Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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).

Dictionary of the History of Ideas | ||