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Dictionary of the History of Ideas | ||

#### 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

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 δ*q* / *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 δ*q*/*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

*dS*= δ

*q*/

*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

Werkinhalt

*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 δ*q*/*T* = 0 implies *SB* ≥ *SA*. 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* = δ*q*/*T*), irreversible (*dS* > δ*q*/*T*), or

impossible (*dS* < δ*q*/*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

*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

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

entropy to history was made by Henry Adams. Trying

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).

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).

*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”

this concept in his popular work

*>Ektropismus oder die*

physikalische Theorie des Lebens(Auerbach, 1910). The

physikalische Theorie des Lebens

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

for some time, as shown in a paper published by W.

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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1966), II, 208-17.

MAX JAMMER

[See also Cosmology; Evolutionism; Indeterminacy inPhysics; Time; Uniformitarianism and Catastrophism.]

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