The Erwin Schrr Odinger International Institute for Mathematical Physics Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?

An overview of the history of Ludwig Boltzmann s more than one hundred year old ergodic hypothesis is given The existing main results the majority of which is connected with the theory of billiards are surveyed and some perspectives of the theory and interesting and realistic problems are also mentioned In Werner Heisenberg was elected a honorary doctor of Lor and E otv os Uni versity Budapest In his inaugural lecture he made a point that sounded something like this A theoretical physicist feels best if there is no rigorously de ned math ematical object behind his considerations Certainly Heisenberg was having the early years of quantum mechanics in his mind but what he said perfectly tted the work of Ludwig Boltzmann as well One could choose several areas of his interest to illustrate this statement out of which the history of the ergodic hypothesis we are going to elaborate on is only one Lecture presented on International Symposium in Honor of Boltzmann s th Birthday Feb ruary Vienna Research supported by the Hungarian National Foundation for Scienti c Research grant No Another striking example perhaps not su ciently widely known is the case with the Boltz mann equation He published it in B and the rst mathematically satisfactory deriva tion of the equation was only obtained more than years later in by Oscar Lanford L though the picture is still not complete Thus needless to say that Boltzmann s original argument was highly intuitive At the same time however it was so much challenging for the great mathe matician David Hilbert that he included among his celebrated collection of problems presented at the International Mathematical Congress held at Paris in the sixth one with the title Math ematical Treatment of the Axioms of Physics see H In its formulation besides requiring an axiomatic approach to the theory of probabilities Hilbert also says it is therefore very desir able that the discussion of the foundations of mechanics be taken up by mathematicians also Thus Boltzmann s work on the principles of mechanics suggests the problem of developing mathematically