Autobiography
In his memorable series "Etudes sur le temps humain", Georges
Poulet devoted one volume to the "Mesure de l'instant".^{1}
There he proposed a classification of authors according to the
importance they give to the past, present and future. I believe that in
such a typology my position would be an extreme one, as I live mostly in
the future. And thus it is not too easy a task to write this
autobiographical account, to which I would like to give a personal tone.
But the present explains the past.
In my Nobel Lecture, I speak much about fluctuations; maybe this is not
unrelated to the fact that during my life I felt the efficacy of
striking coincidences whose cumulative effects are to be seen in my
scientific work.
I was born in Moscow, on the 25th of January, 1917 - a few months before
the revolution. My family had a difficult relationship with the new
regime, and so we left Russia as early as 1921. For some years (until
1929), we lived as migrants in Germany, before we stayed for good in
Belgium. It was at Brussels that I attended secondary school and
university. I acquired Belgian nationality in 1949.
My father, Roman Prigogine, who died in 1974, was a chemical engineer
from the Moscow Polytechnic. My brother Alexander, who was born four
years before me, followed, as I did myself, the curriculum of chemistry
at the Université Libre de Bruxelles. I remember how much I hesitated
before choosing this direction; as I left the classical (Greco-Latin)
section of Ixelles Athenaeum, my interest was more focused on history
and archaeology, not to mention music, especially piano. According to my
mother, I was able to read musical scores before I read printed words.
And, today, my favourite pastime is still piano playing, although my
free time for practice is becoming more and more restricted.
Since my adolescence, I have read many philosophical texts, and I still
remember the spell "L'évolution créatrice" cast on me. More
specifically, I felt that some essential message was embedded, still to
be made explicit, in Bergson's remark:
"The more deeply we study the nature of time, the better we understand
that duration means invention, creation of forms, continuous elaboration
of the absolutely new."
Fortunate coincidences made the choice for my studies at the university.
Indeed, they led me to an almost opposite direction, towards chemistry
and physics. And so, in 1941, I was conferred my first doctoral degree.
Very soon, two of my teachers were to exert an enduring influence on the
orientation of my future work.
I would first mention Théophile De Donder (1873-1957).^{2} What
an amiable character he was! Born the son of an elementary school
teacher, he began his career in the same way, and was (in 1896)
conferred the degree of Doctor of Physical Science, without having ever
followed any teaching at the university.
It was only in 1918 - he was then 45 years old - that De Donder could
devote his time to superior teaching, after he was for some years
appointed as a secondary school teacher. He was then promoted to
professor at the Department of Applied Science, and began without delay
the writing of a course on theoretical thermodynamics for engineers.
Allow me to give you some more details, as it is with this very
circumstance that we have to associate the birth of the Brussels
thermodynamics school.
In order to understand fully the originality of De Donder's approach, I
have to recall that since the fundamental work by Clausius, the second
principle of thermodynamics has been formulated as an inequality: "uncompensated
heat" is positive - or, in more recent terms, entropy production is
positive. This inequality refers, of course, to phenomena that are
irreversible, as are any natural processes. In those times, these
latter were poorly understood. They appeared to engineers and physico-chemists
as "parasitic" phenomena, which could only hinder something: here the
productivity of a process, there the regular growth of a crystal,
without presenting any intrinsic interest. So, the usual approach was to
limit the study of thermodynamics to the understanding of equilibrium
laws, for which entropy production is zero.
This could only make thermodynamics a "thermostatics". In this context,
the great merit of De Donder was that he extracted the entropy
production out of this "sfumato" when related it in a precise way to the
pace of a chemical reaction, through the use of a new function that he
was to call "affinity".^{3}
It is difficult today to give an account of the hostility that such an
approach was to meet. For example, I remember that towards the end of
1946, at the Brussels IUPAP meeting,^{4} after a presentation of
the thermodynamics of irreversible processes, a specialist of great
repute said to me, in substance: "I am surprised that you give more
attention to irreversible phenomena, which are essentially transitory,
than to the final result of their evolution, equilibrium."
Fortunately, some eminent scientists derogated this negative attitude. I
received much support from people such as Edmond Bauer, the successor to
Jean Perrin at Paris, and Hendrik Kramers in Leyden.
De Donder, of course, had precursors, especially in the French
thermodynamics school of Pierre Duhem. But in the study of chemical
thermodynamics, De Donder went further, and he gave a new formulation of
the second principle, based on such concepts as affinity and degree of
evolution of a reaction, considered as a chemical variable.
Given my interest in the concept of time, it was only natural that my
attention was focused on the second principle, as I felt from the start
that it would introduce a new, unexpected element into the description
of physical world evolution. No doubt it was the same impression
illustrious physicists such as Boltzmann^{5} and Planck^{6}
would have felt before me. A huge part of my scientific career would
then be devoted to the elucidation of macroscopic as well as microscopic
aspects of the second principle, in order to extend its validity to new
situations, and to the other fundamental approaches of theoretical
physics, such as classical and quantum dynamics.
Before we consider these points in greater detail, I would like to
stress the influence on my scientific development that was exerted by
the second of my teachers, Jean Timmermans (1882-1971). He was more an
experimentalist, specially interested in the applications of classical
thermodynamics to liquid solutions, and in general to complex systems,
in accordance with the approach of the great Dutch thermodynamics school
of van der Waals and Roozeboom.^{7}
In this way, I was confronted with the precise application of
thermodynamical methods, and I could understand their usefulness. In the
following years, I devoted much time to the theoretical approach of such
problems, which called for the use of thermodynamical methods; I mean
the solutions theory, the theory of corresponding states and of isotopic
effects in the condensed phase. A collective research with V. Mathot, A.
Bellemans and N. Trappeniers has led to the prediction of new effects
such as the isotopic demixtion of helium He^{3}+ He^{4},
which matched in a perfect way the results of later research. This part
of my work is summed up in a book written in collaboration with V.
Mathot and A. Bellemans, The Molecular Theory of Solutions. ^{
8}
My work in this field of physical chemistry was always for me a specific
pleasure, because the direct link with experimentation allows one to
test the intuition of the theoretician. The successes we met provided
the confidence which later was much needed in my confrontation with more
abstract, complex problems.
Finally, among all those perspectives opened by thermodynamcis, the one
which was to keep my interest was the study of irreversible phenomena,
which made so manifest the "arrow of time". From the very start, I
always attributed to these processes a constructive role, in opposition
to the standard approach, which only saw in these phenomena degradation
and loss of useful work. Was it the influence of Bergson's "L'évolution
créatrice" or the presence in Brussels of a performing school of
theoretical biology?^{9} The fact is that it appeared to me that
living things provided us with striking examples of systems which were
highly organized and where irreversible phenomena played an essential
role.
Such intellectual connections, although rather vague at the beginning,
contributed to the elaboration, in 1945, of the theorem of minimum
entropy production, applicable to non-equilibrium stationary states.^{10}
This theorem gives a clear explanation of the analogy which related the
stability of equilibrium thermodynamical states and the stability of
biological systems, such as that expressed in the concept of "homeostasy"
proposed by Claude Bernard. This is why, in collaboration with J.M.
Wiame,^{11} I applied this theorem to the discussion of some
important problems in theoretical biology, namely to the energetics of
embryological evolution. As we better know today, in this domain the
theorem can at best give an explanation of some "late" phenomena, but it
is remarkable that it continues to interest numerous experimentalists.^{12}
From the very beginning, I knew that the minimum entropy production was
valid only for the linear branch of irreversible phenomena, the one to
which the famous reciprocity relations of Onsager are applicable.^{13}
And, thus, the question was: What about the stationary states far from
equilibrium, for which Onsager relations are not valid, but which are
still in the scope of macroscopic description? Linear relations are very
good approximations for the study of transport phenomena (thermical
conductivity, thermodiffusion, etc.), but are generally not valid for
the conditions of chemical kinetics. Indeed, chemical equilibrium is
ensured through the compensation of two antagonistic processes, while in
chemical kinetics - far from equilibrium, out of the linear branch - one
is usually confronted with the opposite situation, where one of the
processes is negligible.
Notwithstanding this local character, the linear thermodynamics of
irreversible processes had already led to numerous applications, as
shown by people such as J. Meixner,^{14} S.R. de Groot and P.
Mazur,^{15} and, in the area of biology, A. Katchalsky.^{16}
It was for me a supplementary incentive when I had to meet more general
situations. Those problems had confronted us for more than twenty years,
between 1947 and 1967, until we finally reached the notion of "dissipative
structure". ^{17}
Not that the question was intrinsically difficult to handle; just that
we did not know how to orientate ourselves. It is perhaps a
characteristic of my scientific work that problems mature in a slow way,
and then present a sudden evolution, in such a way that an exchange of
ideas with my colleagues and collaborators becomes necessary. During
this phase of my work, the original and enthusiastic mind of my
colleague Paul Glansdorff played a major role.
Our collaboration was to give birth to a general evolution criterion
which is of use far from equilibrium in the non-linear branch, out of
the validity domain of the minimum entropy production theorem. Stability
criteria that resulted were to lead to the discovery of critical states,
with branch shifting and possible appearance of new structures. This
quite unexpected manifestation of "disorder-order" processes, far from
equilibrium, but conforming to the second law of thermodynamics, was to
change in depth its traditional interpretation. In addition to classical
equilibrium structures, we now face dissipative coherent structures, for
sufficient far-from-equilibrium conditions. A complete presentation of
this subject can be found in my 1971 book co-authored with Glansdorff.^{18}
In a first, tentative step, we thought mostly of hydrodynamical
applications, using our results as tools for numerical computation. Here
the help of R. Schechter from the University of Texas at Austin was
highly valuable.^{19} Those questions remain wide open, but our
centre of interest has shifted towards chemical dissipative systems,
which are more easy to study than convective processes.
All the same, once we formulated the concept of dissipative structure, a
new path was open to research and, from this time, our work showed
striking acceleration. This was due to the presence of a happy meeting
of circumstances; mostly to the presence in our team of a new generation
of clever young scientists. I cannot mention here all those people, but
I wish to stress the important role played by two of them, R. Lefever
and G. Nicolis. It was with them that we were in a position to build up
a new kinetical model, which would prove at the same time to be quite
simple and very instructive - the "Brusselator", as J. Tyson would call
it later - and which would manifest the amazing variety of structures
generated through diffusion-reaction processes.^{20}
This is the place to pay tribute to the pioneering work of the late A.
Turing,^{21} who, since 1952, had made interesting comments
about structure formation as related to chemical instabilities in the
field of biological morphogenesis. I had met Turing in Manchester about
three years before, at a time when M.G. Evans, who was to die too soon,
had built a group of young scientists, some of whom would achieve fame.
It was only quite a while later that I recalled the comments by Turing
on those questions of stability, as, perhaps too concerned about linear
thermodynamics, I was then not receptive enough.
Let us go back to the circumstances that favoured the rapid development
of the study of dissipative structures. The attention of scientists was
attracted to coherent non-equilibrium structures after the discovery of
experimental oscillating chemical reactions such as the Belusov-Zhabotinsky
reaction;^{22} the explanation of its mechanism by Noyes and his
co-workers;^{23} the study of oscillating reactions in
biochemistry (for example the glycolytic cycle, studied by B. Chance^{24}
and B. Hess^{25}) and eventually the important research led by
M. Eigen.^{26} Therefore, since 1967, we have been confronted
with a huge number of papers on this topic, in sharp contrast with the
total absence of interest which prevailed during previous times.
But the introduction of the concept of dissipative structure was also to
have other unexpected consequences. It was evident from start that the
structures were evolving out of fluctuations. They appeared in fact as
giant fluctuations, stabilized through matter and energy exchanges with
the outer world. Since the formulation of the minimum entropy production
theorem, the study of non-equilibrium fluctuation had attracted all my
attention.^{27} It was thus only natural that I resumed this
work in order to propose an extension of the case of far-from-equilibrium
chemical reactions.
This subject I proposed to G. Nicolis and A. Babloyantz. We expected to
find for stationary states a Poisson distribution similar to the one
predicted for equilibrium fluctuations by the celebrated Einstein
relations. Nicolis and Babloyantz developed a detailed analysis of
linear chemical reactions and were able to confirm this prediction.^{28}
They added some qualitative remarks which suggested the validity of such
results for any chemical reaction.
Considering again the computations for the example of a non-linear
biomolecular reaction, I noticed that this extension was not valid. A
further analysis, where G. Nicolis played a key role, showed that an
unexpected phenomenon appeared while one considered the fluctuation
problem in nonlinear systems far from equilibrium: the distribution law
of fluctuations depends on their scale, and only "small fluctuations"
follow the law proposed by Einstein.^{29} After a prudent
reception, this result is now widely accepted, and the theory of non-equilibrium
fluctuations is fully developing now, so as to allow us to expect
important results in the following years. What is already clear today is
that a domain such as chemical kinetics, which was considered
conceptually closed, must be thoroughly rethought, and that a brand-new
discipline, dealing with non-equilibrium phase transitions, is now
appearing.^{30, 31, 32}
Progress in irreversible phenomena theory leads us also to
reconsideration of their insertion into classical and quantum dynamics.
Let us take a new look at the statistical mechanics of some years ago.
From the very beginning of my research, I had had occasion to use
conventional methods of statistical mechanics for equilibrium situations.
Such methods are very useful for the study of thermodynamical properties
of polymer solutions or isotopes. Here we deal mostly with simple
computational problems, as the conceptual tools of equilibrium
statistical mechanics have been well established since the work of Gibbs
and Einstein. My interest in non-equilibrium would by necessity lead me
to the problem of the foundations of statistical mechanics, and
especially to the microscopic interpretation of irreversibility.^{33}
Since the time of my first graduation in science, I was an enthusiastic
reader of Boltzmann, whose dynamical vision of physical becoming was for
me a model of intuition and penetration. Nonetheless, I could not but
notice some unsatisfying aspects. It was clear that Boltzmann introduced
hypotheses foreign to dynamics; under such assumptions, to talk about a
dynamical justification of thermodynamics seemed to me an excessive
conclusion, to say the least. In my opinion, the identification of
entropy with molecular disorder could contain only one part of the truth
if, as I persisted in thinking, irreversible processes were endowed with
this constructive role I never cease to attribute to them. For another
part, the applications of Boltzmann's methods were restricted to diluted
gases, while I was most interested in condensed systems.
At the end of the forties, great interest was aroused in the
generalization of kinetic theory to dense media. After the pioneering
work by Yvon^{34}, publications of Kirkwodd^{35}, Born
and Green^{36}, and of Bogoliubov^{37} attracted a lot
of attention to this problem, which was to lead to the birth of non-equilibrium
statistical mechanics. As I could not remain alien to this movement, I
proposed to G. Klein, a disciple of Fürth who came to work with me, to
try the application of Born and Green's method to a concrete, simple
example, in which the equilibrium approach did not lead to an exact
solution. This was our first tentative step in non-equilibrium
statistical mechanics.^{38} It was eventually a failure, with
the conclusion that Born and Green's formalism did not lead to a
satisfying extension of Boltzmann's method to dense systems.
But this failure was not a total one, as it led me, during a later work,
to a first question: Was it possible to develop an "exact" dynamical
theory of irreversible phenomena? Everybody knows that according to the
classical point of view, irreversibility results from supplementary
approximations to fundamental laws of elementary phenomena, which are
strictly reversible. These supplementary approximations allowed
Boltzmann to shift from a dynamical, reversible description to a
probabilistic one, in order to establish his celebrated H theorem.
We still encountered this negative attitude of "passivity" imputed to
irreversible phenomena, an attitude that I could not share. If - as I
was prepared to think - irreversible phenomena actually play an active,
constructive role, their study could not be reduced to a description in
terms of supplementary approximations. Moreover, my opinion was that in
a good theory a viscosity coefficient would present as much physical
meaning as a specific heat, and the mean life duration of a particle as
much as its mass.
I felt confirmed in this attitude by the remarkable publications of
Chandrasekhar and von Neumann, which were also issued during the forties.^{39}
That was why, still with the help of G. Klein, I decided to take a fresh
look at an example already studied by Schrödinger,^{ 40} related
to the description of a system of harmonic oscillators. We were
surprised to see that, for all such a simple model allowed us to
conclude, this class of systems tend to equilibrium. But how to
generalize this result to non-linear dynamical systems?
Here the truly historic performance of Léon van Hove opened for us the
way (1955).^{41} I remember, with a pleasure that is always new,
the time - which was too short - during which van Hove worked with our
group. Some of his works had a lasting effect on the whole development
of statistical physics; I mean not only his study of the deduction of a
"master equation " for anharmonic systems, but also his fundamental
contribution on phase transitions, which was to lead to the branch of
statistical mechanics that deals with so-called "exact" results.^{42}
This first study by van Hove was restricted to weakly coupled anharmonic
systems. But, anyway, the path was open, and with some of my colleagues
and collaborators, mainly R. Balescu, R. Brout, F. Hénin and P. Résibois,
we achieved a formulation of non-equilibrium statistical mechanics from
a purely dynamical point of view, without any probabilistic assumption.
The method we used is summed up in my 1962 book.^{43} It leads
to a "dynamics of correlations", as the relation between interaction and
correlation constitutes the essential component of the description.
Since then, these methods have led to numerous applications. Without
giving more detail, here, I will restrict myself to mentioning two
recent books, one by R. Balescu,^{44} the other by P. Résibois
and M. De Leener.^{45}
This concluded the first step of my research in non-equilibrium
statistical mechanics. The second is characterized by a very strong
analogy with the approach of irreversible phenomena which led us from
linear thermodynamics to non-linear thermodynamics. In this tentative
step also, I was prompted by a feeling of dissatisfaction, as the
relation with thermodynamics was not established by our work in
statistical mechanics, nor by any other method. The theorem of Boltzmann
was still as isolated as ever, and the question of the nature of
dynamics systems to which thermodynamics applies was still without
answer.
The problem was by far more wide and more complex than the rather
technical considerations that we had reached. It touched the very nature
of dynamical systems, and the limits of Hamiltonian description. I would
never have dared approach such a subject if I had not been stimulated by
discussions with some highly competent friends such as the late Léon
Rosenfeld from Copenhagen, or G. Wentzel from Chicago. Rosenfeld did
more than give me advice; he was directly involved in the progressive
elaboration of the concepts we had to explore if we were to build a new
interpretation of irreversibility. More than any other stage of my
scientific career, this one was the result of a collective effort. I
could not possibly have succeeded had it not been for the help of my
colleagues M. de Haan, Cl. George, A. Grecos, F. Henin, F. Mayné, W.
Schieve and M. Theodosopulu. If irreversibility does not result from
supplementary approximations, it can only be formulated in a theory of
transformations which expresses in "explicit" terms what the usual
formulation of dynamics does "hide". In this perspective, the kinetic
equation of Boltzmann corresponds to a formulation of dynamics in a new
representation.^{46, 47, 48, 49}
In conclusion: dynamics and thermodynamics become two complementary
descriptions of nature, bound by a new theory of non-unitary
transformation. I came so to my present concerns; and, thus, it is time
to end this intellectual autobiography. As we started from specific
problems, such as the thermodynamic signification of non-equilibrium
stationary states, or of transport phenomena in dense systems, we have
been faced, almost against our will, with problems of great generality
and complexity, which call for reconsideration of the relation of
physico-chemical structures to biological ones, while they express the
limits of Hamiltonian description in physics.
Indeed, all these problems have a common element: time. Maybe the
orientation of my work came from the conflict which arose from my
humanist vocation as an adolescent and from the scientific orientation I
chose for my university training. Almost by instinct, I turned myself
later towards problems of increasing complexity, perhaps in the belief
that I could find there a junction in physical science on one hand, and
in biology and human science on the other.
In addition, the research conducted with my friend R. Herman on the
theory of car traffic^{50} gave me confirmation of the
supposition that even human behaviour, with all its complexity, would
eventually be susceptible of a mathematical formulation. In this way the
dichotomy of the "two cultures" could and should be removed. There would
correspond to the breakthrough of biologists and anthropologists towards
the molecular description or the "elementary structures", if we are to
use the formulation by Lévi-Strauss, a complementary move by the physico-chemist
towards complexity. Time and complexity are concepts that present
intrinsic mutual relations.
During his inaugural lecture, De Donder spoke in these terms:^{51}
"Mathematical physics represents the purest image that the view of
nature may generate in the human mind; this image presents all the
character of the product of art; it begets some unity, it is true and
has the quality of sublimity; this image is to physical nature what
music is to the thousand noises of which the air is full..."
Filtrate music out of noise; the unity of the spiritual history of
humanity, as was stressed by M. Eliade, is a recent discovery that has
still to be assimilated.^{52} The search for what is meaningful
and true by opposition to noise is a tentative step that appears to be
intrinsically related to the coming into consciousness of man facing a
nature of which he is a part and which it leaves.
I have many times advocated the necessary dialogue in scientific
activity, and thus the vital importance of my colleagues and
collaborators in the journey that I have tried to describe. I would also
stress the continuing support that I received from institutions which
have made this work a feasible one, especially the Université Libre de
Bruxelles and the University of Texas at Austin. For all of the
development of these ideas, the International Institute of Physics and
Chemistry founded by E. Solvay (Brussels, Belgium) and the Welch
Foundation (Houston, Texas) have provided me with continued support.
The work of a theoretician is related in a direct way to his whole life.
It takes, I believe, some amount of internal peace to find a path among
all successive bifurcations. This peace I owe to my wife, Marina. I know
the frailty of the present, but today, considering the future, I feel
myself to be a happy man.
References
1. G.
Poulet, Etudes sur le temps humain, Tone 4, Edition 10/18, Paris,
1949.
2. See the
note on De Donder in the Florilège (pedant le XIXe siècle et le début du
XXe), Acad. Roy. Belg., Bull. Cl. Sc., page 169, 1968.
3. Th. De
Donder (Rédaction nouvelle par P. Van Rysselberghe), Paris, Gauthier-
Villars, 1936.
See also:
I. Prigogine and R. Defay: Thermodynamique Chimique conformément aux
méthodes de Gibbs et De Donder (2 Tomes), Liège, Desoer, 1944-1946.
Or the translation in English:
Chemical Thermodynamics, translated by D.H. Everett, Langmans 1954, 1962.
4. See
Colloque de Thermodynamique, Union Intern. de Physique pure et appliquée
(I.U.P.A.P.), 1948.
5.
Bolzmann, L., Wien, Ber. 66, 2275, 1872.
6. Planck,
M., Vorlesaungen über Thermodynamik, Walter de Gruyter, Berlin, Leipzig,
1930.
7.
Timmermans, J., Les Solutions Concentrées, Masson et Cie, Paris, 1936.
Let us also quote his thesis on experimental research on demixtion in liquid
mixtures.
8.
Prigogine, I., The Molecular Theory of Solutions, avec A. Bellemans et V.
Mathot; North-Holland Publ. Company, Amsterdam, 1957.
See also: Prigogine and Defay, Ref. 3.
9. Let us
quote some remarkable works of this School:
Barchet, A., La Vie créatrice des formes, Alcan, Paris, 1927.
Dalcq, A., L'Oeuf et son dynamisme organisateur, Alban Michel. Paris, 1941.
Barchet, J., Embryologie Chimique, Desoer, Liège et Masson, Paris, 1946.
I was also much interested in the beautiful book by Marcel Florkin:
L'Evolution biochimique, Desoer, Liège, 1944.
10.
Prigogine, I., Acad. Roy. Belg. Bull. Cl. Sc. 31, 600, 1945.
- Etude thermodynamique des phénomènes irréversibles. Thèse d'agrégation
présentée en 1945 à l'Université Libre de Bruxelles.
Desoer, Liège, 1947.
- Introduction à la Thermodynamique des processus irréversibles, traduit de
l'anglais par J. Chanu, Dunod, Paris, 1968.
11.
Prigogine, I., and Wiame, J.M., Experientia, 2, 451, 1946.
12.
Nicolis, G. and Prigogine, I., Self Organization in Non-Equilibrium Systems
(Chaps. III and IV), J. Wiley and Sons, New York, 1977.
13.
Onsager, L. , Phys. Rev., 37, 405, 1931.
14.
Meixner, J., Ann. Physik, (5), 35, 701, 1939; 36, 103, 1939;
39, 333, 1941; 40, 165, 1941;
Zeitsch Phys. Chim. B 53, 235, 1943.
15. de
Groot, S.R. and Mazur, P., Non-Equilibrium Thermodynamics, North-Holland,
Amsterdam, 1962.
16.
Katchalsky, A. and Curran, P.F., Non-Equilibrium Thermodynamics in
Biophisics, Harvard Univ. Press, Cambridge, Mass., 1946.
17.
Prigogine, I., Structure, Dissipation and Life.
Theoretical Physics and Biology, Versailles, 1967.
North-Holland Publ. Company, Amsterdam, 1969.
It is in this communication that the term "structure dissipative" is used
for the first time.
18.
Glansdorff, P. and Prigogine, I., Structure, Stabilité et Fluctuations,
Masson, Paris, 1971.
- Thermodynamic Theory of Structure Stability and Fluctuations, Wiley and
Sons, London, 1971.
- Traduction en langue russe: Mir, Moscou, 1973.
- Traduction en langue japonaise; Misuzu Shobo, 1977.
This book presents in detail the original work by the two authors, which led
to the concept of dissipative structure. For a brief historical account, see
also:
Acad. Roy. Belg., Bull. des Cl. Sc., LIX, 80, 1973.
19.
Schechter, R.S., The Variational Method in Engineering, McGraw-Hill, New
York, 1967.
20. Tyson,
J., Journ. of Chem. Physics, 58, 3919, 1973.
21. Turing,
A., Phil. Trans. Roy. Soc. London, Ser B, 237, 37, 1952.
22.
Belusov, B.P., Sb. Ref. Radiat. Med. Moscow, 1958.
Zhabotinsky, A.P., Biofizika, 9, 306, 1964.
Acad. Sc. U.R.S.S. Moscow (Nauka), 1967.
23. Noyes,
R.M. et al., Ann. Rev. Phys. Chem. 25, 95, 1974.
24. Chance,
B., Schonener, B. and Elsaesser, S., Proc. Nat. Acad. Sci. U.S.A. 52,
337-341, 1964.
25. Hess,
B., Ann. Rev. Biochem. 40, 237, 1971.
26. Eigen,
M., Naturwissenschaften, 58, 465, 1971.
27.
Prigogine, I. and Mayer, G., Acad. Roy. Belg. Bull. Cl. Sc., 41, 22,
1955.
28.
Nicolis, G. and Babloyantz, A., Journ. Chem. Phys., 51, 6, 2632,
1969.
29.
Nicolis, G. and Prigogine, I., Proc. Nat. Acad. Sci. U.S.A., 68,
2102, 1971.
30.
Prigogine, I., Proc. 3rd Symp. Temperature, Washington D.C., 1954.
Prigogine, I. and Nicolis, G., Proc. 3rd. Intern. Conference: From
Theoretical Physics to Biology, Versailles, France, 1971.
31.
Nicolis, G. and Turner, J.W., Proc. of the Conference on Bifurcation Theory,
New York, 1977. To Appear.
32.
Prigogine, I. and Nicolis, G., Non-Equilibrium Phase Transitions and
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51. For the
reference, see note 2.
52. Mircéa
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From
Nobel Lectures, Chemistry 1971-1980, Editor-in-Charge Tore Frängsmyr,
Editor Sture Forsén, World Scientific Publishing Co., Singapore, 1993
This
autobiography/biography was first published in the book series
Les Prix Nobel. It was later edited and republished in
Nobel Lectures. To cite this document, always state the source as
shown above.
Ilya Prigogine died on May 28, 2003.
Copyright
© The Nobel Foundation 1977