Boltzmann was awarded a doctorate from the University of Vienna in 1866 for a thesis on the kinetic theory of gases supervised by Jožef Stefan and subsequently became his assistant. In 1869 Boltzmann was appointed to a chair of theoretical physics at Graz. He held this post for four years; then, in 1873, he accepted the chair of mathematics at Vienna. He did not stay very long in any place, and after three years he moved back in Graz, this time to the chair of experimental physics. In 1894, Boltzmann moved back to Vienna, this time to the chair of theoretical physics which became vacant on the death of his teacher Jožef Stefan. However, the following year Ernst Mach was appointed to the chair of history and philosophy of science at Vienna. Boltzmann had many scientific opponents but, to Boltzmann, Mach was more than a scientific opponent - the two were on bad personal terms. Dissatisfied with having to work with Mach, Boltzmann in 1900 moved to Leipzig, where he joined his strongest scientific opponent, Wilhelm Ostwald who, despite their scientific disagreements, was on good personal terms with Boltzmann. In 1901 Mach retired from Vienna due to ill health, which prompted Boltzmann to return to Vienna from his self-styled exile in 1902. In addition to his teaching in mathematical physics, Boltzmann was appointed as the reader of Mach's philosophy course. His philosophy lectures quickly became famous, with the audience soon being too large for the biggest lecture hall available. It was the fame of his lectures on philosophy rather than his work in physics than earned Boltzmann an invitation to the court of the Austrian Emperor, Franz Josef I.
Boltzmann is best known for laying the foundations of statistical mechanics, along, but independently, with Willard Gibbs. Their theories connected the properties and behaviour of atoms and molecules with the large-scale properties and behaviour of the substances of which they are the building blocks. In 1871, Boltzmann derived the Maxwell-Boltzmann distribution, in 1884, he derived Stefan-Boltzmann law theoretically.