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).
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
thesis. In his reply Boltzmann admitted the mathe-
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).
