Philosophy of Science, 75 (December 2008) pp. 682–696. 0031-8248/2008/7505-0016$10.00
Copyright 2008 by the Philosophy of Science Association. All rights reserved.

682

Boltzmann, Gibbs, and the Concept
of Equilibrium

David A. Lavis†‡

The Boltzmann and Gibbs approaches to statistical mechanics have very different
definitions of equilibrium and entropy. The problems associated with this are discussed,
and it is suggested that they can be resolved, to produce a version of statistical me-
chanics incorporating both approaches, by redefining equilibrium not as a binary prop-
erty (being/not being in equilibrium) but as a continuous property (degrees of equilibrium)
measured by the Boltzmann entropy and by introducing the idea of thermodynamic-like
behavior for the Boltzmann entropy. The Kac ring model is used as an example to test
the proposals.

1. Introduction. The object of study in the Gibbs formulation of statistical
mechanics is an ensemble of systems, and the Gibbs entropy is a functional
of the ensemble probability density function. Equilibrium is defined as
the state where the probability density function is a time-independent
solution of Liouville’s equation. The development of this approach has
been very successful, but its extension to nonequilibrium presents con-
tentious problems.1

To implement the Boltzmann approach (Lebowitz 1993; Bricmont 1995;
Goldstein 2001) the phase space is divided into a set of macrostates. The
Boltzmann entropy at a particular point in phase space is a measure of
the volume of the macrostate in which the phase point is situated. The
system is understood to be in equilibrium when the phase point is in a

†To contact the author, please write to: Department of Mathematics, King’s College,
London WC2R 2LS, U.K.; e-mail: david.lavis@kcl.ac.uk.

‡This is a modified version of the paper presented at the twentieth biennial meeting
of the Philosophy of Science Association, Vancouver, Canada, November 2006. I am
grateful to the organizers for giving me the opportunity to contribute to the conference
and to Roman Frigg for many useful discussions.

1. The most developed program for doing this is that of the Brussels-Austin School
of the late Ilya Prigogine. (For a comprehensive review, see Bishop 2004.)



THE CONCEPT OF EQUILIBRIUM 683

particular region of phase space. The entropy and equilibrium are thus
properties of a single system.

The purpose of this article is to attempt to produce a synthesis of the
Gibbs and Boltzmann approaches, which validates the Gibbs approach,
as currently used in ‘equilibrium’ statistical mechanics and solid state
physics, while at the same time endorsing the Boltzmann picture of the
time evolution of entropy, including ‘the approach to equilibrium’. In
order to do this we need to resolve in some way three questions, to which
the current versions of the Gibbs and Boltzmann approaches offer ap-
parently irreconcilable answers: (a) What is meant by equilibrium? (b)
What is statistical mechanical entropy? and (c) What is the object of study?
The attempt to produce conciliatory answers to a and b will occupy most
of this article. However, we shall at the outset deal with c. As indicated
above, ensembles are an intrinsic feature of the Gibbs approach (see, e.g.,
Prigogine 1994, 8). However we follow the Neo-Boltzmannian view of
Lebowitz (1993, 38) that we “neither have nor do we need ensembles.”
The object of study in statistical mechanics is a single system and all talk
of ensembles can be understood as just a way of giving a relative frequency
flavor to the probabilities of events occurring in that system.

We now describe briefly the dynamics and thermodynamics of the sys-
tem together with the statistical approach of Gibbs. The Boltzmann ap-
proach is described in greater detail in Section 2.

At the microscopic (dynamic) level, the system (taken to consist of N
microsystems) is supposed to have a one-to-one autonomous dynamics

, on its phase space . The system is reversible, meaning thatx r f x Gt N
there exists a self-inverse operator on , such that ′J x � G f x p x rN t

. Then . On the subsets of there is a′ �1f Jx p Jx f p (f ) p Jf J Gt �t t t N
sigma-additive measure , such that (a) is finite, (b) is absolutelym m(G ) mN
continuous with respect to the Lebesque measure on , and (c) isG mN
preserved by ; that is , and measurable . This{f } m(fg) p m(g) G t g O Gt t N
means that there will be no convergence to an attractor (which could in
the dynamic sense be taken as an equilibrium state).

At the phenomenological (thermodynamic) level, equilibrium is a state
in which there is no perceptible change in macroscopic properties. It is
such that a system (a) either is or is not in equilibrium (a binary property),
(b) never evolves out of equilibrium, and (c) when not in equilibrium
evolves toward it.

At the statistical level (in the Gibbs approach), the phase point x �
is distributed according to a probability density function in-G r(x; t)N



684 DAVID A. LAVIS

variant under , meaning that it is a solution of Liouville’s equation.{f }t
At equilibrium the Gibbs entropy is the functional

S [r] :p �k r(x) ln [r(x)]dm (1)G B �
GN

of a time-independent probability density function. Problems arise when
an attempt is made to extend the use of (1) to nonequilibrium situations,
which are now perceived as being those where is time dependent.r

2. The Macroscopic Level–Boltzmann Approach. Here we must introduce
a set of macroscopic variables at the observational level that give moreY
detail than the thermodynamic variables, and a set of macrostates {m}
defined so that: (i) every is in exactly one macrostate denoted byx � GN

, (ii) each macrostate corresponds to a unique set of values for , (iii)m Yx
is invariant under all permutations of the microsystems, and (iv) themx

phase points and are in macrostates of the same size.2 The Boltzmannx Jx
entropy, which is a function on the macrostates, and consequently also a
function on the phase points in , isGN

S (x) p S (m ) :p k ln [m(m )]. (2)B B x B x

This is, of course, an extensive variable, and the quantity of interest is
the dimensionless entropy per microsystem , which for thes :p S /(Nk )B B B
sake of brevity we shall refer to as the Boltzmann entropy. Along a tra-
jectory will not be a monotonically increasing function of time. RathersB
we should like it to exhibit thermodynamic-like behavior, defined in an
informal (preliminary) way as follows:

Definition (TL1): The evolution of the system will be thermodynamic-
like if spends most of the time close to its maximum value, fromsB
which it exhibits frequent small fluctuations and rarer large fluctu-
ations.

This leads to two problems which we discuss in Sections 2.1 and 2.2.

2.1. How Do We Define Equilibrium? Is it possible to designate a part
of as the equilibrium state? On the grounds of the system’s reversibilityGN
and recurrence we can, of course, discount the possibility that such a

2. When is the direct product of the configuration space and the momentum space,GN
and the macrostates are generated from a partition of the one-particle configuration
space, the points and are in same macrostate. However, for discrete-time systemsx Jx
phase space is just configuration space and the points and are usually in differentx Jx
macrostates.



THE CONCEPT OF EQUILIBRIUM 685

region is one which, once entered by the evolving phase point, will not
be exited. As was well understood by both Maxwell and Boltzmann,
equilibrium must be a state which admits the possibility of fluctuations
out of equilibrium. Lebowitz (1993, 34) and Goldstein (2001, 8) refer to
a particular macrostate as the equilibrium macrostate, and the remark by
Bricmont (1995, 179), that “by far the largest volumes [of phase space]
correspond to the equilibrium values of the macroscopic variables (and
this is how ‘equilibrium’ should be defined)” is in a similar vein. So is
there a single equilibrium macrostate? If so it must be that in which the
phase point spends more time than in any other macrostate and, if the
system were ergodic, it would be the largest macrostate (see Sec. 2.2),m Max
with largest Boltzmann entropy. There is one immediate problem asso-
ciated with this. Suppose we consider the set of entropy levels ,{s (m)}B

. Then, as has been shown by Lavis (2005) for the baker’s gas,G m O GN
associated with these levels there may be degeneracies , such that, forq(m)
some with , . The effect of this ism m(m) ! m(m ) m(m)q(m) 1 m(m )Max Max
that the entropy will be likely, in the course of evolution, to spend more
time in a level less than the maximum (see Lavis 2005, Figs. 4 and 5).

Another example, which has been used to discuss the evolution of
Boltzmann’s entropy (see Bricmont 1995, App. 1) and which we shall use
as an illustrative example in this article, is the Kac ring model (Kac 1959,
99).3 Macrostates in this model can be indexed ,k p 0, 1, . . . , (1/2)N
where is the macrostate with up spins and downm (1/2)N � k (1/2)N � kk
spins, giving

N!
Nm(m ) p with m(G ) p 2 . (3)k N([N/2] � k)!([N/2] � k)!

Then, of course, with monotonically decreasing withm p m m(m )Max 0 k
increasing . But although the maximum macrostate is unique,FkF

, and the entropy level corresponding to the largestq(m ) p 2 G k ( 0k
volume of is given, for , by the macrostate pair . It may beG N 1 2 mN �1
supposed that this question of degeneracy is an artifact of relevance only
for small . It is certainly the case, both for the baker’s gas and KacN
ring, that, if is a macrostate which maximizes , although′ ′ ′m m(m )q(m )

3. This model consists of up or down spins distributed equidistantly around a circle.N
Randomly distributed at some of the midpoints between the spins are spin flippers.m
The dynamics consists of rotating the spins (but not the spin-flippers) one spin-site in
the clockwise direction, with spins changing their direction when they pass through a
spin flipper. consists of the points, corresponding to all combinations of the twoNG 2N
spin states, and is decomposable into dynamically invariant cycles. If is even, them
parity of is preserved along a cycle which has maximum size . If is odd, thek N m
parity of alternates with steps along a cycle and the maximum cycle size is .k 2N



686 DAVID A. LAVIS

, , as . So, maybe the union′ ′ ′m(m )q(m ) r� m(m ) s (m ) r s (m ) N r �Max B B Max
of and all the equally sized macrostates with measure can be′m m(m )Max
used as the equilibrium state. To test this possibility take the Kac ring
and consider the partial sum

k k

m(B(N, k)) :p m(m ) p m(m ) � 2 m(m ), (4)� �j 0 j
jp�k jp1

where is the union of all with . Then it is not difficultB(N, k) m j � [�k, k]j
to show that, for fixed ,k

m(B(N, k))
r 0, as N r �. (5)

m(G )N

The proportion of contained within the band of macrostatesG B(N, k)N
decreases with .4 If we want the equilibrium region to satisfy the propertyN
described in the quotation given above from Bricmont (1995), then this
cannot be done by designating a band of a fixed number of macrostates
in this way. To ensure that as becomes large “by far the largest volumesN
[of phase space] correspond to the equilibrium values of the macroscopic
variables” we need to choose a value of increasing with . Thus fork N
example to create an ‘equilibrium band of macrostates’ containing
99.999% of we must choose , for ; , forG k p 22 N p 100 k p 70 N pN

; and for . An entropy profile for this last case1,000 k p 221 N p 10,000
is shown in Figure 1. The putative equilibrium state, representing 99.999%
of is given by the region bounded by the horizontal lines in theG10,000
figure. If the system were ergodic (like the baker’s gas) we would expect
the system to spend almost all of the 10,000 time steps in this region,
whereas in this particular simulation only 4,658 steps satisfied this con-
dition. This is to be expected as the Kac ring is not ergodic but has an
ergodic decomposition into cycles of which this figure represents half a
cycle.5 The proportion of ‘equilibrium states’ will differ between cycles.
So the problems with defining an equilibrium region are:

(i) Just choosing the largest macrostate as the equilibrium region does
not guarantee that this region becomes an increasing proportion of
phase space as increases. In fact the reverse is the case for theN
Kac ring.

4. For later reference we note that, in particular, this result applies to withk p 0
for large . The proportion of phase space in the largest ma-�m(m )/m(G ) � 2/Np N0 N

crostate decreases with .N

5. This is because is odd. The second half of the cycle in which the spins are reversem
has an identical entropy profile.



THE CONCEPT OF EQUILIBRIUM 687

Figure 1. Half a cycle of the plot of against time, for a Kac ring of 10,000 spins.sB
There are spin-flippers, distributed randomly around the ring, and the evo-m p 509
lution is initiated in the low-entropy state . The upper and lower horizontalk p 4,993
lines correspond, respectively, to and .s (m ) p 0.69266408 s (m ) p 0.69168704B Max B 221

(ii) Any choice of a collection of macrostates to represent equilibrium
is:
(a) Arbitrary: leading to an arbitrary division between fluctuations

within and out of equilibrium, as is shown in the profile in Figure
1.

(b) Difficult: as we have shown in the Kac ring. Except for ergodic
systems there is no clear way to make a choice which guarantees
that the system will spend most of its time in equilibrium. The
choice of a region consisting of 99.999% of phase space still
yields an evolution where only about 47% of the points on the
trajectory lie within it.

But why define equilibrium in this binary way? We suggest that the quality
which we are trying to capture is a matter of degree, rather than the two-
valued property of either being in equilibrium or not in equilibrium. We,
therefore, make the following proposal:

Definition (C): All references to a system being, or not being, in
equilibrium should be replaced by references to the commonness of
the state of the system, with this property being measured by (some
possibly-scaled form of ) the Boltzmann entropy.

2.2. We Need Thermodynamic-Like Behavior to Be Typical. By this we
mean that most initial states of the system should lead to thermodynamic-



688 DAVID A. LAVIS

like behavior. Before discussing the dynamic properties needed for this,
we shall refer briefly to the more limited notion of typicality employed
by the Neo-Boltzmannians and contained in the assertion (Lebowitz 1999,
S348) “that will typically be increasing in a way which explains andSB
describes qualitatively the evolution towards equilibrium of macroscopic
systems.” The conditions for this to be the case were first discussed by
Ehrenfest and Ehrenfest-Afanassjewa (1992, 32–34).

Consider a macrostate divided into four parts , , ,(��) (��) (��)m m m m
, where consists of those points in which have evolved from(��) (��)m m m

a smaller macrostate and which evolve into a larger macrostate, with the
other parts defined in a similar way.6 On grounds of symmetry, if andx

are both in , then ; otherwise the macrostates will(��) (��)Jx m m(m ) p m(m )
be in reversal pairs with plus and minus signs interchanged. In order for
forward evolution to a larger macrostate to be typical it must be the case
that the overwhelmingly largest part of is . This was asserted without(��)m m
proof by Ehrenfest and Ehrenfest-Afanassjewa (1992). More recent ar-
guments have been advanced from the point of view that a macrostate is
more likely to be surrounded by larger macrostates or that it is easier to
‘aim at’ a larger, rather than a smaller, neighboring macrostate. Even
accepting this argument, it can at the most explain how, if the state of a
system is assigned randomly to a point in a macrostate, then the subse-
quent first transition is to a larger macrostate. As was pointed out by
Lavis (2005), it gives no explanation for the entropy direction at the next
transition, since the part of the macrostate occupied after the first tran-
sition will be determined by the dynamics. In any case, I wish to argue
that this is a too narrowly defined version of typicality, which should be
applied to thermodynamic-like behavior over the whole evolution.

I have already argued in Section 2.1 that commonness or ‘equilibrium-
ness’ is a matter of degree, and it is clear that thermodynamic-like behavior
is also a matter of degree, for which we need to propose a measure. A
difference between these properties is that commonness is something which
can be assessed at an instant of time, whereas thermodynamic-like be-
havior is a temporally global property assessed over the whole trajectory.

Let be a trajectory in identified (uniquely) by the property thatL Gx N
it passes through the point and let be the proportion of time whichx T (g)x
the phase point evolving along spends in the some .7 DefinitionL g O Gx N
TL1 is an informal qualitative definition of thermodynamic-like behavior,

6. For simplicity we have excluded evolutions between macrostates of the same size.

7. It was shown by Birkhoff (1931) that exists and is independent of the locationT (g)x
of on for almost all (f.a.a.) ; that is, except possibly for a set of -measurex L x � G mx N
zero. From this it follows (see, e.g., Lavis 1977) that is a constant of motionT (g)x
f.a.a. .x � GN



THE CONCEPT OF EQUILIBRIUM 689

for which we need the entropy profile of , not only to be quite close tosB
for most of its evolution but also for fluctuations(s ) :p s (m )B Max B Max

around this value to be fairly small. We, therefore, propose the following
definition:

Definition (TL2): The degree to which the evolution of the system is
thermodynamic-like along is measured by the extent to whichLx

D [s ] :p FAs S � (s ) F, (6)x B B x B Max

and

2� [ ]W [s ] :p A s � As S S , (7)x B B B x x
are small, where

As S :p T (m)s (m), (8)�B x x B
{m}

is the time average of along and is the standard deviations L W [s ]B x x B
with respect to the time distribution.

Of course, it could be regarded as unsatisfactory that two parameters are
used as a measure of the degree of a property and it is a matter of judgment
which is more important. For the Kac ring of 10,000 spins with the entropy
profile shown in Figure 1,

�2 �1D [s ] p 0.58122724 # 10 , W [s ] p 0.31802804 # 10 , (9)x B x B

and, as a comparison, for the same ring with the flippers placed at every
tenth site

D [s ] p 0.20078055, W [s ] p 0.20632198. (10)x B x B

It is clear (and unsurprising) that the random distribution of spin flippers
leads to more thermodynamic-like behavior.

To explore the consequences of TL2 we distinguish between four aspects
of a system:

(i) The number of microsystems and their degrees of freedom, to-N
gether giving the phase space , with points representing micro-GN
states.

(ii) The measure on .m GN
(iii) The mode of division of into the set of macrostates.G {m}N
(iv) The -measure preserving dynamics of the system.m

Having chosen i and ii, the choices for iii and then iv are not unique. In
the case of the baker’s gas (Lavis 2005), is a -dimensional unitG 2NN
hypercube with volume measure. Macrostates are specified by partitioning



690 DAVID A. LAVIS

each square face of the hypercube. With such a setup it would now be
possible to choose all manner of discrete-time dynamics.8

Whether a system is ergodic will be determined by i, ii, and iv and, if
it is,

m(g)
T (g) p t(g), where t(g) :p , G g O G (11)x Nm(G )N

is both the proportion of in and of the time spent in , f.a.a.G g g L �N x
. This will be the case for the baker’s gas but not the Kac ring. ForGN

any specification of i–iii we denote the results of computing ,As SB x
, and using (11) by , , and ; that is to say, weD [s ] W [s ] As S D[s ] W[s ]x B x B B B B

omit the unnecessary trajectory-identifying subscript . If we were ablex
to devise a model with i–iii the same as the Kac ring, butN p 10,000
with an ergodic dynamics, we would have

As S p 0.69261408,B (12)�4 �4D[s ] p 0.49997497 # 10 , W[s ] p 0.70707142 # 10 ,B B

and it is not difficult to show that and are monotonicallyD[s ] W[s ]B B
decreasing functions of . Ergodicity leads to more thermodynamic-likeN
behavior, which becomes increasingly thermodynamic-like with increasing

.9 This behavior is also typical, since it occurs f.a.a. .N L � Gx N
Of course, the results (12) are not simply dependent on the putative

ergodic dynamics of the system, but also on the way that the macrostates
have been defined. If the time average of is to be close tos (x) (s )B B Max
and if the fluctuations in are to be small, then most of must lies (x) GB N
in macrostates with close to . In the case of the Kac ring,s (m) (s )B B Max
with , 99.98% of lies in macrostates withN p 10,000 G s (m) 110000 B

. However, of course, the Kac ring, although not ergodic, gives0.999(s )B Max
every appearance, at least in the instances investigated (see Fig. 1), of
behaving in a thermodynamic-like manner. Although ergodicity, with a
suitable macrostate structure, is sufficient for thermodynamic-like behav-
ior, it is clearly not necessary.

Consider the case where can be ergodically decomposed; meaningGN
that

G p w G , (13)N a
{a}

where is invariant and indecomposable under . Then the time spentG {f }a t

8. Discrete time because of the absence of a momentum component in the phase space.
(See note 2 above.)

9. This latter result can be proved for any division of into macrostates with theGN
measure given by a combinatorial quantity like (3).



THE CONCEPT OF EQUILIBRIUM 691

Figure 2. Plot of for the cycle of the Kac ring shown in Figure 1 togetherT (m ) aa k
with the corresponding curve of for a system with ergodic dynamics.t(m )k

in isg O GN

m(g ∩ G )aT (g) p , f.a.a. L � G , (14)a x am(G )a
and we, henceforth, identify a time average along a trajectory in usingGa

with also replacing the subscript in (6) and (7). In the case of. . .A S a xa
the Kac ring the index labels the cycles which form the ergodic decom-a
position of . For a particular cycle, such as that shown in Figure 1,GN

is obtained simply by counting the number of times the phase pointT (m)a
visits each macrostate in a complete cycle. The data are then used to
compute the results given in (9). A plot of against the microstateT (m )a k
index is shown in Figure 2 with a comparison made with , thek t(m )k
corresponding curve for a system with ergodic dynamics. This gives a
graphic illustration of the suggestion that ergodic systems, with a suitable
choice of macrostates, are likely to be more thermodynamic-like in their
behavior than nonergodic systems. It might also be speculated that –�
ergodic systems (Vranas 1998) show increasingly thermodynamic-like be-
havior with decreasing .�

Another advantage of an ergodic system is that, f.a.a. , the levelL � Gx N



692 DAVID A. LAVIS

of thermodynamic-like behavior will be the same. This contrasts with an
ergodic decomposition characterized by (13), where it is possible for dif-
fering levels of thermodynamic-like behavior to be exhibited within dif-
ferent members of the decomposition. To be precise, take small positive

and and regard behavior along a trajectory as thermodynamic-like� �D W
if and only if both and . Let be the union of(T)D [s ] ! � W [s ] ! � Ga B D a B W N
all in which the behavior is thermodynamic-like with .(A) (T)G G p G \Ga N N N

In the discussion of the Boltzmann approach, reference to probabilities
has so far been avoided. To complete the discussion in this section and
to relate the arguments to the Gibbs approach we shall need (see Lavis
2005) to introduce two sets of probabilities for a system with the ergodic
decomposition (13). The first of these is . Thenp :p Pr (x � L ), G Ga a a
thermodynamic-like behavior will be typical for the system if

p K 1. (15)� a
(A)G OGa

Thus we have two levels of degree, the first represented by the choices of
and and the second concerning the extent to which (15) is satisfied.� �D W

3. Reconciling Gibbs and Boltzmann. As we saw in Section 1, the Gibbs
approach depends on defining a probability density function on ,r(x) GN
for a system ‘at equilibrium’. Thus we must address more directly the
question of probabilities. Assuming the ergodic decomposition (13), we
take the time-average definition of von Plato (1989), for which

Prob(x � gFx � G ) :p T (g), G g O G , (16)a a N

where is given by (14). Thus the probability density function forT (g)a
isGa

1/m(G ) x � Ga ar (x) p , (17)a {0 otherwise
from which we have

r(x) p ED r (x) :p r (x)p . (18)�a a a
{a}

If we assume that all points of are equally likely, then on Bayesian/GN
Laplacean grounds, and consonant with the approach of Bricmont (2001),
we should choose

p p m(G )/m(G ), (19)a a N

giving, from (18), , which is the microcanonical distributionr(x) p 1/m(G )N



THE CONCEPT OF EQUILIBRIUM 693

and for which, from (1),

s :p S /(Nk ) p ln [m(G )] p s (G ). (20)G G B N B N

It also follows, from (2), (8), (11), and (19) that

ED As S p As S. (21)B a B

Lavis (2005) has proposed a general scheme for relating a phase function
, defined on to a macro-function defined on the macrostatesf x � G FN

and then to a thermodynamic function . The first step is to course{m} F
grain over the macrostates to produce .10 The second step is tof (x) F(m)
define the thermodynamic variable along the trajectory as . InF L AFSx x
the case of the Boltzmann entropy, which is both a phase function and
a macro-function, the first step in this procedure is unnecessary since it
already, by definition, course grained over the macrostates. Then we pro-
ceed to identify the dimensionless thermodynamic entropy per microsys-
tem with . In the case of a system with an ergodic decomposition thisAs SB x
definition would yield a different thermodynamic entropy for eachsa
member of the decomposition, with, from (6),

s :p As S p (s ) � D [s ]. (22)a B a B Max a B

In the case where the behavior is thermodynamic-like in , differs fromG sa a
by at most some small and, if (15) holds, this will be the case(s ) �B Max D

for measurements along most trajectories. In the case of the Kac ring
with and the trajectory investigated for Figures 1 and 2 theN p 10,000
actual difference is given in (9), a value which is likely to decrease with
increasing .N

It is often said that in “equilibrium [the Gibbs entropy] agrees with
Boltzmann and Clausius entropies (up to terms that are negligible when
the number of particles is large) and everything is fine” (Bricmont 1995,
188). Interpreted within the present context this means that the good
approximation , for the entropy per microsystem of a system fors (m )B Max
which thermodynamic-like behavior is typical, can be replace by s pG

. The advantage of this substitution is obvious, since the first ex-s (G )B N
pression is dependent on the division into macrostates and second is not.
However, a little care is needed in justifying this substitution. It is not
valid because, as asserted in the quotation from Bricmont (2001, 3),

occupies an increasing proportion of as increases. Indeed, wem G NMax N
have shown for the Kac ring that the reverse is the case. That proportion
becomes vanishingly small as increases. However, the required substi-N

10. It is argued that is a good approximation to for the phase functions relevantF f
to thermodynamics since their variation is small over the points in a macrostate.



694 DAVID A. LAVIS

tution can still be made, since for that model

s (m ) ln (N )B Max � 1 � , as N r �. (23)
s (G ) 2N ln (2)B N

Although it may seem that the incorrect intuition on the part of Bricmont
(2001) concerning the growth in the relative size of the largest macrostate,
leading as it does to the correct conclusion with respect to entropy, is
easily modified and of no importance, we have shown in Section 2.1 that
it has profound consequences for the attempt to define equilibrium in the
Boltzmann approach.

It should be emphasized that the Gibbs entropy (20) is no longer taken
as that of some (we would argue) nonexistent equilibrium state, but as
an approximation to the true thermodynamic entropy which is the time
average over macrostates of the Boltzmann entropy. The use of a time-
independent probability density function for the Gibbs entropy is not
because the system is at equilibrium but because the underlying dynamics
is autonomous.11 The thermodynamic entropy approximated by the Gibbs
entropy (20) remains constant if the phase space remains unchanged but
changes discontinuously if a change in external constraints leads to a
change in . An example of this, for a perfect gas in a box when a partitionGN
is removed, is considered by Lavis (2005), who shows that the Boltzmann
entropy follows closely the step change in the Gibbs entropy.

4. Conclusion. In our program for reconciling the Boltzmann and Gibbs
approaches to statistical mechanics we have made use both of ergodicity
and ergodic decomposition, and there is deep (and justified) suspicion of
the use of ergodic arguments, particularly among philosophers of physics.
Earman and Rédei (1996, 75) argue that “ergodic theory in its traditional
form is unlikely to play more than a cameo role in whatever the final
explanation of the success of equilibrium statistical mechanics turns out
to be.” In its ‘traditional form’ the ergodic argument goes something like
this: (a) Measurement processes on thermodynamic systems take a long
time compared to the time for microscopic processes in the system and
thus can be effectively regarded as infinite time averages. (b) In an ergodic
system the infinite time average can be shown, for all but a set of measure
zero, to be equal to the macrostate average with respect to an invariant
normalized measure which is unique.12 The traditional objections to this
argument are also well known: (i) Measurements may be regarded as time

11. A nonautonomous dynamic system will not yield a time-independent solution to
Liouville’s equation.

12. In the sense that it is the only invariant normalized measure absolutely continuous
with respect to the Lebesque measure.



THE CONCEPT OF EQUILIBRIUM 695

averages, but they are not infinite time averages. If they were one could
not, by measurement, investigate a system not in equilibrium. In fact,
traditional ergodic theory does not distinguish between systems in equi-
librium and not in equilibrium. (ii) Ergodic results are all to within sets
of measure zero and one cannot equate such sets with events with zero
probability of occurrence. (iii) Rather few systems have been shown to
be ergodic. So one must look for a reason for the success of equilibrium
statistical mechanics for nonergodic systems and when it is found it will
make the ergodicity of ergodic systems irrelevant as well.

This use of ergodicity differs substantially from that described above,
and it thus escapes wholly or partly the strictures applied to it. In respect
of the question of equilibrium/nonequilibrium I argue that the reason this
does not arise in ergodic arguments is that equilibrium does not exist.
The phase point of the system, in its passage along a trajectory, passes
through common (high entropy) and uncommon (low entropy) macro-
states and that is all. So we cannot be charged with ‘blurring out’ the
period when the system was not in equilibrium. The charge against ergodic
arguments related to sets of measure zero is applicable only if one wants
to argue that the procedure always works; that is that non-thermody-
namic-like behavior never occurs. But we have, in this respect, taken a
Boltzmann view. We need thermodynamic-like behavior to be typical, and
I have proposed conditions for this to be the case. But I admit the pos-
sibility of atypical behavior occurring with small but not-vanishing prob-
ability. While the class of systems admitting a finite or denumerable er-
godic decomposition is likely to be much larger than that of the purely
ergodic systems, there remains the difficult question of determining general
conditions under which the temporal behavior along a trajectory, mea-
sured in terms of visiting times in macrostates, approximates, in most
members of the ergodic decomposition, to thermodynamic-like behavior.

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