## Game solution

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.

This gentleman is the other founder of kinetic theory, **James Clerk Maxwell**, also famous for contributions to all fields of physics including of course the Maxwell equations. Maxwell was extremely good at using mathematical equations to explain physical phenomena. Some information about him can be found here.

I will only say a few words about the small portion of Maxwell’s work which I have looked at: his 1867 and 1879 papers about kinetic theory. In the former, Maxwell essentially establishes all the framework for kinetic theory from scratch, including (in a disguised form) the equation which would later be known as Boltzmann’s equation. He uses this theory to study heat conduction and computes many coefficients numerically. Thomson said that Maxwell gave everybody an impression of strong “power”: this is exactly the feeling that I had when consulting the abovementioned papers.

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.

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.)

This is the most well-known portrait of **Alan Turing**, monster mathematician and one of the fathers of computer science. He is mostly remembered for his contributions to logic (especially his works on indecidability), and to theoretical computer science, with the study of Turing machines, algorithms and proofs. During World War II he had a crucial role by breaking the code of the Enigma encoding machine, used by the Germans. In spite of this he was tried by the British government for being homosexual, and had to undergo regular hormon injections to avoid prison. He died by poison, possibly in a suicide. Because they were classified confidential, some of his major contributions to computer science were revealed long after his death. More details can be found here and here.

Less known is that Turing was interested in probability theory (his dissertation was about central limit theorem; and, after all, his code-breaking used statistical methods) and partial differential equations. During the last part of his life he worked a lot on morphogenesis, and developed the first attempts of mathematical explanation of pattern formation via the study of reaction-diffusion equations; this study led him to the discovery of a phenomenon now known as Turing instability.

Although the choice of these great guys was not done via logical thinking, it is clear that they have several common features. All of them have been very innovative in applying mathematics to another scientific field in which this had not been done before (be it a part of physics, economics or biology), and they all tried hard to understand the world by the study of mathematical equations. As for their topics of interest, four of them worked on/with statistical mechanics and/or entropy, and all of them have spent a lot of energy studying some properties of diffusion processes.