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ENTROPY

1. The Thermodynamic Definition. The idea of
entropy as a measure of the mechanical unavailability
of energy—that is, of that part of a given amount of

113

energy which cannot be transformed into useful me-
chanical work and is therefore lost or dissipated for
all practical purposes—originated in early nine-
teenth-century studies on the efficiency of steam en-
gines. The first to raise the question of maximum effi-
ciency in steam-power engineering was Nicolas
Léonard Sadi Carnot (1769-1832). In a famous memoir
he asked: “How can we know that the steam is used
in the most advantageous way possible to produce
motive power?” (Carnot, 1824). Carnot studied heat
engines whose thermal interaction with their sur-
roundings consists only in the exchange (absorption or
rejection) of heat with appropriate reservoirs of fixed
temperatures, and he showed that the reversibly oper-
ating engine is more efficient than its irreversible
counterpart when working between the same temper-
atures. He derived this conclusion, which became
known as “Carnot's Theorem,” on the basis of the
caloric theory according to which heat is regarded as
a fluid. As scattered statements in the literature of that
period indicate, Carnot and some of his contemporaries
have been fully aware that in any practical trans-
formation of heat into mechanical work part of the
stored energy is always dissipated. The exact answer
as to precisely what part is necessarily lost could not
be given before the Second Law of Thermodynamics
was explicitly stated.

During the years 1840-48 J. P. Joule, J. R. Mayer,
and H. von Helmholtz discovered the equivalence
between heat and work and thus established the First
Law of Thermodynamics (conservation of energy in
a closed system). It invalidated Carnot's assumptions
but not his conclusions. Referring to the conflict be-
tween Carnot and Joule, William Thomson (Lord
Kelvin, 1824-1907) declared that further experiments
were needed to resolve this dilemma (Thomson, 1849).
But only one year later Rudolf Julius Emmanuel
Clausius (1822-88) showed without the benefit of fur-
ther experimentation that the issue can be resolved by
either taking “Carnot's Theorem” as an independent
principle or by deriving it from the First Law of
Thermodynamics in conjunction with the premiss that
“heat always shows a tendency to equalize temperature
differences and therefore to pass from hotter to colder
bodies” (Italics in the original; Clausius, 1850).

With these words Clausius introduced rather casually
the Second Law of Thermodynamics, namely, that it
is impossible for a self-acting cyclic machine, unaided
by any external agency, to convey heat from one body
at a given temperature to another at a higher tempera-
ture, a statement which Max Planck later called “the
Clausius Formulation of the Second Law.” Gibbs
rightly remarked that with Clausius' memoir of 1850
“the science of thermodynamics came into existence”
(Gibbs, 1889). In fact, it was the first paper to contain
the two principal laws of thermodynamics. Thomson
derived Carnot's Theorem from the First Law and the
premiss that “it is impossible by means of inanimate
material agency to derive mechanical effect from any
portion of matter by cooling it below the temperature
of the coldest of the surrounding objects” (Thomson,
1851). It is easy to show that the Kelvin Formulation
of the Second Law, as Planck called the preceding
statement (impossibility of perpetual motion of the
second kind), is fully equivalent to the Clausius formu-
lation (Huang, 1963). In a second paper Thomson
discussed the cosmological implications of the Second
Law and concluded that “within a finite period of time
past, the earth must have been, and within a finite
period of time to come the earth must again be, unfit
for the habitation of man as at present constituted,
unless operations have been, or are to be performed,
which are impossible under the laws to which the
known operations going on at present in the material
world are subject” (Thomson, 1852).

Although the preceding formulations of the Second
Law as well as Thomson's sweeping generalization
expressed essentially what subsequently became known
as “the entropy principle,” the concept of entropy as
such was still unknown. Its definition was made possible
only after Clausius demonstrated the following
theorem: if in a cyclic transformation qi denotes the
quantity of heat drawn from (positive), or rejected by
(negative), a heat reservoir at the (absolute) tempera-
ture Ti, then the expression ∑qi/Ti is equal to zero
for reversible cycles and negative for irreversible ones.
The first part of this statement was found inde-
pendently also by Thomson. In fact, as Planck showed
in a critical analysis of Clausius' paper (Planck, 1879),
Thomson's derivation of the so-called “Clausius Equal-
ity” ∑qi/ Ti = 0, or in the limit of infinitesimal quanti-
ties of heat
ϕ ʃ δq / T = 0 (1)
for reversible (more precisely, quasi-static) closed-cycle
processes, was logically superior. Nevertheless, it was
Clausius who first realized in the same paper (Clausius,
1854) that for reversible processes δq/T is a total (or
exact) differential, or equivalently that T-1 is an inte-
grating factor. The line integral of a total differential,
as shown in the calculus, depends only on the limits
of integration and not on the particular path chosen
for the integration. In other words, it defines a point
function or, in thermodynamics, a state function; that
is, a function which depends on the thermodynamic
variables, like volume or temperature, of the state
under consideration.

It took another eleven years for Clausius to realize
the importance of the state function defined by the

114

above-mentioned total differential. Writing dS = δq/T
and integrating, he obtained

where the path of integration corresponds to a reversi-
ble transformation from the thermodynamic state A
to the thermodynamic state B. By combining an irre-
versible transformation from A to B with a reversible
one from B to A and taking notion of the “Clausius
Theorem,” he concluded that

Looking for an appropriate name for the state function
S, Clausius remarked that just as the (inner) energy
U signifies the heat and work content (Wärme- und
Werkinhalt
) of the system, so S, in view of the pre-
ceding results, denotes its “transformation content”
(Verwandlungsinhalt). “But as I hold it to be better,”
he continued “to borrow terms for important magni-
tudes from the ancient languages, so that they may
be adopted unchanged in all modern languages, I pro-
pose to call the magnitude S the entropy of the body,
from the Greek word τροπή, transformation. I have
intentionally formed the word entropy so as to be as
similar as possible to the word energy, for the two
magnitudes to be denoted by these words are so nearly
allied in their physical meanings, that a certain simi-
larity in designation appears to be desirable” (Clausius,
1865).

Clausius' thermodynamic definition of entropy, based
as it is ultimately on a certain existence theorem in
the theory of differentials, is obviously rather abstract
and far removed from visualizability, in spite of the
fact that the differential expression under discussion
reflects an operational result in steam-power engineer-
ing. As the preceding equations (1) and (2) show, the
entropy of a closed (adiabatically isolated—change of
state without transfer of heat) system can never de-
crease, for δq/T = 0 implies SBSA. Extrapolating
this result for the universe as a whole, Clausius con-
cluded his paper with the famous words: “The energy
of the universe is constant—the entropy of the uni-
verse tends toward a maximum.”

That irreversibility indeed entails increase of en-
tropy—the so-called “entropy principle”—follows
logically from the two statements, (1) that the entropy
of the universe never decreases, and (2) that a process,
accompanied by entropy increase, is irreversible
(Gatlin, 1966).

The fundamental importance of the entropy concept
was soon understood to lie in the fact that it makes
it possible to predict whether an energy transformation
is reversible (dS = δq/T), irreversible (dS > δq/T), or
impossible (dS < δq/T), even if the total energy in-
volved is conserved. Moreover, with the help of the
entropy concept other thermodynamic state functions
could be defined, such as the Helmholtz free energy
or the Gibbs thermodynamic potential, which proved
extremely useful for the calculation of the maximum
attainable work under conditions of constant tempera-
ture or constant temperature and constant pressure,
respectively.

2. A Modernized Version of the Thermodynamic
Definition of Entropy.
With the extension of thermo-
dynamics at the end of the last century to electrical
and magnetic phenomena, to elastic processes, phase
changes, and chemical reactions—the result of re-
searches by J. W. Gibbs, Helmholtz, H. A. Lorentz,
P. Duhem, W. H. Nernst, and others—it was felt unsat-
isfactory that a science of such astounding generality,
and especially such central conceptions as that of
entropy, should be based on engineering experience
with heat engines and their cycles. L. J. Henderson's
critical remark, that “the steam engine did much more
for science than science ever did for the steam engine,”
served as a serious challenge for those concerned with
foundational research. Stimulated by Max Born, who
as a student had criticized the conventional approach
as deviating “too much from the ordinary methods of
physics” (Born, 1921), Constantin Carathéodory re-
placed this approach in 1908 by a purely axiomatic
treatment, based on the integrability properties of
Pfaffian differentials (Carathéodory, 1909). His “prin-
ciple of adiabatic unattainability”—according to which
adiabatically inaccessible equilibrium states exist in the
neighborhood of any equilibrium state, this being a
mathematical reformulation of the impossibility of a
perpetual motion of the second kind—implies the
existence of an integrating divisor only tempera-
ture-dependent and hence of the entropy S. Due to
the mathematical intricacies of Carathéodory's ideas,
they were generally ignored in spite of the enthusiastic
acceptance by Born, A. Landé, S. Chandrasekhar, and
H. A. Buchdahl. Since 1958, however, primarily after
having been simplified by L. A. Turner, F. W. Sears,
and P. T. Landsberg, Carathéodory's approach became
more popular, and his definition of entropy is now
presented even in textbooks of thermodynamics (P. T.
Landsberg, 1961; I. P. Bazarov, 1964).

3. The Kinetic Definition of Entropy. Another
definition of entropy, which we owe to Ludwig Boltz-
mann, has been provided by the kinetic theory of gases.
Elaborating on J. C. Maxwell's famous statistical deri-
vation of the velocity distribution of gas molecules
under equilibrium conditions, Boltzmann studied the

115

change of this distribution f(v) under equilibrium ap-
proach and showed that f(v) always tends toward the
Maxwellian form (Boltzmann, 1872). Boltzmann ob-
tained this result by introducing a certain one-valued
function of the instantaneous state distribution of the
molecules, which he called the E-function and later
the H-function (Burbury, 1890), and of which he could
show, apparently on the basis of pure mechanics alone,
that it decreases until f(v) reaches the Maxwellian
form. His proof relied on the simple fact that the
expression (x - y) (log y - log x) is always negative
for positive numbers x and y. Since under equilibrium
conditions E turned out to be proportional to the
thermodynamic entropy, Boltzmann realized that his
E-function (or H-function) provides an extension of the
definition of entropy to nonequilibrium states not
covered by the thermodynamic definition.

4. The Statistical Definition of Entropy. Boltz-
mann's H-Theorem, that is, his conclusion that, for
nonequilibrium systems, H is a decreasing function in
time, was bound to raise questions concerning the
nature of irreversibility of physical systems and its
compatibility with the principles of mechanics. Boltz-
mann, fully aware of these problems, tried therefore
to base his conclusion on more general grounds by
taking into consideration the relative frequencies of
equilibrium states compared to nonequilibrium distri-
butions. In 1877 he showed that if W denotes the
number of states in which each molecule has a specified
position and velocity (so-called “micro-states”) which
describe the same given macroscopic state defined by
measurable thermodynamic variables like pressure or
temperature (so-called “macro-states”), then the
entropy of the system (gas) is proportional to the loga-
rithm of W (Boltzmann, 1877a). The introduction of
the logarithm followed from the fact that for two
independent systems the total entropy is the sum of
the individual entropies while the total probability is
the product of the individual probabilities:
S = S1 + S2 = f(W1) + f(W2 = f(W1 W2)
implies that S = constant log W. It is clear that a given
macro-state can be realized by a large number of
different micro-states, for the interchange of two mol-
ecules, for example, does not alter the density distribu-
tion in the least. If therefore the number W of micro-
states corresponding to a given macro-state is regarded
as a measure of the probability of occurrence of the
thermodynamic state, this statistical conception of
entropy provides an immediately visualizable inter-
pretation of the concept: it measures the probability
for the occurrence of the state; and the fact that in
adiabatically closed systems S increases toward a maxi-
mum at thermodynamic equilibrium means that the
system tends toward a state of maximum probability.
Finally, since ordered arrangements of molecules (e.g.,
when the molecules in one part of a container all move
very fast—corresponding to a high temperature—and
those in another part all move very slowly—corres-
ponding to a low temperature ζ) have a much smaller
probability of occurrence than disordered or random
arrangements, the increase of entropy signifies increase
of disorganization or of randomness (equalization of
temperature).

5. Immediate Consequences of the Entropy Con-
ception.
Boltzmann's statistical interpretation of
entropy, based as it was on probabilistic considerations,
had to regard the principle of entropy increase as a
statistical law rather than as a strict law of nature as
originally maintained by thermodynamics. The con-
clusion that a spontaneous change toward a state of
smaller entropy or increased order, though extremely
improbable, is no longer an impossibility had, as we
shall see, important implications for the cosmological
applications of the concept of entropy.

The first to draw cosmological conclusions from
thermodynamics was, as we have seen, Thomson in
1852. Two years later, Helmholtz discussed the dissi-
pation principle and formulated the so-called “theory
of thermal death,” or “heat death” (Helmholtz, 1854).
Eventually, Helmholtz declared, the universe would
run down to a state of uniform temperature and “be
condemned to a state of eternal rest.” These arguments
by Thomson and Helmholtz implied also the existence
of an initial state of minimum entropy and hence a
distinctive beginning “which must have been produced
by other than the now acting causes”; they challenged
therefore uniformitarian geology (James Hutton,
Charles Lyell) and its denial of large-scale catastrophic
changes in the past. Since, moreover, the Darwinian
theory of biological evolution relied at that time con-
siderably on the uniformitarian doctrine, it was only
natural that the religious controversy about Darwinism
embraced also the discussion on entropy.

The principle of entropy increase was also carried
over into social philosophy, primarily by Herbert
Spencer. His First Principles (1862), published three
years after Darwin's Origin of Species and aimed at
interpreting life, mind, and society in terms of matter,
motion, and energy (force), had as its central thesis the
instability of homogeneity and its trend toward
heterogeneity as a characteristic of evolution in all its
phases, whether of individual organisms, groups of
organisms, the earth, the solar system, or the whole
universe. These developments, however, were held as
incidental to a more fundamental process, namely, “the
integration of matter and the concomitant dissipation
of motion” (Spencer, 1862). That Spencer's social
physicalism and its alleged implications for human

116

history had been directly influenced by the principle
of energy dissipation is shown by the fact that in the
fifth edition of First Principles (1887) Spencer made
an explicit reference to Helmholtz' essay on “The
Interaction of Natural Forces.” Spencer's conclusion
that the total degradation of energy in the cosmos is
followed, due to a process of concentration under
gravitation, by a renewed dispersion and evolution so
that eras of dissolution and evolution alternate, found
but little approval by contemporary scientists like John
Tyndall and James Clerk Maxwell (Brush, 1967).

A most remarkable application of the notion of
like others of his generation to establish history as a
science, and prompted by the conceptual similarity
between history and irreversibility, Adams attempted
to describe human history in terms of socio-physical
or rather socio-thermodynamical laws (Adams, 1919).
Stimulated by Andrew Gray's study of Lord Kelvin
(Samuels, 1964) and, in particular, by Gray's discussion
of the social implications of “Kelvin's great generaliza-
tion” (energy dissipation) and the idea of the ultimate
heat death which had been popularized meanwhile by
H. G. Wells in The Time Machine (1895), Adams re-
ferred to Auguste Comte's teachings that the human
mind had passed through three phases, the theological,
metaphysical, and positive; these phases Adams com-
pared with the chemical phases of solid, liquid, and
gaseous, subject to Gibbs's phase'rule, and claimed that
these “three phases always exist together in equilib-
rium; but their limits on either side are fixed by changes
of temperature and pressure, manifesting themselves
in changes of Direction or Form.” The Renaissance,
for example, with its marked change in direction, form,
and level of what Adams called “spiritual energy,” was
for him but a phase transition in accordance with
Gibbs's rule. Adams concluded that the future historian
“must seek his education in the world of mathematical
physics” and, in particular, in the teachings of Kelvin,
Maxwell, and Gibbs.

6. Restrictions of the Conception. Even before the
appearance of Boltzmann's statistical interpretation of
entropy, which, as we have seen, questioned the uni-
versal validity of the entropy principle, doubts had
been voiced whether the principle applies unre-
strictedly to small-scale phenomena. One of the earliest
devices conceived to this effect was the “sorting
demon,” first mentioned by Maxwell in a letter of 11
December 1867 to P. G. Tait (Knott, 1911) and pub-
lished in Maxwell's Theory of Heat (1871). Referring
to a vessel containing a gas at thermodynamic equilib-
rium, and “divided into two portions A and B, by a
division in which there is a small hole,” Maxwell
imagined a being “whose faculties are so sharpened
that he can follow every molecule on its course,” and
who “opens and closes this hole, so as to allow only
the swifter molecules to pass from A to B, and only
the slower ones to pass from B to A. He will then,
without expenditure of work, raise the temperature of
B and lower that of A, in contradiction to the second
law of thermodynamics” (Maxwell, 1871). The gist of
this device, which Kelvin “nicknamed” “Maxwell's
Demon,” was of course the idea that through the inter-
vention of an intelligent being, capable of sorting
physical systems of molecular size merely “by simple
inspection,” as Maxwell put it, the entropy principle
could be violated.

The problem raised by Maxwell's demon became the
subject of much discussion (Whiting, 1885), especially
when it was subsequently generalized to molecular
fluctuations and quasi-macroscopic manipulations
(Smoluchowski, 1914). After the rise of quantum me-
chanics John Slater claimed that the idea of Maxwell's
demon must become nugatory through W. Heisenberg's
indeterminacy relations (Slater, 1939). However, N. L.
Balazs showed that for nondegenerate systems of rela-
tively heavy particles with small concentrations and
high temperatures quantum effects do not affect the
demon's mode of operation and that, consequently,
Slater's view was erroneous (Balazs, 1953). Leo Szilard
offered a satisfactory solution of the problem raised
by Maxwell's demon. He showed that the process of
“inspection” (observation or measurement), necessarily
preceding the sorting operation, is not at all so “sim-
ple” as Maxwell believed; rather it is inevitably associ-
ated with an entropy increase which, at least, compen-
sates the decrease under discussion (Szilard, 1929).
Szilard's investigation was followed by a series of
studies on the relation between entropy and measure-
ment which culminated in Claude Shannon's funda-
mental contribution (Shannon, 1948) to the modern
theory of information and the notion of “negentropy”
(negative entropy) as a measure of information, just
as entropy measures lack of information about the
structure of a system. In 1951 Leon Brillouin proposed
an information theoretical refutation of Maxwell's
demon (Brillouin, 1951), and since then entropy, as a
logical device for the generation of probability distri-
butions, has been applied also in decision theory, reli-
ability engineering, and other technical disciplines. By
regarding statistical mechanics as a form of statistical
inference rather than as a physical theory E. T. Jaynes
greatly generalized the usage of the concept of entropy
(Jaynes, 1957). Moreover, M. Tribus demonstrated the
possibility of retrieving the thermodynamical concept
of entropy from the information-theoretical notion of
entropy for both closed and open systems (Tribus,
1961).

117

A few years after Maxwell's invention of the demon
another attempt to avoid the consequences of the
entropy principle was advanced, first by Thomson
(Thomson, 1874), and two years later, in greater detail,
by Josef Loschmidt, with whose name this so-called
“reversibility objection” (Umkehreinwand) is usually
associated (Loschmidt, 1876). It emphasized the incon-
sistency of irreversibility with the time reversal invari-
ance of Newtonian mechanics and its laws of (molecu-
lar) collisions which underlie Boltzmann's derivation
of the H-Theorem. It claimed that for any motion or
sequence of states of the system in which H decreases
there exists, under time reversal, another motion in
precisely the opposite way in which H increases. Con-
sequently, Loschmidt declared, a purely mechanical
proof of the Second Law of Thermodynamics or of
the principle of entropy increase cannot be given. To
counter this objection Boltzmann argued statistically
that of all state distributions having the same energy,
the Maxwell distribution corresponding to equilibrium
has an overwhelming probability, so that a randomly
chosen initial state is almost certain to evolve into the
equilibrium state under increase of entropy (Boltz-
mann, 1877b). In fact, Boltzmann's statistical definition
of entropy (Boltzmann, 1877a) was a by-product of his
attempt to rebut Loschmidt's objection. Later on, when
the problem of mechanics and irreversibility became
a major issue before the British Association for the
Advancement of Science at its Cardiff meeting (August
1891), and its Oxford meeting (August 1894) which
Boltzmann attended, he revised the result of his
H-Theorem by ascribing to the H-curve certain dis-
continuity properties (Boltzmann, 1895). In a cele-
brated Encyklopädie article on the foundations of
statistical mechanics Paul and Tatiana Ehrenfest
demonstrated by a profound analysis of the problem
that Boltzmann's arguments could not be considered
as a rigorous proof of his contention (Ehrenfest, 1911).

Meanwhile Henri Poincaré had published his famous
prize essay on the three-body problem (Poincaré,
1890), in which he proved that a finite energy system,
confined to a finite volume, returns in the course of
a sufficiently long-interval to an arbitrarily small
neighborhood of almost every given initial state.
Poincaré saw in this theorem support for the thesis
of the stability of the solar system in the tradition of
Lagrange, Laplace, and Poisson; in spite of his great
interest in fundamental questions in thermodynamics
he does not seem to have noticed its applicability to
systems of molecules and the mechanical theory of
heat. It was only in 1896 that Ernst Zermelo made
use of Poincaré's theorem for his so-called “recurrence
objection” (Wiederkehreinwand) to challenge Boltz-
mann's derivation of the entropy principle. Zermelo
claimed that in view of Poincaré's result all molecular
configurations are (almost) cyclic or periodic and hence
periods of entropy increase must alternate with periods
of entropy decrease. The ancient idea of an eternal
recurrence, inherited from primitive religions, resusci-
tated by certain Greek cosmologies, such as the
Platonic conception of the “Great Year” or Pythago-
rean and Stoic cosmology, and revived in the nine-
teenth century especially by Friedrich Nietzsche, now
seemed to Zermelo to be a scientifically demonstrable
matical
correctness of Poincaré's theorem and of
Zermelo's contention, but rejected their physical sig-
nificance on the grounds that the recurrence time
would be inconceivably long (Boltzmann, 1896). In
fact, as M. Smoluchowski showed a few years later,
the mean recurrence time for a one per cent fluctuation
of the average density in a sphere with a radius of
5 × 10-5 cm. in an ideal gas under standard conditions
would amount to 1068 seconds or approximately
3 × 1060 years. The time interval between two large
fluctuations, the so-called “Poincaré cycle,” turned out
to be 101023 ages of the universe, the age of the universe
taken as 1010 years (Smoluchowski, 1915).

7. Applicability Limits of the Concept of Entropy.
These time intervals, though enormously great, are yet
finite and cannot therefore be ignored in cosmological
considerations. In applying the notion of entropy to
the universe at large, Boltzmann, following a sugges-
tion made by his long-time assistant L. Schuetz, de-
scribed the universe as follows: though generally in
thermal equilibrium “and therefore dead,” it contains
“here and there small regions of the same size as our
galaxy which, during the relative short time of eons,
fluctuate noticeably from thermal equilibrium” and in
which the entropy “will be equally likely to increase
or decrease” (Boltzmann, 1896-98). In recent years the
applicability of the entropy concept to such cosmolog-
ical considerations has been repeatedly questioned
(Plotkin, 1950; Milne, 1952). Whether the introduction
of an upper boundary to the applicability of the
entropy concept—like that of its lower boundary
(Maxwell's demon)—will eventually be refuted remains
an open question.

Nor has a unanimous answer been obtained to the
problem whether the notion of entropy fully applies
also to biology. Helmholtz had envisaged the possibility
of cytological processes being associated with entropy
decrease, a thesis which subsequently was given limited
support by H. Zwaardemaker, but rejected by the
majority of contemporary biologists. It gained a revival
of interest when in 1910 Felix Auerbach, an ardent
proponent of biological entropy decrease, adopted
from G. Hirth (Hirth, 1900) the notion of “ectropy”

118

as the biological antithesis of entropy, and promulgated
this concept in his popular work >Ektropismus oder die
physikalische Theorie des Lebens
(Auerbach, 1910). The
issue is of course intimately connected with the conflict
between biological mechanism, according to which
biological phenomena can be exclusively explained in
physicochemical terms, and vitalism, according to
which the processes of life have a character sui generis.

8. The Boltzmann Problem. What may be called
“the Boltzmann problem,” namely the question as to
the minimum additional assumption, if any, necessary
to derive macroscopic irreversibility from pure me-
chanics, is also philosophically of great importance. On
the solution of this problem depends decisively whether
a purely mechanistic explanation of nonelectromag-
netic phenomena is possible. Until quite recently the
conceptual difficulties were usually overcome by the
introduction of probabilistic assumptions a priori, such
as Boltzmann's hypothesis of a molecular chaos in his
Stosszahlansatz. It was soon understood, however, that
these assumptions, though not inconsistent with the
principles of pure mechanics, are nevertheless not
derivable from them. A general tendency arose to
banish probability from statistical mechanics as far as
possible.

With the advent of quantum mechanics, which like
classical mechanics is time reversal invariant, it seemed
Pauli in 1928, that the above-mentioned probabilistic
hypotheses could be derived from the statistical aspects
inherent in the very foundations of the quantum theory.
A certain equation, derived on the basis of Dirac's
perturbation theory, which describes the transition
probabilities between quantum mechanical states,
appeared appropriate for the treatment of irreversible
processes. This so-called “master equation” (T. Prigo-
gine, P. Résibois) in conjunction with the Hermiticity
assumption of perturbation operators, made it possible
to derive all laws of thermodynamics as well as the
phenomenological equations for thermal conduction,
diffusion, and even the Onsager reciprocity relations,
without major difficulties (R. T. Cox, 1950, 1952;
E. C. G. Stueckelberg, 1952; J. S. Thomsen, 1953; N. C.
van Kampen, 1954).

Since 1958, however, the logical legitimacy of using
perturbation theory in this context has been seriously
questioned, and new attempts to solve this problem
were made in the so-called “thermodynamics of irre-
versible processes.” In 1968 it became apparent that
irreversibility is intimately connected with the coexist-
ence of phases in equilibrium and occurs whenever a
thermodynamic variable is coupled through an equi-
librium process to another independent variable. (See
D. G. Schweitzer, “The Origin of Irreversibility from
Conventional Equilibrium Concepts,” Physics Letters
27a
[1968], 402-04.)

Contemporary investigations of the “Boltzmann
problem” are very important also for foundational
research on quantum mechanics and, especially, on its
theory of measurement, since here quantal phenomena
are coupled with macroscopic irreversible processes
that occur in the measuring device (G. Ludwig, P.
Bocchieri, A. Loinger, G. M. Prospèri). Also Louis de
Broglie's (1964) reinterpretation of quantum mechanics
as a hidden thermodynamics (thermodynamique cachée
des particules
) will be greatly affected by the outcome
of these investigations.

9. Extrascientific Applications. In summary, it
should be remarked that the notion of entropy or,
equivalently, the Second Law of Thermodynamics,
“the most metaphysical law of nature” (Henri Bergson,
Creative Evolution), had considerable influence also on
extrascientific considerations. Because of their prox-
imity to cosmological speculations, philosophy and
theology were of course most affected.

Ever since Boltzmann (1895), in his rebuttal of
Zermelo's recurrence objection, reduced (local)
anisotropy of time (the “arrow of time”) to statistical
irreversibility, the entropy concept has played an im-
portant role in philosophical discussions on the nature
of time (A. S. Eddington, H. Reichenbach, A. Grün-
baum, H. Mehlberg, P. W. Bridgman, K. R. Popper).
The concept became also a battleground between
idealism (Jeans, 1930) and materialism (Kannegiesser,
1961).

The dysteleological tenet of the energy dissipation
principle and its gloomy prediction of a heat death
touched upon profound religious issues, and was bound
to provoke theological polemics. Examples of such
controversies on the acceptability of theological con-
sequences from the entropy principle are the discussion
between Abel Rey (Rey, 1904) and Pierre Duhem
(Duhem, 1906) and the published correspondence be-
tween Arnold Lunn and J. B. S. Haldane (Lunn, 1935).

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MAX JAMMER