627 0015-9018/02/0400-0627/0 © 2002 Plenum Publishing Corporation Foundations of Physics, Vol. 32, No. 4, April 2002 (© 2002) Caratheodory and the Foundations of Thermodynamics and Statistical Physics Ioannis E. Antoniou 1 1 Department of Mathematics, Aristoteles University of Thessaloniki, Thessaloniki, Greece 54006; International Solvay Institutes for Physics and Chemistry, ULB-CP 231, Blvd. du Triomphe, 1050 Brussels, Belgium; e-mail: iantonio@vub.ac.be Received July 30, 2001; revised September 13, 2001 Constantin Caratheodory offered the first systematic and contradiction free for- mulation of thermodynamics on the basis of his mathematical work on Pfaff forms. Moreover, his work on measure theory provided the basis for later improved for- mulations of thermodynamics and physics of continua where extensive variables are measures and intensive variables are densities. Caratheodory was the first to see that measure theory and not topology is the natural tool to understand the dif- ficulties (ergodicity, approach to equilibrium, irreversibility) in the Foundations of Statistical Physics. He gave a measure-theoretic proof of Poincaré’s recurrence theorem in 1919. This work paved the way for Birkhoff to identify later ergodicity as metric transitivity and for Koopman and von Neumann to introduce spectral analysis of dynamical systems in Hilbert spaces. Mixing provided an explanation of the approach to equilibrium but not of irreversibility. The recent extension of spectral theory of dynamical systems to locally convex spaces, achieved by the Brussels–Austin groups, gives new nontrivial time asymmetric spectral decomposi- tions for unstable and/or non-integrable systems. In this way irreversibility is resolved in a natural way. KEY WORDS: Caratheodory; thermodynamics; statistical physics; irreversi- bility; dynamical dystems; spectral analysis. 1. INTRODUCTION The life and contributions of Constantin Caratheodory in mathematics, physics and education have been reported by several authors. (1–4) The following table indicates Caratheodory’s contributions in Mathematical