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This of course is Ludwig Boltzmann, one of the fathers of
Statistical Mechanics. Among other contributions,
Boltzmann had the idea of the famous
formula for the entropy S=k log W, identified its counterpart
in kinetic theory and proved that it can only increase in time
along solutions of the Boltzmann equation: this is the H-Theorem.
This was the first time that one had obtained a
justification of the second principle of thermodynamics
starting from the basic laws of physics.
It is well-known that Boltzmann had to fight continuously about
the relevance of his equation and its probabilistic interpretation,
about his conclusions, and about the existence of atoms
as well. Apparently he believed that we would never have direct
evidence of atoms, although he was aware of Brownian motion.
He was depressive and committed suicide in 1906, a few years before the atomic
hypothesis became accepted by most people. Usually Boltzmann
is represented with a long beard and a frigthening look, but
I prefer this portrait of him as a young scientist.
More information, as well as pictures (including one
of his grave with the entropy formula on it) can be
found here. Carlo Cercignani recently wrote a nice
biography:
Ludwig Boltzmann, The Man Who Trusted Atoms
(reviews can be found
here and
here).
Although notoriously difficult to read, Boltzmann's scientifical writings
have been of lasting value for decades and decades. In 1959 the
great mathematician Mark Kac wrote ``Boltzmann summarized not (but not all)
of his work in a two volume treatise, Vorlesungen über
Gastheorie. This is one of the greatest books in the history
of exact sciences and the reader is strongly advised to consult it.''
For me as for many specialists, the theory of the Boltzmann equation
remains one of the most fascinating areas in partial
differential equations --- a meeting point between kinetic theory,
fluid mechanics, statistical mechanics and information theory.
This was my PhD subject, and still remains my favourite topic.
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Here comes John Nash, well-known mathematical genius,
most famous for his work about strategic games, and for his
record of mental illness. Every researcher in population dynamics
today knows about Nash equilibria. Although it is his work about strategic
games which gained him the Nobel Prize in economics, mathematicians
more often recall his important contributions in Riemannian geometry
(his works about isometric embedding, including the Nash fixed-point
theorem) and partial differential equations (independently of De Giorgi,
he proved what is now known as the De Giorgi-Nash-Moser theorem on
the regularity of solutions of elliptic and parabolic equations
with discontinuous coefficients). A lot of information about
Nash can be found
here, here (including links to his moving Nobel Prize speech
and to his own Homepage)
and here. Sylvia Nasar wrote a successful
biography of Nash, A Beautiful Mind; here is a
review
of this book by John Milnor. The book later inspired a movie,
which was attacked by many, partly because it distorted
the reality in several respects.
I personally have extreme admiration for Nash's work on partial
differential equations. He wrote just one paper on the subject,
in 1958 (Continuity of solutions of parabolic and elliptic equations),
but this one of the most astonishing works in the history
of partial differential equations. His proof has been often
described as complicated, but I find it extremely attractive,
and I also like a lot the way the paper is written: with a lot
of explanations about his intuition and the way he arrived at
the result. The genesis of the paper is fascinating, as
discussed in Nasar's book. By the way, one of the ingredients in
the proof is Boltzmann's entropy functional.
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This is Mark Kac, one of the most famous probabilists ever.
Of course he is mostly known for the Feynman-Kac formula,
for his paper about the ``shape of a drum'', and for the
works about ``probabilistic number theory'' which he did together
with Paul Erdöos. See here for a short biography and
references.
It is less known that Kac was one of the first mathematicians
to be interested in kinetic theory, and more generally
applications of probability theory to various problems of
statistical mechanics. I was extremely impressed by
his fascinating broad-audience book
Probability and related topics in physical science.
His 1956 paper about the ``Foundations of kinetic theory''
is still worth reading today. In spite of some misconceptions,
this paper paved the way towards the study of ``chaos'' in kinetic
theory and more generally in limits of mean-field types.
Kac's attempts to study the Boltzmann equation and the $H-Theorem
were all the more remarkable that his mathematical style was at
the opposite of what you would traditionally expect from
a specialist of partial differential equations: he always
tried to think in terms of linear operators, combinatorics,
operator series, group representations, etc.
He himself said in his lecture notes
Integration in Function Spaces and Some of Its Applications:
``Mathematicians, or rather mathematical analysts, are divided
roughly speaking into two classes: the "calculators",
i.e. those who look for exact formulas, and the
"estimators", i.e. those who live by inequalities.
I belong to the first class''. [As for me, I would rather
belong to the second class.]
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