Hans Capel

Hans Willem Capel passed away in Haarlem, Netherlands, on 7 April 2023 after a short illness.

Hans was born in Sceaux, France, on 3 June 1936. He studied theoretical physics at the University of Leiden, where he obtained his doctorate in 1965 for a thesis, “The hole-equivalence principle, the Van Vleck relation and the application to the theory of d-ions in Ligand fields,” conducted under the supervision of Sybren de Groot. This sparked Hans’s interest in statistical mechanics, and in particular in theoretical solid-state physics, where his main contributions were in exactly solvable models. An outstanding early achievement was his introduction of what is now known as the Blume–Capel model, a generalization of the Ising model in a magnetic field that has found numerous applications in the field of phase transitions.

In 1970 Hans was appointed lector in theoretical physics at the Lorentz Institute in Leiden, where he served until 1983. His contributions to statistical mechanics were many and varied. In fruitful collaborations with both his PhD students and colleagues and visitors to the institute, he investigated exact Landau expansions for long-range interaction models, such as those occurring in the description of superfluid helium-3. He also made seminal contributions to the computation and asymptotics of correlation functions in the XY model. That gave rise to exact difference equations for the spin–spin correlation functions, which in the thermodynamic limit led to Painlevé-type equations, forming a bridge to his later work on solitons.

The emphasis on rigorous results and exact solvability was a theme that continued throughout Hans’s career, leading his interests in the early 1980s toward soliton systems, in particular the solitonic behavior of classical spin systems and associated nonlinear Schrödinger equations. Working closely with his PhD students as always, he set up a new methodology (coined direct linearization) for obtaining solutions of exactly solvable evolution equations and, moreover, to find a framework in which a whole hierarchy of integrable equations and their interconnections was embedded. This turned out to be a veritable gold mine: Many novel soliton systems were discovered, including anisotropic NLS equations, lattice Korteweg–de Vries and Boussinesq systems, an integrable lattice version of the massive Thirring model, and various discrete analogues of the Kadomtsev–Petviashvili equation.

In the 1990s, along the same thread of exact solvability, Hans was involved in applying reductions to the newly found systems, leading to integrable dynamical mappings. He and his collaborators of that period developed an elegant, systematic machinery to derive and study integrable maps both in the classical and the quantum domain.

When in 1983 he was appointed professor of theoretical physics at the University of Amsterdam, Hans decided to also delve into nonintegrable dynamical systems and the burgeoning field of chaos and fractals. He used to say that chaos and integrability should not be viewed as separate subject areas, but instead as two sides of the same coin, intrinsically linked together. True to his word, Hans became one of the very few scientists to have explored both sides at a deep level.

With PhD students, colleagues, and guests—who often became good friends—he worked on a wide variety of topics, ranging from sequences of period doubling bifurcations to intermittent bursts of chaos, and from random walks on fractal lattices to nonconservative maps with time-reversal symmetry. He had an exceptional ability to find his way in every subject he embraced, invariably arriving at some original and significant insight—and always in his endearing, quite inimitable personal style. Hans was like a father figure for his students: equable, genuinely encouraging, with a gentleman’s sense of humor, and an unwavering readiness to “battle"—that is, to discuss any fresh idea straight away on the blackboard in his office.

In 1990 he was elected a member of the Royal Netherlands Academy of Arts and Sciences. He was a longtime editor of Physica A, and he served as dean of the faculty of science at the University of Amsterdam. The occasion of his retirement in 2001 was marked by a memorable international symposium called Chaos, Demons, and Wavelets.

Hans will be remembered with fondness by colleagues and students alike (including his 21 PhD students) and by everybody who ever had the good fortune to “battle” with him. He lives on in all his scientific children and grandchildren.

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