Derivation of the Boltzmann principle
Michele Campisia兲
Institute of Physics, University of Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany
Donald H. Kobeb兲
Department of Physics, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427
共Received 22 September 2009; accepted 4 January 2010兲
We derive the Boltzmann principle SB = kB ln W based on classical mechanical models of
thermodynamics. The argument is based on the heat theorem and can be traced back to the second
half of the 19th century in the works of Helmholtz and Boltzmann. Despite its simplicity, this
argument has remained almost unknown. We present it in a contemporary, self-contained, and
accessible form. The approach constitutes an important link between classical mechanics and
statistical mechanics. © 2010 American Association of Physics Teachers.
关DOI: 10.1119/1.3298372兴
I. INTRODUCTION
One of the most intriguing equations of contemporary
physics is Boltzmann’s celebrated principle
SB = kB ln W,
共1兲
where kB is Boltzmann’s constant. Despite its unquestionable
success in providing a means to compute the thermodynamic
entropy of isolated systems based on counting the number W
of available microscopic states, its theoretical justification
remains obscure and vague in most statistical mechanics
textbooks. In this respect Khinchin has commented:1 “All
existing attempts to give a general proof of this postulate
must be considered as an aggregate of logical and mathematical errors superimposed on a general confusion in the
definition of the basic quantities.” The lack of a crystal-clear
proof of Boltzmann’s principle puts physics students, teachers, and all physicists in the uncomfortable position of being
forced to accept the relation as a postulate which is necessary
to link thermodynamic entropy to microscopic dynamics.
Recent studies in the history and foundations of statistical
mechanics2 have drawn attention to the fact that a similar
relation,
S = kB ln ⌽,
共2兲
emerges naturally from classical mechanics if the ergodic
hypothesis is made, the properties that entropy should satisfy
are appropriately set, and the basic quantities are consistently
defined. The quantity ⌽ is the volume in phase space enclosed by a hypersurface of constant energy E.
Equation 共2兲 is valid for both large and small systems and
coincides with the Boltzmann formula for large systems.
Hence, the derivation of Eq. 共2兲 provides the missing link for
Eq. 共1兲. The basic argument underlying the derivation of Eq.
共2兲 can be traced to as early as the second half of the 19th
century in the work of Helmholtz and Boltzmann.3,4
The purpose of this article is to provide an accessible and
contemporary presentation of the original argument of Helmholtz and Boltzmann3,4 and of its recent developments.2 We
derive Boltzmann’s principle from classical mechanics with
one simple guiding principle—the heat theorem 共see statement 1兲 and one central assumption, namely, the ergodic hypothesis.
In Sec. II we briefly review the basics of thermodynamics.
We give concise formulations of the first and second laws of
thermodynamics and introduce the heat theorem. We con608
Am. J. Phys. 78 共6兲, June 2010
http://aapt.org/ajp
struct a one-dimensional mechanical model of thermodynamics in Sec. III according to the work of Helmholtz.3 The
concepts of ergodicity and microcanonical probability distribution emerge naturally in this model and are introduced in
Sec. IV. In Sec. V we generalize the one-dimensional model
to more realistic Hamiltonian systems of N-particles in three
dimensions. At this stage the ergodic hypothesis is assumed
and Eq. 共2兲 is derived. In Sec. VI we point out that the
mechanical entropy of Eq. 共2兲 does not change during quasistatic processes in isolated systems, in agreement with the
second law of thermodynamics. Non-quasi-static processes
that can lead to an increase in entropy have been treated
elsewhere.5,6 In Sec. VII the Boltzmann principle is derived.
A summary and some remarks concerning the validity of the
ergodic hypothesis are given in Sec. VIII.
We present the line of reasoning and main results in the
text, while proofs and problems are provided in Appendices
B–E.
II. CLAUSIUS ENTROPY
We first review the first and second laws of thermodynamics in the formulation given by Clausius,7 which gives the
definition of thermodynamic entropy.
The differential form of the first law of thermodynamics
is8
dE = ␦Q + ␦W,
共3兲
where dE is the change in internal energy, ␦Q is the energy
added to the system due to heating, and ␦W is the work done
on the system during an infinitesimal transformation. The
first law is the energy conservation law applied to a system
in which there is an exchange of energy by both work and
heating. It is essential to realize that ␦Q and ␦W are inexact
differentials, whereas dE is an exact differential. The internal
energy E is a state variable, a quantity that characterizes the
thermodynamic equilibrium state of the system. In contrast,
W and Q characterize thermodynamic energy transfers only
and are not properties of the state of the system.9
A differential is exact if and only if the integral along a
path in the system’s state space depends only on the end
points. In contrast, the corresponding integral of an inexact
differential with the same end points depends on the path
taken. A summary of the formal definition of differential
forms 共more precisely, 1-form兲 and their major properties is
given in Appendix A.
© 2010 American Association of Physics Teachers
608
␦Q
T
共4兲
= dS.
The function S is called the thermodynamic entropy of the
system.
Statement 1 can also be stated in an equivalent integral
form. The integral of ␦Q / T along a path connecting a state A
to a state B in the system’s state space does not depend on
the path, but only on its end points A and B.11 This statement
means that there exists a state function S 共the thermodynamic
entropy兲 such that
冕
B
A
␦Q
T
= S共B兲 − S共A兲.
ϕ
(b)
ϕ(x; V2 )
E2
E1
(c)
E1 , V1
(d)
E2 , V2
x− (E1 , V1 )
0
x+ (E1 , V1 )
x
x− (E2 , V2 ) 0 x+ (E2 , V2 )
x
Fig. 1. Point particle in the U-shaped potential ␸共x ; V兲 = mV2x2 / 2. 共a兲 Shape
of the potential for a V = V1. 共b兲 Shape of the potential for a V = V2. 共c兲 Phase
space orbit corresponding to the potential ␸共x ; V1兲 at energy E1. 共d兲 Phase
space orbit corresponding to the potential ␸共x ; V2兲 at energy E2. The two
quantities E , V, uniquely determine one “state,” that is, one closed orbit in
phase space.
共5兲
From Eq. 共3兲, the heat transfer is ␦Q = dE − ␦W. In general,
work can be performed by changing external parameters ␭i,
such as the volume, magnetic field, and the electric field. The
work ␦W done is given by −兺iFid␭i, where Fi denotes the
corresponding conjugate forces, pressure, magnetization, and
electric polarization, respectively. Therefore, the first law can
be written as
␦Q = dE + 兺 Fid␭i .
(a)
ϕ(x; V1 )
p
The second law of thermodynamics can be conveniently
summarized in three statements.
Statement 1. The differential ␦Q / T is exact. This statement
is one of the most important statements of thermodynamics.
Although ␦Q is not an exact differential, an exact differential
is obtained when it is divided by the absolute temperature T.
An equivalent statement is that there exists a state function S
such that
共6兲
A crucial point in statements 2 and 3 is that they are for
thermally isolated systems. Thermally isolated systems are
those that are not in contact with a thermal bath, by means of
which either their temperature or energy can be controlled.
Thus, these processes are those in which only the external
parameter V is varied in a controlled way and the variable E
is uncontrolled. In statement 2 a quasi-static process is one in
which the change in the parameter V is so slow that at any
instant of time the system is almost in equilibrium. In statement 3, this requirement is relaxed.
i
Without loss of generality, we will restrict ourselves in the
following to only one external parameter V with conjugate
force P:12
␦Q = dE + PdV.
共7兲
In this case, statement 1 can be expressed as
共dE + PdV兲/T = exact differential = dS.
共8兲
Equation 共8兲 is known in literature as the heat theorem.10
Because by definition dS = 共⳵S / ⳵E兲dE + 共⳵S / ⳵V兲dV, the heat
theorem can be expressed in equivalent terms as there exists
a function S共E , V兲 such that13
⳵S 1
= ,
⳵E T
⳵S P
= .
⳵V T
共9兲
Any inexact differential, such as ␦Q = dE + PdV, does not
enjoy the same property: It is impossible to find a function of
state Q共E , V兲 such that ⳵Q / ⳵E = 1 and ⳵Q / ⳵V = P.
The following two statements, regarding the function S,
complete Clausius’s form of the second law.
Statement 2. For a quasi-static process occurring in a thermally isolated system, the entropy change between two equilibrium states is zero,
⌬S = 0.
共10兲
Statement 3. For a non-quasi-static process occurring in a
thermally isolated system, the entropy change between two
equilibrium states is non-negative,
⌬S ⱖ 0.
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Am. J. Phys., Vol. 78, No. 6, June 2010
共11兲
III. ONE-DIMENSIONAL MECHANICAL MODELS
OF THERMODYNAMICS
In this section we construct a one-dimensional classical
mechanical analog of Clausius thermodynamic entropy. This
construction dates back to Helmholtz2,3,10 and is based on the
heat theorem 共8兲.
Consider a particle of mass m and coordinate x moving in
a U-shaped potential ␸共x兲, as illustrated in Fig. 1. To allow
for the possibility of doing work on the particle by means of
an external intervention, we assume that the potential ␸ depends on some externally controllable parameter V so that
␸ = ␸共x ; V兲. The Hamiltonian of this system is
H共x,p;V兲 = K共p兲 + ␸共x;V兲,
共12兲
where K共p兲 = p2 / 2m is the kinetic energy and p is the momentum. Given this mechanical system, we need to specify
its internal energy E, “temperature” T, and the “force” P
conjugate to the external parameter V.
For the internal energy, we take the energy E given by the
Hamiltonian. For a fixed V, the particle’s energy E is a constant of motion. For simplicity, we define the zero of energy
in such a way that the minimum of the potential is 0, regardless of the value of V.
Once V and E are specified, the orbit of the particle in
phase space is fully determined. We call E and V the system’s “thermodynamic” state variables.14 The particle moves
between the two turning points x⫾共E , V兲 and forms closed
orbits in phase space with a period ␶共E , V兲 共see Fig. 1兲.
Now that the state variables are fixed, we have to define
M. Campisi and D. H. Kobe
609
the corresponding temperature and conjugate force. In agreement with the common understanding that temperature for
classical systems is a measure of the kinetic energy of the
particles, we take the temperature to be proportional to the
kinetic energy averaged over a period
T共E,V兲 ⬅
2
kB␶共E,V兲
冕
␶共E,V兲
K共p共t;E,V兲兲dt.
共13兲
0
The Boltzmann constant kB ensures the correct dimensions
for the temperature T. For the conjugate force, we take the
time average of −⳵␸ / ⳵V,15
P共E,V兲 ⬅ −
1
␶共E,V兲
冕
␶共E,V兲
0
⳵␸共x共t;E,V兲;V兲
dt.
⳵V
共14兲
In Eqs. 共13兲 and 共14兲 x共t ; E , V兲 and p共t ; E , V兲 are the solution
of Hamilton’s equations of motion with a fixed V and arbitrary initial condition x0 , p0 such that H共x0 , p0 ; V兲 = E.
Having identified the mechanical analogs of internal energy, external parameter, temperature, and conjugate force
with the quantities E, V, T, and P, respectively, we can now
ask whether a mechanical analog of the entropy S exists. To
answer this question, we must find, according to statement 1,
as expressed in Eq. 共9兲, a function S共E , V兲 such that
1
⳵ S共E,V兲
=
,
⳵E
T共E,V兲
⳵ S共E,V兲 P共E,V兲
=
.
⳵V
T共E,V兲
共15兲
That such a function exists is given by the following theorem.
Helmholtz’s theorem. A function S共E , V兲 satisfying Eq.
共15兲 exists and is given by
S共E,V兲 = kB log 2
冕
x+共E,V兲
x−共E,V兲
冑2m共E − ␸共x;V兲兲dx.
共16兲
The proof of the theorem is given in Appendix B and Ref.
10, pp. 45–46. The entropy S共E , V兲 can be rewritten compactly as
冖
S共E,V兲 = kB log
pdx,
共17兲
where p = 冑2m共E − ␸共x ; V兲兲 is the momentum of the particle
when it is located at x: The integral 养pdx is called the reduced action.15 It is the area ⌽ in phase space enclosed by
the orbit of energy E and parameter V,
S共E,V兲 = kB log ⌽共E,V兲,
where
⌽共E,V兲 =
冕
dpdx.
共18兲
共19兲
simple to model a real thermodynamic system composed of
as many as 1023 particles. Thus, it is necessary to generalize
the Helmholtz theorem to multidimensional systems.
IV. ERGODICITY AND THE MICROCANONICAL
ENSEMBLE
To generalize our model to more degrees of freedom, we
need ergodicity and the microcanonical ensemble. The onedimensional example of Sec. III can be used to introduce
these important concepts.
The time average of a phase function f共x , p兲 over the orbit
specified by E and V is
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Am. J. Phys., Vol. 78, No. 6, June 2010
冕
␶
f共x共t兲,p共t兲兲dt.
共20兲
0
For simplicity, we have dropped the explicit dependence on
E, V of ␶, x共t兲, p共t兲, and 具f典t. Because p = mv = mdx / dt, the
differential dt is
dx
.
p共x兲
dt = m
共21兲
Using this differential, we obtain
具f典t =
2m
␶
冕
x+
x−
dx
f共x,p共x兲兲.
p共x兲
共22兲
The factor 2 is due to the particle going from x− to x+ in a
half period ␶ / 2. Now consider the integral
冕
dp␦共p2/2m + ␸共x;V兲 − E兲,
共23兲
where ␦ denotes the Dirac delta function. If we use the expression ␦共f共p兲兲 = 兺i␦共p − pi兲 / 兩f ⬘共pi兲兩, where the pi’s are the
zeroes of f共p兲 and 兰␦共p − pi兲dp = 1, we obtain for the integral,
冕
␦共p2/2m + ␸共x;V兲 − E兲dp = 2m/p共x兲.
共24兲
Then, Eq. 共22兲 becomes
具f典t =
1
␶
冕 冕
dx
dp␦共p2/2m + ␸共x;V兲 − E兲f共p,x兲,
共25兲
where it is unnecessary to specify the integration limits x⫾
because they are implied by the Dirac delta function. The
period ␶ is given by
␶=
冕 冕
冕 冕
␶
=
x+
dt = 2
0
H共x,p;V兲ⱕE
Helmholtz’s theorem shows that there exists a mechanical
counterpart of the entropy given by the logarithm of the
phase space volume enclosed by the curve of constant energy
H共x , p ; V兲 = E.
That there exists a function S satisfying Eq. 共15兲 is a remarkable result that shows there is a consistent onedimensional mechanical model of thermodynamics. Although this model suggests the deep connection between
classical mechanics and thermodynamic entropy, it is too
1
␶
具f典t ⬅
x−
dx
dx
p共x兲
dp␦共p2/2m + ␸共x;V兲 − E兲.
共26兲
Hence, we arrive at
具f典t =
冕 冕
dx
dp␳共x,p;E,V兲f共p,x兲,
共27兲
where we have introduced the phase space probability density function
M. Campisi and D. H. Kobe
610
␳共x,p;E,V兲 =
1
␦共p2/2m + ␸共x;V兲 − E兲.
␶共E,V兲
共28兲
From Eq. 共26兲, it is clear that ␳ is properly normalized. The
function ␳共x , p ; E , V兲 is the microcanonical distribution.
Equation 共27兲 shows that the time average of a phase space
function f共x , p兲 over a period is equal to its microcanonical
average. This property is called ergodicity. All onedimensional systems with a U-shaped potential are ergodic.
1
⳵ S共E,V兲N
=
,
⳵E
TN共E,V兲
We now extend the treatment for one degree of freedom to
systems of N-particles in three dimensions with 3N degrees
of freedom. The Hamiltonian for an interacting system of
N-particles of mass m is
HN共q,p;V兲 = KN共p兲 + ␸N共q;V兲,
共29兲
3N 2
pi / 2m is the kinetic energy and ␸N is the
where KN共p兲 = 兺i=1
3N
potential energy. The coordinates q = 兵qi其i=1
and their conju3N
gate canonical momenta p = 兵pi其i=1 are 3N-dimensional vectors.
In analogy with one-dimensional systems with a U-shaped
potential, we define the microcanonical probability distribution as
␳N共q,p;E,V兲 =
1
␦共E − HN共q,p;V兲兲,
⍀N共E,V兲
共30兲
where ⍀N共E , V兲 is the normalization
⍀N共E,V兲 =
冕 冕
¯
␦共E − HN共q,p;V兲兲dqdp.
共31兲
The microcanonical average 具f典␮ of a phase function f共q , p兲
is defined as
具f典␮ ⬅
冕 冕
¯
␳N共q,p;E,V兲f共q,p兲dqdp.
共32兲
Continuing the analogy with one-dimensional systems, we
make the following crucial assumption.
Ergodic hypothesis. For a given E and V, the time average
具f典t of any function f共q , p兲 is uniquely determined and is
equal to its microcanonical average 具f典␮,
具f典t = 具f典␮ .
共33兲
In analogy with Eqs. 共13兲 and 共14兲, we define the temperature as
TN共E,V兲 ⬅
2
具KN典t ,
3NkB
共34兲
which is twice the average kinetic energy per degree of freedom. The conjugate force is defined as
PN共E,V兲 ⬅ −
冓 冔
⳵␸N
⳵V
.
共35兲
t
We ask, in agreement with statement 1 as expressed in Eq.
共9兲, does there exist a function SN共E , V兲 such that
611
Am. J. Phys., Vol. 78, No. 6, June 2010
共36兲
The following theorem gives the answer.
Generalized Helmholtz theorem. A function SN共E , V兲 satisfying Eq. 共36兲 exists and is given by
SN共E,V兲 = kB log ⌽N共E,V兲,
where
⌽N共E,V兲 ⬅
V. MULTIPARTICLE MECHANICAL MODELS
OF THERMODYNAMICS
⳵ S共E,V兲N PN共E,V兲
=
?
⳵V
TN共E,V兲
冕 冕
¯
共37兲
dqdp.
HN共q,p兲ⱕE
共38兲
The proof, which is based on the equipartition theorem, is
given in Appendix C and Ref. 2. Unlike the temperature and
conjugate force, SN is not a time average of some phase
function f共q , p兲.
This theorem shows that ergodic systems constitute ideal
mechanical models of thermodynamics. The state variables E
and V define their thermodynamic state. In addition, their
temperature and conjugate force can be defined as functions
of the state variables. Surprisingly, the heat differential 共dE
+ PNdV兲 / TN is exact, allowing for a consistent and logical
definition of entropy SN.
VI. ADIABATIC INVARIANCE
According to the generalized Helmholtz theorem, SN is
consistent with the first law of thermodynamics and statement 1 of the second law of thermodynamics. Is this construction also consistent with statements 2 and 3 of the second law of thermodynamics?
For SN to be consistent with statement 2, it is necessary for
it to satisfy the following condition. If the parameter V is
varied very slowly in time from a V0 = V共t0兲 to V f = V共t f 兲, the
corresponding change in the entropy SN is zero. The time of
variation must be much slower than any time scale of the
system’s dynamics. By allowing for a time dependence of V,
the Hamiltonian function of the system is also time dependent, and hence energy is not conserved. Consider the initial
system at t = t0 with phase space point q0 , p0. The system
then evolves under the time-dependent Hamiltonian
HN共q,p;V共t兲兲 = KN共p兲 + ␸N共q;V共t兲兲
共39兲
to a new phase space point q f 共q0 , p0兲 , p f 共q0 , p0兲, where we
have made explicit the dependence on the initial condition of
the evolved phase space point. The energy at time t f is E f
= HN共q f 共q0 , p0兲 , p f 共q0 , p0兲 ; V共t f 兲兲. It is known16 that for ergodic systems the energy E f reached at the end of a very
slow change depends only on the initial energy E0
= HN共q0 , p0 ; V共t0兲兲. The final energy E f is determined by
solving the equation
⌽N共E0,V0兲 = ⌽N共E f ,V f 兲.
共40兲
Thus, the quantity ⌽N共E , V兲 does not change when V is varied infinitely slowly in time. In classical mechanics this
property is called adiabatic invariance. Because the entropy
is SN共E , V兲 = kB log ⌽N共E , V兲, and ⌽N共E , V兲 is an adiabatic
invariant, it is evident that SN is also an adiabatic invariant.
Therefore, the entropy does not change if V is varied very
slowly in time and thus SN complies with statement 2. A
proof of adiabatic invariance of ⌽N is given in Appendix D
and Ref. 16, pp. 27–30.
M. Campisi and D. H. Kobe
611
It is also possible to prove that, in an averaged sense, SN
complies with statement 3 as well.5,6 In this case the average
change in entropy must be considered because the final energy is not uniquely determined by the initial energy for fast
transformations. Depending on the initial conditions, different final energies can be obtained and hence different final
entropies.
VII. THE BOLTZMANN PRINCIPLE
For a system composed of a very large number N of particles that interact via short-range forces, the phase space
volume ⌽N共E兲 approaches an exponential dependence on E,
⌽N共E兲 ⬀ eE. Because ⍀N = ⳵⌽N / ⳵E 关see Eq. 共31兲兴, we also
have ⌽N ⬀ ⍀N 共see Ref. 17, p. 148兲.
The quantity ⍀N in Eq. 共31兲 is the measure of the shell of
constant energy HN共q , p ; V兲 = E. As such, it is proportional to
the number W of microstates compatible with a given energy
E. According to semiclassical theory each microstate occupies a volume h3N of phase space, where h is Planck’s
constant.18 By introducing an arbitrary energy scale ⌬E, the
number of microstates W is given by W = ⍀N⌬E / h3N兲. Thus,
for very large N we obtain the proportionalities,
⌽N ⬀ ⍀N ⬀ W
共N Ⰷ 1兲.
共41兲
We take the logarithm and obtain
SN ⯝ kB ln W = SB
共N Ⰷ 1兲,
共42兲
except for an irrelevant constant. Equation 共42兲 shows that
for large ergodic systems composed of many particles interacting via short range forces, the differential of the Boltzmann entropy is equal to the differential ␦Q / T. Hence, it can
be identified with the Clausius entropy, which gives a proof
of the Boltzmann principle.
compliance with the equipartition theorem and the fact that it
is a positive and increasing function of the energy.27,28 The
entropy in Eq. 共37兲 also appears in Gibbs seminal book.29
However, its connection with the heat theorem has not been
recognized previously.
The most crucial point in the derivation of the Boltzmann
principle is the introduction of the ergodic hypothesis. Although this hypothesis is generally believed to hold for real
macroscopic systems, its proof is a formidable challenge
which has been achieved only in few special cases.30 A proof
that a gas of elastically colliding hard spheres is ergodic was
announced in 1963 by Sinai.31 However, the full proof was
not published and the problem is still open 共see Ref. 32 for a
more detailed discussion兲. Nevertheless, the ergodicity of
hard spheres systems is plausible as indicated by recent numerical simulations 共see Sec. IV of Ref. 33兲.
In regard to these difficulties, we note that the present
derivation of the Boltzmann principle does not use the fact
that the average of any arbitrary phase function is equal to its
microcanonical average, as required by the ergodic hypothesis. It only uses the fact that the time average of K and
−⳵␸ / ⳵V is equal to their microcanonical averages 共see the
proof of the generalized Helmholtz theorem in Appendix C兲.
Thus, the ergodic hypothesis can be replaced by the following hypothesis.
Weak ergodic hypothesis. For given E and V, the time
averages 具K典t and 具⳵␸ / ⳵V典t are uniquely determined and are
equal to their microcanonical averages, that is,
具K典t = 具K典␮ ,
共44兲
具⳵␸/⳵ V典t = 具⳵␸/⳵ V典␮ .
共45兲
In summary, ergodicity is too stringent a condition and the
weaker ergodic condition for only the temperature and pressure is sufficient to obtain the Boltzmann principle.34
VIII. CONCLUSIONS
Given an ergodic system, it is possible to specify its thermodynamic state by means of the total energy E and the
external parameter V. Given the state E , V, we can unambiguously define the quantities TN共E , V兲 and PN共E , V兲. Once
these are identified with the system temperature and conjugate force, we can ask whether, as prescribed by the heat
theorem, the combination
dE + PNdV
TN
共43兲
is an exact differential. Surprisingly, the answer is positive,
meaning that there exists a function SN共E , V兲 that can be
identified with the thermodynamic entropy of the system.
The generalized Helmholtz theorem says that this function is
given by the logarithm of the volume ⌽N共E , V兲 of phase
space enclosed by the hypersurface of energy H共q , p ; V兲 = E.
For macroscopic systems, this entropy coincides with the
Boltzmann entropy, thus revealing the rationale of the Boltzmann principle.
The entropy in Eq. 共37兲 is sometimes referred to as the
Hertz entropy.19,20 Hertz21,22 derived it from the requirement
of adiabatic invariance 共see also Refs. 16, 23, and 24兲,
whereas we have derived it from the heat theorem. The fundamental character of the entropy in Eq. 共37兲 is also recognized in Ref. 25, where its property of being a canonical
invariant is emphasized, and in Ref. 26, which highlights its
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Am. J. Phys., Vol. 78, No. 6, June 2010
ACKNOWLEDGMENTS
The authors wish to thank the Texas Section of the American Physical Society for the Robert S. Hyer Recognition for
Exceptional Research award presented at its Fall 2008 meeting. They also thank Cosimo Gorini for reading the manuscript. They are grateful to Professor Randall B. Shirts for
providing the translation of Ref. 25 and for his comments.
Valuable remarks from anonymous referees are also gratefully acknowledged.
APPENDIX A: DIFFERENTIAL FORMS: BRIEF
REVIEW OF DEFINITIONS AND MAIN RESULTS
A differential form ␻ 共more precisely, a 1-form兲 in a connected subset A of R2 is formally written as
␻ = M共x1,x2兲dx1 + N共x1,x2兲dx2 ,
共A1兲
where M共x1 , x2兲 , N共x1 , x2兲 are two functions on A and
共x1 , x2兲 are the coordinates in R2.35
Given a curve ␺ : 关s0 , s1兴 → A,
␺共s兲 = 共␺1共s兲, ␺2共s兲兲,
共A2兲
the integral of ␻ along the curve ␺ is defined as
M. Campisi and D. H. Kobe
612
冕 冕
␺
␻=
s1
关M共␺共t兲兲␺1⬘共t兲 + N共␺共t兲兲␺2⬘共t兲兴dt,
共A3兲
s0
where ␺1,2
⬘ are the derivatives of ␺1,2.
A differential form ␻ is said to be exact if there exist a
function G : A → R such that
␻ = dG,
共A4兲
that is,
⳵G
共x1,x2兲 = M共x1,x2兲,
⳵ x1
⳵G
共x1,x2兲 = N共x1,x2兲,
⳵ x2
共A5兲
where G is called a primitive for the differential form.
Let ⌺共A , B兲 be the set of all curves connecting the point
A ⬅ 共a1 , a2兲 to the point B ⬅ 共b1 , b2兲 in A. A differential form
is exact if and only if for any two points A and B in A and
curves ␺ and ␾ in ⌺共A , B兲 it satisfies
冕 冕
␺
␻=
␾
␻.
共A6兲
The integral of an exact differential form along any curve ␺
connecting A to B does not depend on the curve ␺, but only
on the ending points, and is given by
冕 冕
␺
2具K典t =
␻=
dG = G共B兲 − G共A兲.
␺
共A7兲
The following statement also holds: A differential form is
exact if and only if its integral along any closed curve is
zero. If the functions M and N are of class C1 共that is, they
are differentiable兲, then a necessary condition for the form ␻
to be exact is that
⳵M
⳵ x2
=
⳵N
⳵ x1
.
共A8兲
In this case the differential form ␻ is said to be closed.
⌽
,
␶
共B3兲
from which we obtain
⳵S
kB
=
⳵ E 2具K典t
共B4兲
using Eq. 共B2兲. Similarly, we also obtain
冓 冔
⳵S
⳵␸
kB
=−
⳵V
2具K典t ⳵ V
共B5兲
.
t
From Eqs. 共13兲 and 共14兲, we obtain
⳵S 1
= ,
⳵E T
共B6兲
⳵S P
= .
⳵V T
共B7兲
APPENDIX C: PROOF OF THE GENERALIZED
HELMHOLTZ THEOREM
The proof of generalized Helmholtz theorem makes use of
the multidimensional version of Eq. 共B1兲,
⍀N =
⳵ ⌽N
,
⳵E
共C1兲
which can be proved, in a way similar to Eq. 共B1兲, by expressing ⌽N as 兰␪共E − H共q , p ; V兲兲dqdp and using the relation
␦共x兲 = d␪共x兲 / dx. The equipartition theorem,17
2具K典␮ ⌽N
=
3N
⍀N
共C2兲
is the generalization of Eq. 共B3兲 to many dimensions. We use
Eqs. 共C2兲 and 共C1兲 with Eq. 共37兲 and obtain
3NkB
⳵ SN
=
.
⳵ E 2具KN典␮
共C3兲
冓 冔
In a similar way, we obtain
3NkB ⳵␸N
⳵ SN
=−
⳵V
2具KN典␮ ⳵ V
APPENDIX B: PROOF OF HELMHOLTZ’S
THEOREM
For a one-dimensional system confined in a U-shaped potential the period ␶ of the orbit is equal to the derivative with
respect to energy of the phase space area ⌽ enclosed by the
orbit15
␶=
⳵⌽
.
⳵E
共B1兲
A simple way to prove this relation is by expressing the area
as ⌽共E , V兲 = 兰␪共E − H共x , p ; V兲兲dpdx, where ␪共x兲 is the Heaviside step function 关␪共x兲 = 1 if x ⱖ 0, ␪共x兲 = 0 if x ⬍ 0兴. If we
take the derivative with respect to E and use the relation
␦共x兲 = d␪共x兲 / dx, we obtain ␶ 关see Eq. 共26兲兴. By using Eqs.
共B1兲 and 共18兲, we have
⳵S
␶
= kB .
⌽
⳵E
From Eq. 共22兲, we obtain the relation
613
Am. J. Phys., Vol. 78, No. 6, June 2010
共B2兲
.
␮
共C4兲
By using Eqs. 共34兲 and 共35兲 with the ergodic hypothesis, we
finally arrive at
⳵ SN 1
=
,
⳵ E TN
共C5兲
⳵ SN PN
=
.
⳵ V TN
共C6兲
APPENDIX D: PROOF OF ADIABATIC
INVARIANCE OF ⌽N
To prove that ⌽N in Eq. 共38兲 is an adiabatic invariant, we
use the time-dependent Hamiltonian
HN共q,p;V共t兲兲 = K共p兲 + ␸共q;V共t兲兲.
共D1兲
We take the total time derivative of the Hamiltonian
HN共q , p ; V共t兲兲 in Eq. 共D1兲,
M. Campisi and D. H. Kobe
613
共D2兲
where the terms involving q̇ and ṗ cancel by Hamilton’s
equations.36 The derivative dV / dt changes slowly in time,
but dHN / dt and ⳵HN / ⳵V can change rapidly because of their
dependence on q共t兲 and p共t兲. To eliminate the fast variables
q , p, we take the average of Eq. 共D2兲 with respect to the
microcanonical ensemble, which gives
冓 冔 冓 冔
dHN
dt
⳵ HN
⳵V
=
␮
dV
.
␮ dt
共D3兲
By Liouville’s theorem36 the average on the left-hand side of
Eq. 共D3兲 is
冓 冔
dHN
dt
=
␮
dE
.
dt
共D4兲
The microcanonical average in Eq. 共33兲 on the right-hand
side of Eq. 共D3兲 is
冓 冔
dHN
dV
=
␮
冕 冕
¯
=−
␳N共q,p,E,V兲
⳵ HN共q,p,V兲
dqdp
⳵V
1 ⳵ ⌽N
,
⍀N ⳵ V
共D5兲
where ⌽N and ⍀N are given in Eqs. 共31兲 and 共38兲, respectively. If we substitute Eqs. 共D4兲 and 共D5兲 into Eq. 共D3兲 and
use ⍀N = ⳵⌽N / ⳵E, we obtain
d⌽N ⳵ ⌽N dE ⳵ ⌽N dV
+
= 0,
⬅
dt
⳵ E dt
⳵ V dt
共D6兲
which shows that ⌽N is constant and therefore an adiabatic
invariant.
APPENDIX E: SUGGESTED PROBLEM
Consider the following Hamiltonian of a one-dimensional
harmonic oscillator with angular frequency V 共see Fig. 1兲:
H共x,p;V兲 =
共a兲
共b兲
共c兲
共d兲
p2 mV2x2
.
+
2
2m
共E1兲
Calculate the area ⌽共E , V兲 enclosed by the trajectory of
energy E and angular frequency V. Use Eq. 共B1兲 and
check that the period of the orbit is, as expected, given
by ␶共E , V兲 = 2␲ / V.
Use Eqs. 共13兲 and 共14兲 to show that kBT共E , V兲 = E and
P共E , V兲 = −E / V.
Show that the differential form dE + PdV, with P共E , V兲
as in part 共b兲 is not exact. 关Hint: Show that Eq. 共A8兲 is
not satisfied.兴 Show that the integral of dE + PdV over
the rectangular path with corners 共E0 , V0兲, 共E0 , V1兲,
共E1 , V1兲, 共E1 , V0兲, and E0 ⫽ E1, V0 ⫽ V1, is not zero.
Consider the differential form ␻ = 共1 / T兲dE + 共P / T兲dV,
with P共E , V兲 and T共E , V兲 as in part 共b兲. Find a primitive function S共E , V兲 for ␻. Show that, apart from an
additive constant, it is S共E , V兲 = log ⌽共E , V兲, as dictated
by Theorem 1. Check that Eq. 共A8兲 is satisfied.
a兲
Electronic mail: michele.campisi@physik.uni-augsburg.de
Electronic mail: kobe@unt.edu
b兲
614
A. Khinchin, Mathematical Foundations of Statistical Mechanics 共Dover,
New York, 1949兲, p. 142.
2
M. Campisi, “On the mechanical foundations of thermodynamics: The
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290 共2005兲.
3
H. Helmholtz, “Principien der statik monocyklischer Systeme,”
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162, 179–202.
4
L. Boltzmann, “Über die Eigenschaften monocyklischer und anderer
damit verwandter Systeme,” Crelles Journal 98, 68–94 共1884兲; also L.
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共J. A. Barth, Leipzig, 1909兲, Vol. 3, pp. 122–152, reissued Chelsea, New
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5
M. Campisi, “Statistical mechanical proof of the second law of thermodynamics based on volume entropy,” Stud. Hist. Philos. Mod. Phys. 39,
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M. Campisi, “Increase of Boltzmann entropy in a quantum forced harmonic oscillator,” Phys. Rev. E 78, 051123-1–10 共2008兲.
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J. Uffink, “Bluff your way in the second law of thermodynamics,” Stud.
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H. B. Callen, Thermodynamics 共Wiley, New York, 1960兲.
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D. Chandler, Introduction to Modern Statistical Mechanics 共Oxford U. P.,
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10
G. Gallavotti, Statistical Mechanics: A Short Treatise 共Springer-Verlag,
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11
A path in the space of state variables corresponds to a transformation that
leads the system from A to B through a sequence of almost equilibrium
states. Consequently, the change ␦Q is understood to be a quasi-static
change.
12
For the common case of work due to expansion or compression the external parameter V is the volume and its conjugate force P is the pressure.
In general, V and P stand for any conjugate pair of “displacement” and
force depending on the nature of the work done on the system 共for example, magnetic field and magnetization兲.
13
ជ ⬅ 共1 / T , P / T兲 is a conservative vector field.
In other words, the field F
ជ =ⵜ
ជ S, where
That is, there exists a potential function S共E , V兲 such that F
ជ
ⵜ is the gradient operator in the space 共E , V兲. For this reason, the entropy
can be understood as a thermodynamic potential.
14
The thermodynamic state variables 共E , V兲 should not be confused with
the mechanical state variables 共p , x兲.
15
L. Landau and E. Lifshitz, Mechanics 共Pergamon, Oxford, 1960兲.
16
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L. Landau and E. Lifshitz, Statistical Physics, 2nd ed. 共Pergamon, Oxford, 1969兲.
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P. Hertz, “Über die mechanischen Grundlagen der Thermodynamik,”
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22
P. Hertz, “Über die mechanischen grundlagen der thermodynamik,” Ann.
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26
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28
The conditions ⌽ ⱖ 0 , ⳵E⌽ = ⍀ ⱖ 0 ensure that the temperature derived
from the Hertz entropy, that is, T = 共⳵E log ⌽兲−1 = ⌽ / ⍀ is positive definite.
29
J. Gibbs, Elementary Principles in Statistical Mechanics 共Yale U. P., New
Haven, 1902兲, reprinted by Dover, New York, 1960.
1
dHN共q,p;V兲 ⳵ HN共q,p;V兲 dV
,
=
dt
dt
⳵V
Am. J. Phys., Vol. 78, No. 6, June 2010
M. Campisi and D. H. Kobe
614
30
J. L. Lebowitz and O. Penrose, “Modern ergodic theory,” Phys. Today 26
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共1963兲.
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具philsci-archive.pitt.edu/archive/
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33
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34
The simplest example of a nonergodic system that satisfies this weaker
condition is a one-dimensional particle in a symmetric double well potential. For energies below a certain critical value Ec, the curve of constant energy splits into two disjoint curves ␥l and ␥r, with the phase space
trajectory covering only one of them. 共For energy above Ec there is only
one trajectory and ergodicity holds.兲 Because ␥l and ␥r are mirror images
of each other, the time averages of even functions of x, such as K and
−⳵␸ / ⳵V, are not affected by which of the two curves the motion follows.
Thus, such averages can be calculated as microcanonical averages over
the curve ␥l 艛 ␥r.
35
In the main text the role of 共x1 , x2兲 is played by the state variables 共E , V兲.
36
H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed.
共Addison-Wesley, San Francisco, 2002兲, pp. 337, 419–421.
Dribble Glasses. More properly, these two glasses in the collection of Union College in Schenectady, New York, are
examples of Tantalus cup or the automatic siphon. When the water level reaches the bend in the internal glass tube, the
siphon starts and the glass empties itself of water. 共Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon
College兲
615
Am. J. Phys., Vol. 78, No. 6, June 2010
M. Campisi and D. H. Kobe
615