Entropy and time
Vinay Ambegaokar and Aashish A. Clerk
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853

共Received 9 April 1999; accepted 9 June 1999兲
The emergence of a direction of time in statistical mechanics from an underlying
time-reversal-invariant dynamics is explained by examining a simple model. The manner in which
time-reversal symmetry is preserved and the role of initial conditions are emphasized. An extensionof the model to finite temperatures also is discussed. © 1999 American Association of Physics Teachers.

I. INTRODUCTION
The second law of thermodynamics is usually stated as theinequality ⌬S⭓0 for an isolated system, where S is the entropy. Implicit in thermodynamics is thus a direction of time
determined by the evolution toward equilibrium of a macroscopic system with no external influences. The extent towhich this notion emerges from statistical mechanics basedon an underlying time-reversal-invariant dynamics is the
topic of this paper.
The point of view taken here is not controversial; it has
been accepted since the work of Boltzmann1 was understood.
Including the topic in this special issue may thus seem unnecessary. However, our impression is that many undergraduate 共and even many graduate兲 courses do not cover thisthought-provoking topic adequately. The reason may be thatthe message can be lost in subtleties described by words suchas Stosszahlansatz, Umkehreinwand, and Wiederkehreinwand, and by discussing the topic in the context of Boltzmann’s equation, Liouville’s theorem, and coarse graining.
An alternative is to discuss simple concrete models. We willconsider one that is easy to simulate and has a long history:
the 1907 double-urn model of Paul and Tatiana Ehrenfest.2
Our treatment is based in part on Chapter 10 of Ref. 3.
The outline of the paper is as follows. After a brief general
discussion, the way in which a direction of time follows fromthe statistics of large numbers is illustrated using the Ehrenfest model 共in its dog–flea version兲. The difference between
time symmetric fluctuations and the time evolution of a macroscopically identifiable nonequilibrium initial condition isemphasized. Calculations are done for a single system andfor an ensemble of systems, the latter being described by aMarkovian equation. With the passage of time 共no pun intended兲, the model has become more topical than its conceivers could have imagined. Here it is used to describe the approach to equilibrium of two-level quantum systems such as
spins. Temperature is introduced into the model via a METROPOLIS algorithm, and the approach to equilibrium at constant temperature is discussed, including a populationinversion 共negative temperature兲 initial condition. To ourknowledge, the Ehrenfest model has not been used in thisway, especially as regards the introduction of temperature.
II. BACKGROUND
From the point of view of thermal physics, the state of an
isolated physical system with many degrees of freedom is
specified by its energy and other macroscopic parameterssuch as volume and magnetization. The assumption of manydegrees of freedom implies a dense spectrum of excitations,and thus a very large number of microscopic states in a small
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Am. J. Phys. 67 共12兲, December 1999

energy interval consistent with the given macroscopic parameters. The starting point of a statistical analysis of a mechanical system is the enumeration 共allowed in principle byboth quantum and classical mechanics兲 of these microscopic
states. The fundamental postulate of equilibrium statisticalmechanics is that each of them is equally probable in equilibrium; the logarithm of their number is the entropy associated with the thermodynamic equilibrium of the isolated system.
We are interested in how equilibrium is reached. A reasonable, but incorrect, expectation is that if the microscopic
states are not equally likely at some instant, the evolution
will be toward a situation in which they are. If the number ofmicrostates explored by the system increased with time, its
logarithm or entropy would also increase, giving a statisticalunderpinning to the rule that an increase of entropy characterizes spontaneous processes in isolated systems.
The trouble with this too simple idea is that not every oneof the microscopic configurations consistent with a given
macroscopic nonequilibrium state tends, under the action of
the laws of mechanics, toward equilibrium, although most ofthem do. As a result the strict inequality in the second law
has to be replaced by a statement of overwhelming likelihood in statistical mechanics, thereby allowing the latter tobe consistent with the time-symmetric equations of motion.
A tiny loophole now opens, with the consequence that there
is no longer a strict logical connection between the directionof time and the increase of entropy: One can never rule out
the overwhelmingly unlikely possibility that a low entropy
initial condition is a time-symmetric giant fluctuation caught
midway. To close the loophole, we have to make the extremely reasonable hypothesis that macroscopic deviations
from equilibrium are due to externally imposed initial conditions.
The word ‘‘overwhelming’’ is not being used lightly. To
illustrate its meaning, consider the ratio of the number ofmicroscopic configurations for a gas filling all or 99.99% of
a container. If we treat the N molecules of the gas as very
weakly interacting, an estimate for this ratio is 0.9999⫺N ,because each molecule has 0.01% fewer available states inthe smaller volume. For a liter, N is order 1022, so that thelogarithm of the ratio is 1018. The reciprocal of the ratio isthus very small indeed. Yet, this unimaginably tiny number
is the probability that a gas in equilibrium in the entire container would be found to be occupying 99.99% of its volume,and thus to have undergone a small but macroscopic entropyreducing spontaneous fluctuation.
III. THE DOG–FLEA MODEL
Treating the evolution of a reasonably realistic statisticalsystem is technically difficult. Even for a weakly interacting© 1999 American Association of Physics Teachers1068

Fig. 1. Early time development of the number of fleas on Anik due torandom jumps. The horizontal lines indicate two standard deviations aboveand below the mean.

Fig. 2. Long time behavior of fleas on Anik showing fluctuations in equilibrium. The horizontal lines indicate two standard deviations above andbelow the mean.

gas, collisions cannot be ignored, because they are themechanism that shuffles a particular molecule between its
states of motion. The approach to equilibrium thus depends
on the details of the motion, and is a less general phenomenon than equilibrium itself. In order not to lose the woods
for the trees, it is useful to look for simple but illustrative
examples. One such example is a collection of ‘‘two-level’’systems, that is, systems described by two possible outcomes.
As a physical realization of this model, one could consider
a collection of weakly interacting quantum spin 21 systems
each of which has up and down states. A more whimsicalillustration is based on the model proposed in Ref. 2: Consider a system consisting of a subsystem of 50 fleas whose‘‘states’’ are residence on dog A or dog B, which we shallcall Anik and Burnside, sleeping side by side. To simulate
molecular agitation, we suppose that the fleas jump back andforth between the dogs. Now we need something that plays
the role of the rest of the system. Let us suppose that the fleas
are each equipped with a number, and have been trained tojump when their number is called. The ‘‘environment’’ agitating the fleas, which is like a heat reservoir, will be something that calls out numbers at random, and our closed system will be the fleas and the reservoir.
To simulate the approach to equilibrium, it is necessary tostart from a configuration that almost never occurs in the
maximally random state of affairs. Suppose that in the beginning Anik has no fleas at all. We agitate the fleas byhaving a computer generate random numbers between 1 and
50 and transferring the flea with this number from one dog tothe other. At every step we record the total number 共between
0 and 50兲 of fleas on Anik, but not their labels. This way has
the practical advantage that we do not have to keep track of
the 2 50 ways of assigning 50 labeled fleas to the two dogs. It
also means that we are following only the ‘‘macrostate.’’A typical sequence is shown in Figs. 1 and 2. In the firststep it is certain that the number called will belong to a flea
on Burnside. In the second step the probability of this happening again is 49/50. Thus, there initially seems to be asteady march toward an equal partition of the fleas between
the dogs. The early time development is shown in Fig. 1.

After about 100 steps we reach a situation where sometimes
Anik and sometimes Burnside has more fleas. In this region
we would expect that every one of the 2 50 configurations
mentioned above would be equally likely. This expectationtranslates into a binomial distribution corresponding to therepeatable random event of tossing 50 fair coins, namely

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Am. J. Phys., Vol. 67, No. 12, December 1999

P共 m 兲⫽

冉 冊

1 50
,
2 50 m共1兲

where P(m) is the probability of the macrostate with m fleas
50
on Anik, and ( m) is the combinatorial coefficient.
An examination of Fig. 2 confirms this expectation. The
horizontal lines have been drawn to include two standard
deviations on either side of the mean, which corresponds
approximately to the 95% range for the distribution Eq. 共1兲.
共The standard deviation is 冑50/2⬇3.5.兲 Our eye tells us that,except for the initial transient, fluctuations outside this range
are indeed rare.
We can be more quantitative. Figure 3 is a histogram of
the relative durations of the possible outcomes, constructedfrom Fig. 2 with the first 100 steps omitted. Superimposedon the histogram is the binomial distribution. The agreement
is very good.4 The dance of the fleas in Fig. 2 has thus very
quickly forgotten its unusual starting point and become theendless jitterbug of ‘‘equilibrium,’’ in which an event as unlikely as a flea-less dog simply never happens again without
outside intervention.
Figure 2 also illustrates the role of motion reversal. The
model is time-reversal invariant because a string of random
numbers in reverse order is just as random. After the first 100
or so steps, the evolution has no sense of time. If we were to
expand the region near one of the reasonably large fluctuations away from the mean, we would find that there is no
characteristic feature of the buildup preceding the maximum
deviation to distinguish it from the time reverse of the decay
following the maximum. There is also no conclusive argument to rule out the possibility that the start of the trace
shown in Fig. 1 has captured a truly giant fluctuation midway. Of course, we know that the figure was not produced inthis way.
V. Ambegaokar and A. A. Clerk

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Fig. 3. Histogram generated from Fig. 2 with the first 100 time steps omitted.

Fig. 4. The Boltzmann entropy associated with Fig. 2, showing fluctuations
in equilibrium.

The point of the previous paragraph is worth reemphasizing. The time asymmetry expressed in the second law is not
simply the consequence of applying time-symmetric microscopic dynamics to systems having many degrees of freedom. Time-reversal symmetry is indeed preserved in our system. If at some instant the system is in a highly improbablestate 共that is, no fleas on Anik兲, it is overwhelmingly likelythat it will be in a more evenly distributed state at some latertime. However, if the improbable state were due to a giant
fluctuation, precisely the same argument could be made regarding the prior history of the system—it would then be
overwhelmingly likely that at earlier times the system wasalso in a more evenly distributed state. In this sense, there isperfect symmetry between past and future.
The notion of a statistical ‘‘arrow of time’’ thus depends
on the added ingredient of imposed initial conditions. When
we see a system in a highly unlikely state, we justifiably
assume that this state is the result of a prepared starting condition, and not of an overwhelmingly improbable fluctuationfrom equilibrium. As has been particularly emphasized byPeierls,5 this setting of initial conditions at some specified
time breaks the symmetry between past and future.
In fact it is virtually impossible to wait long enough forthe initial configuration in Figs. 1 and 2 to occur as a fluctuation in equilibrium, where it has a probability of 2 ⫺50. To
have a reasonable chance of witnessing such a fluctuation,
we would have to allow a number of time steps approximately equal to the reciprocal of this probability—about
1015—to elapse. Thus, to recover the unlikely configurationof a totally clean Anik by random shuffling of fleas between
equally dirty dogs, even for this very small system of 50
fleas, we would need a plot roughly two hundred thousand
million times as long as Fig. 2, which extends for only 5000
time steps. Because Fig. 2 is about 5 cm wide, the length of
the required trace would be about 10 mil km. In comparison,
the distance to the moon is only about 400 000 km. The lawof large numbers is at work, here making an unlikely event
overwhelmingly unlikely. Though not logically certain, it isroughly 99.999 999 999 999 999% probable that the time in
Fig. 1 is running in the direction of increasing disorder.
Ehrenfests’ dogs bring into focus the essential characteristics of time in statistical mechanics. 共i兲 A starting point

macroscopically distinct from equilibrium is overwhelmingly
likely to evolve to greater disorder, that is, toward equilibrium. 共ii兲 In equilibrium, fluctuations have no sense of time.
共iii兲 Giant fluctuations from equilibrium to extremely unlikely states are extremely rare. 共iv兲 A statistically determined direction of time follows from the assumption that asystem in a highly unlikely ordered state has been so prepared by external influences. Even for the rather small system we considered, these uses of the words ‘‘overwhelmingly’’ and ‘‘extremely’’ are very conservative.
The word entropy has not appeared in this section. As amatter of fact, there is more than one way to introduce that
notion here, as will be seen in more detail in Sec. IV. The
essential point can be made by noting that the states with nfleas on Anik are ‘‘macrostates,’’ each allowing for ( 50n ) assignments of distinct fleas or ‘‘microstates.’’ We could simply call the logarithm of the latter number, the entropy, that
is, S(n)⫽ln(50n ). The combinatorial coefficients have a maximum half way, at n⫽25, and become smaller in either direction. In Fig. 4, the values of n plotted in Fig. 2 have been
converted to a plot of entropy versus time steps using thisrule. As in Fig. 2, S starts from zero, because S(0)⫽ln(500 )⫽ln 1⫽0, and has fluctuations.
This entropy, sometimes called the Boltzmann entropy,
can be associated with a single time trace such as in Fig. 2.
Although it fluctuates in equilibrium, the fluctuations diminish as the size of the system increases. In a sufficiently large
system the Boltzmann entropy increases steadily as equilibrium is approached.
Note that we have ignored any contribution to the entropyof the closed system from the reservoir which is responsible
for the hopping of the fleas. This assumption is justified here
because energy has not entered into our considerations, making the model slightly artificial. It is probably best to think of
the reservoir as having a very high temperature 共in energy
units兲 compared to the characteristic energies of the subsystem. As a result, heat exchanges with the subsystem occur
with no change in the entropy of the reservoir, and only theflea entropy changes with time. The introduction of energy
and temperature in Sec. V will lead to an interesting difference.

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V. Ambegaokar and A. A. Clerk

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IV. GIBBS ENTROPY
The dog–flea model is simple enough to allow the solutionof several other interesting problems in time-dependent statistical mechanics. We first reexamine the assignment of entropy to our subsystem of fleas. The usual expression for theentropy in statistical mechanics isS⫽⫺

兺i P i ln P i ,共2兲

where the sum is over microstates labeled by the index i and
P i is the probability of i. This entropy is associated with a
distribution describing an ensemble of systems, whereas theBoltzmann entropy introduced earlier is defined for the macroscopic time development of a single system. If there are M
equally likely microstates, each of the P i s would equal 1/M ,and Eq. 共2兲 reduces to ln M. The Boltzmann entropy has
exactly this form if the macrostate with n fleas has ( 50n )equally likely microstates. As we saw, the equilibrium Boltzmann entropy fluctuates for the subsystem plus reservoir.
It is possible to assign a constant entropy to equilibrium. A
system of 50 two-level systems at a temperature much higher
than the level spacing is commonly assigned an entropy of
50 ln 2⫽34.657. Expression 共2兲 gives this result if each of
the 2 50 microstates is taken to be equally probable. As wesaw from Fig. 3, the fluctuations in Figs. 1 and 2 are an
expression of equal likelihood of all these microstates.
Even when the probabilities P(m) of the macrostates corresponding to m fleas on Anik are not given by Eq. 共1兲, the50probabilities of the equally likely ( m) microstates i(m) associated with m are
P i(m) ⫽ P 共 m 兲

冒冉 冊

50
.
m

共3兲

If we substitute Eq. 共3兲 into Eq. 共2兲 and do the sum overi(m), we obtain
50

S⫽⫺

50

冉 冊
50

兺 P 共 m 兲 ln P 共 m 兲 ⫹ m⫽0兺 P 共 m 兲 ln m .

m⫽0

共4兲

The second term on the right-hand side of Eq. 共4兲 arises, as
was just shown, from the fact that a macrostate having mfleas on Anik has an additional contribution to the entropy
coming from the equally likely microstates which make up
the macrostate. If expression 共1兲 is substituted into Eq. 共4兲,the result is the previously mentioned 50 ln 2. We shall callthis new entropy the Gibbs entropy, because it is analogous
to the entropy in Gibbs’s canonical ensemble.
To associate a Gibbs entropy with the early, and consequently nonequilibrium, part of the time development wehave been discussing, we need more information than the
single time trace we have been discussing. Because this entropy is a property of a distribution, we need to assign probabilities to every time step of the process, which means that
we have to contemplate an ensemble of subsystems and define probabilities in terms of occurrences in the ensemble.
One way to proceed would be to create a very large numberof traces such as the one in Fig. 1, all of them starting withthe same configuration. Because the sequence of random
numbers would be different in each run, these traces would
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Am. J. Phys., Vol. 67, No. 12, December 1999

Fig. 5. The probabilities of fleas on Anik with the passage of time.

differ from one another. At any given time, we could calculate a histogram like Fig. 3.
Obtaining reliable distributions by this method would require a very large number of runs. Fortunately, there is amuch simpler way of implementing the idea, which does not
require a random number generator. Let us calculate a distribution function P(m), with m running from 0 to 50, whichchanges from step to step, and reflects the random transfer offleas from dog to dog. At the start of the process we know
with certainty that there are no fleas on Anik. In the languageof probability, the distribution at t⫽0 is P 0 (0)⫽1,P 0 (1)⫽ P 0 (2)⫽¯⫽ P 0 (50)⫽0. 共To indicate the time we need
another label, which we shall write as a subscript.兲 Now weargue that the probability distribution at time t determines
the probability distribution at time t⫹1. The assumption thatthe fleas are being called at random implies thatP t⫹1 共 m 兲 ⫽m⫹1
50⫺ 共 m⫺1 兲
P t 共 m⫹1 兲 ⫹P t 共 m⫺1 兲 .
50
50
共5兲

Equation 共5兲 can be understood by saying it in words. Anikcan have m fleas at time t⫹1 either because she had m⫹1 attime t and one jumped off, which has a probability proportional to m⫹1, or because she had m⫺1 and one jumped on,which has a probability proportional to the 50⫺(m⫺1)⫽51⫺m fleas that were on Burnside at time t.
We can write a program to develop the distribution corresponding to the initial certainty forward in time using Eq.
共5兲. However, there is one artificiality in this time evolution:
At odd 共even兲 times only odd 共even兲 numbers of fleas can beon Anik. This artificiality can be remedied by averaging Eq.
共5兲 over two forward steps. The resulting evolution is shownin Fig. 5.
The three-dimensional plot in Fig. 5 is obtained by stacking together the distributions at successive times. It veryclearly shows the initial certainty evolving to a
distribution—which, not surprisingly, can be shown to be thebinomial in Eq. 共1兲—with the outer regions in the range ofpossibilities being extremely unlikely. At each step we cancalculate an entropy using Eq. 共4兲. The result, shown in Fig.
6, shows that the entropy rises steadily from zero to 50 ln 2.
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Fig. 6. Entropy of Anik’s fleas computed by averaging over many trials, via
the time-dependent probabilities generated via Eq. 共5兲.

By imagining that the system can be restarted at will, we
have, as in Sec. III, insisted on the possibility of the external
imposition of an initial condition that is overwhelmingly unlikely in equilibrium. Time-symmetric dynamics applied tosuch an initial condition is overwhelmingly likely to evolve
toward equilibrium.5 If it were possible to repeatedly arrange
for such a state to be a final condition, we could use abackwards-in-time evolution equation relating the distribution function at time t⫺1 to the distribution function at timet. This equation would be identical to Eq. 共5兲, except for the
replacement of t⫹1 by t⫺1:
P t⫺1 共 m 兲 ⫽

m⫹1
51⫺m
P t 共 m⫹1 兲 ⫹P t 共 m⫺1 兲 .
50
50

共6兲

Equation 共6兲 would predict the opposite of what is shownin Fig. 5—as one moved back in time through the history of
the system, the entropy would increase monotonically. But
such repeated occurrences of low entropy states as fluctuations in equilibrium are unimaginable. Such states do not
typically arise in this fashion, nor can we arrange for them todo so. Time asymmetry in this context thus originates
through our use of Eq. 共5兲 and rejection of Eq. 共6兲.5

V. ENERGY AND TEMPERATURE
Up to now, the word temperature has only appeared at theend of Sec. III where it was argued that the standard Ehrenfest model describes equilibration at high temperature. We
have confirmed this argument by showing that the entropy
evolves to the situation in which all microscopic configurations are equally likely. It is, however, not difficult to introduce temperature in this context. Suppose that Anik iscleaner than Burnside, providing a less friendly environment
for fleas. We may model this environment by assuming anenergy cost ⑀ to be paid by a flea jumping from Burnside toAnik. Let the fleas be at an effective temperature T 共in energy units兲, and define ⌬⫽ ⑀ /T. We argue that Eq. 共5兲 should
be changed to1072

Am. J. Phys., Vol. 67, No. 12, December 1999

Fig. 7. Evolution of the Gibbs entropy for various temperatures, startingfrom a state where all fleas are on the ‘‘clean’’ dog Anik.

P t⫹1 共 m 兲 ⫽

m⫹1
50⫺m
P t 共 m⫹1 兲 ⫹关 1⫺e ⫺⌬ 兴 P t 共 m 兲50
50⫹50⫺ 共 m⫺1 兲 ⫺⌬e P t 共 m⫺1 兲 .
50

共7兲For the new conditions, we expect that in equilibrium any
particular flea will spend more time on 共dirty兲 Burnside thanon 共clean兲 Anik. This expectation is implemented in Eq. 共7兲,
which implies that when the number of one of the fleas on
Anik is called, it jumps to Burnside with probability unity
共term 1兲, but if one of the fleas on Burnside is called, it either
stays put with probability 1⫺e ⫺⌬ 共term 2兲, or jumps withprobability e ⫺⌬ 共term 3兲.
Making the jump-probability from Burnside to Aniksmaller than the reverse process by the factor e ⫺⌬ does infact achieve equilibrium in the steady state. In equilibrium, attemperature T, the probability p of a flea being on Anik, andthe probability 1⫺ p of one being on Burnside should begiven by the Gibbs distribution
p⫽

e ⫺⌬
,
1⫹e ⫺⌬1⫺ p⫽

1.
1⫹e ⫺⌬

共8兲

For 50 fleas the equilibrium probability distribution shouldbe the binomial corresponding to 50 tosses of an unfair coinwith outcome probabilities p and 1⫺ p:
P eq共 m 兲 ⫽

冉 冊

50 m
p 共 1⫺ p 兲 50⫺m .
m

共9兲

It can be verified that Eqs. 共8兲 and 共9兲 are a stationary solution of Eq. 共7兲, namely that substituting this form on the right
reproduces it on the left. In short, the effective temperatureof the fleas determines how many are willing to put up with
Anik’s cleanliness. In the high temperature limit, ⌬Ⰶ1, Eq.
共5兲 is recovered. At very low temperatures, ⌬Ⰷ1 and few
fleas leave the snug comfort of Burnside.
Several interesting and informative computations can be
performed with the evolution equation—Eq. 共7兲. We will
focus on one which we find particularly illuminating. Shownin Fig. 7 are three entropy versus time traces, each at a different temperature but with the same initial condition of all
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VI. CONCLUSION

Fig. 8. Evolution of the free energy F 共in units of ⑀兲 for various temperatures, starting from a state where all fleas are on the ‘‘clean’’ dog Anik.

50 fleas on the ‘‘clean’’ 共energetically unfavorable兲 dog
Anik. The curves have been generated by computing theGibbs entropy 共4兲 at successive time steps. We observe thatat low temperatures, the entropy of the system does not increase monotonically in time—after a certain critical time, it
actually starts to decrease. Have we managed to violate thesecond law?
A little thought shows that there is no violation. The second law requires only that the total entropy of the dog–flea
system plus reservoir increase with time. The entropy of thereservoir is insensitive to changes in the configuration of thefleas only at temperatures much greater than the energy cost⑀. In general, changes in entropy and energy of the reservoir
at temperature T are related bydS res⫽dU
dU res
⫽⫺
,
T
T

共10兲

where the last equality is the result of conservation of energy, and U is ⑀ times the number of fleas on Anik. 共Notethat the energy transfer is at at fixed ⑀, which implies that nowork is done.3,6兲 Using Eq. 共10兲, an increase of the total
entropy translates as usual into a decrease of the Helmholtz
free energy F of the flea subsystem, defined by F⫽U⫺TS, where T is the temperature of the reservoir and S isthe entropy defined in Eq. 共4兲. In Fig. 8, we plot the time
evolution of the free energy for the same initial conditions
and temperatures used in Fig. 7. We see that in all cases, thefree energy decreases monotonically with time.
Note that the initial condition for Figs. 7 and 8 共all fleas on
Anik兲 corresponds to having a negative temperature for thedog–flea system. Consequently, reducing the internal energy
共moving fleas from Anik to Burnside兲 initially increases theentropy above its equilibrium low temperature value.

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We have demonstrated that an understanding of time instatistical mechanics can be obtained by carefully examining
the simple Ehrenfest dog–flea model. The model has thevirtues of offering qualitative insights and yielding easily toquantitative analysis. Our study has emphasized the mannerin which time-reversal invariance is maintained in the model,
and the role of initial conditions in establishing a direction oftime. We have also shown that the model can be extended tofinite temperatures, where it may be used to explore interesting issues.
Finally, we list some suggestions for further reading. Excellent elementary discussions are to be found in Ref. 7. The
subject is also treated in many textbooks accessible to advanced undergraduates.8 Whereas the topic is often underplayed in courses on thermal physics, the opposite may betrue in specialized books. Several thought-provoking articles
as well as discussions of the deep implications of the ideas
presented here are to be found in Ref. 9.
Note added in proof. Our colleague Ben Widom remarksthat there are ‘‘purists’’—among whom he does not include
himself—who think that the Ehrenfest model is not a firstprinciples explanation of irreversibility because there is a‘‘stochastic element’’ in the model, which makes it ‘‘notdeterministic, as real dynamics is...’’ To any such purists
among our readers, we point out that our implementation of
the model uses computer generated pseudo-random numbers
which are completely deterministic. 共See Chapter 11 of Ref.
3 for an elementary introduction to deterministic chaos.兲
ACKNOWLEDGMENT
This work has been partially supported by the National
Science Foundation under Grant No. DMR-9805613.
1

L. Boltzmann, ‘‘On Certain Questions of the Theory of Gases,’’ Nature
共London兲 51, 413–415 共1895兲.
2
P. and T. Ehrenfest, ‘‘Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem,’’ Phys. Z. 8, 311–314 共1907兲.
3
V. Ambegaokar, Reasoning about Luck: Probability and Its Uses in Physics 共Cambridge U.P., Cambridge, 1996兲.
4
But not perfect. How reasonable the disagreement is, given the number oftrials, is a question we desist from asking, because ultimately that would
include among other things a test of the random number generator.
5
R. E. Peierls, Surprises in Theoretical Physics 共Princeton U.P., Princeton,
NJ, 1979兲, p. 73.
6
F. Reif, Fundamentals of Statistical and Thermal Physics 共McGraw–Hill,
New York, 1965兲.
7
R. P. Feynman, The Character of Physical Law 共MIT, Cambridge, MA,1967兲, pp. 108–126; A. J. Leggett, The Problems of Physics 共Oxford U.P.,
New York, 1987兲, pp. 148–157; J. L. Lebowitz, ‘‘Boltzmann’s entropy
and time’s arrow,’’ Phys. Today 46, September 32–38 共1993兲.
8
K. Huang, Statistical Mechanics 共Wiley, New York, 1987兲; G. Wannier,Statistical Physics 共Wiley, New York, 1966兲; C. J. Thompson, Mathematical Statistical Mechanics 共Macmillan, New York, 1972兲; J. D. Walecka,
Fundamentals of Statistical Mechanics 共Stanford U.P., Stanford, 1989兲; J.
Waldram, Theory of Thermodynamics 共Cambridge U.P., Cambridge,
1985兲.
9
P. T. Landsberg, ed., The Enigma of Time 共Hilger, London, 1982兲; J. J.
Halliwell et al., eds., Physical Origins of Time Asymmetry 共Cambridge
U.P., Cambridge, 1994兲.

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