤, with the contour of integration in Eq. 共19兲.
The contour passes through the point ␤s located on the real axis going
through it in the vertical 共imaginary兲 direction.
We define
T⬅
1
.
k B␤ s
共28兲
Then, Eqs. 共18兲, 共24兲, and 共26兲 can be rewritten as
F共T兲 = − T ln
冋兺
册
S=−
共29兲
e−En/kBT ,
n
⳵F
,
⳵T
共30兲
共31兲
E = F + TS.
Equations 共29兲–共31兲 define the canonical ensemble, with T
being the temperature. Thus, we have shown, using Eq. 共15兲,
that the thermodynamic quantities derived from a microcanonical distribution are equivalent to the thermodynamic
quantities derived using the canonical distribution. Moreover, ⌬E has disappeared from all the equations.
III. CRITERIA OF THE APPLICABILITY
OF THE DERIVATION
In order to clarify whether Eq. 共25兲 is a good approximation to Eq. 共19兲, we now examine the integral in Eq. 共19兲 in
more detail. The saddle point ␤s is on the real axis, so we
deform the contour of integration to pass through that point,
as shown in Fig. 1, and expand the expression in the exponent of Eq. 共19兲 in a Taylor series. We introduce the notation
␤ = ␤s + ix and find from Eq. 共19兲
eS/kB = ⌬Ee␤sE−␤sF共␤s兲
冕
⬁
−⬁
dx −k CT2x2/2
e B
,
2␲
共32兲
where C = −T共⳵2F / ⳵T2兲 ⬎ 0 is the heat capacity. The saddle
point is a good approximation if the function in the integral
goes to zero quickly with increasing x, that is, if C is sufficiently large. To make this statement more precise, let us use
the criterion Eq. 共23兲, which gives
冏冉 冉
kB3 6C + T 6
d 2C
dC
+T 2
dT
dT
冊冊冏
kB2 T4C2 .
共33兲
Typically the heat capacity of a system is proportional to its
size, C ⬃ NkB, where N is the number of particles in the
system. Because the left-hand side of Eq. 共33兲 is proportional
to N, and the right-hand side is proportional to N2, the criterion is satisfied if N 1.
750
Am. J. Phys., Vol. 75, No. 8, August 2007
The fact that large N makes the saddle point a good approximation is a manifestation of the law of large numbers,
which states that the fluctuations of an additive quantity in a
system with a large number of degrees of freedom grow as
the square root of that number. Thus, in the limit of a very
large system, the fluctuations become negligible compared to
the mean. What we accomplished is another application of
this law: the temperature in a closed system can be well
defined 共its fluctuations can be neglected兲 only when the system is large.
We recall that the temperature is defined as T = 1 / ␤s. In
practice, ␤ = ␤s + ix. The contribution of nonzero values of x
to the integral in Eq. 共32兲 is a manifestation of the fact that
temperature is well defined only if the values of x that contribute substantially to the integral in Eq. 共32兲 are not too
large. Indeed, the relevant values of x are close to zero if C is
sufficiently large. Nonzero values of x contributing to the
integral in Eq. 共32兲 should be interpreted as fluctuations in
the temperature if we identify 1 / kBT = ␤s + ix. However, it is
clear that the temperature fluctuations are purely imaginary
due to the factor of i in this definition of T.
It is well known that in the canonical ensemble there are
thermodynamic fluctuations of temperature, given by1,3
具⌬T2典 = T2 / C. These fluctuations are due to the exchange of
energy between the system and the reservoir and become
smaller as the system becomes larger 共C increases兲. A similar
relation holds here, except that the fluctuations are imaginary. The reason for this interesting difference is that we are
considering an isolated system whose energy does not fluctuate, but we are nevertheless applying the canonical ensemble to it. Because the energy is kept fixed, its thermodynamically conjugate variable, the temperature, must fluctuate
in the imaginary direction. These fluctuations cannot be measured, because they are present only in a totally insulated
system, while to measure the temperature we must bring the
system in contact with a thermometer, which would then
make the system interact with its environment.
The integral over x in Eq. 共32兲 gives
eS/kB =
⌬E
冑2␲kBCT2 e
␤sE−␤sF共␤s兲
共34兲
.
This relation represents a more careful derivation of entropy
as compared to the previously used Eq. 共25兲.
We can now proceed in two possible ways. One is to observe that in the definition of entropy we thus obtained
冋
冉冑
S = kB ␤sE − ␤sF共␤s兲 + log
⌬E
2␲kBCT2
冊册
.
共35兲
The term under the sign of logarithm depends on the size of
the system N only weakly 共as log N via C ⬃ N兲. At the same
time, E and F are proportional to the size of the system.
Thus, at large N, the logarithm can be completely neglected
and we go back to Eq. 共26兲. This argument parallels those in
the textbooks: for large systems the terms which include the
unknown energy interval ⌬E can be neglected.
Another way would be to choose the energy interval
共which up to this moment is arbitrary兲 to be
⌬E = T冑2␲kBC.
共36兲
We see that in this approach the energy interval ⌬E acquires
V. Gurarie
Downloaded 09 Oct 2013 to 130.239.76.10. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
750
a physical meaning. Choosing a different energy interval will
not change the thermodynamics of very large systems. However, the terms we obtained due to careful integration over x
in the saddle-point approximation Eq. 共32兲 uniquely fix ⌬E
in terms of the properties of the system. We can also, in
principle, calculate Eq. 共19兲 as an asymptotic expansion in
powers of 1 / C, by continuing to expand about the saddle
point. This analysis will provide a better approximation of
the magnitude of ⌬E.
This discussion shows that we have come full circle in
looking for the meaning of ⌬E. ⌬E is arbitrary if we care
only about the thermodynamics of very large systems. For
smaller systems, the choice Eq. 共36兲 ensures the compatibility of the microcanonical and canonical ensembles.
We have shown that the microcanonical definition of entropy Eq. 共1兲 and its canonical definition Eq. 共29兲 are equivalent. A large system has the same thermodynamics regardless
of whether it is insulated or it freely exchanges energy with
its environment. The energy interval, which appears in any
reasonable definition of entropy, is related to the temperature
751
Am. J. Phys., Vol. 75, No. 8, August 2007
and heat capacity. We observe that the energy can be approximated very roughly by E ⬃ CT. Thus, for large systems
where C kB,
⌬E
⬃
E
冑
kB
1.
C
共37兲
In other words, the energy interval is small.
ACKNOWLEDGMENT
This work was supported by the NSF Grant DMR0449521.
a兲
Electronic mail: victor.gurarie@colorado.edu
R. K. Pathria, Statistical Mechanics 共Butterworth-Heinemann, Oxford,
1996兲.
2
D. V. Schroeder, An Introduction to Thermal Physics 共Addison-Wesley
Longman, Boston, 1999兲.
3
L. D. Landau and E. M. Lifshitz, Statistical Physics 共ButterworthHeinemann, Oxford, 1984兲.
4
J. D. Murray, Asymptotic Analysis 共Springer-Verlag, New York, 1984兲.
1
V. Gurarie
Downloaded 09 Oct 2013 to 130.239.76.10. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
751