Boltzmann studied physics at the University of Vienna, starting in 1863. Among his teachers were Josef Loschmidt, Joseph Stefan, Andreas von Ettingshausen and Jozef Petzval. Boltzmann received his PhD degree in 1866 working under the supervision of Stefan; his dissertation was on kinetic theory of gases. In 1867 he became a Privatdozent (lecturer). After obtaining his doctorate degree, Boltzmann worked two more years as Stefan’s assistant. It was Stefan who introduced Boltzmann to Maxwell's work.
In 1872, long before women were admitted to Austrian universities, he met Henriette von Aigentler, an aspiring teacher of mathematics and physics in Graz. She was refused permission to unofficially audit lectures, and Boltzmann advised her to appeal; she did, successfully. On July 17, 1876 Ludwig Boltzmann married Henriette von Aigentler; they had three daughters and two sons. Boltzmann went back to Graz to take up the chair of Experimental Physics. Among his students in Graz were Svante Arrhenius and Walther Nernst. He spent 14 happy years in Graz and it was there that he developed his statistical concept of nature. In 1885 he became a member of the Imperial Austrian Academy of Sciences and in 1887 he became the President of the University of Graz.
Boltzmann was appointed to the Chair of Theoretical Physics at the University of Munich in Bavaria, Germany in 1890. In 1893, Boltzmann succeeded his teacher Joseph Stefan as Professor of Theoretical Physics at the University of Vienna.
In Vienna, Boltzmann not only taught physics but also lectured on philosophy. Boltzmann’s lectures on natural philosophy were very popular, and received a considerable attention at that time. His first lecture was an enormous success. Even though the largest lecture hall had been chosen for it, the people stood all the way down the staircase. Because of the great successes of Boltzmann’s philosophical lectures, the Emperor invited him for a reception at the Palace.
Boltzmann was subject to rapid alternation of depressed moods with elevated, expansive or irritable moods, likely the symptoms of undiagnosed bipolar disorder. He himself jestingly attributed his rapid swings in temperament to the fact that he was born during the night between Mardi Gras and Ash Wednesday. Meitner relates that those who were close to Boltzmann were aware of his bouts of severe depression and his suicide attempts.
On September 5, 1906, while on a summer vacation in Duino, near Trieste, Boltzmann hanged himself during an attack of depression. He is buried in the Viennese Zentralfriedhof; his tombstone bears the inscription S=k. log W.
Around the turn of the century, Boltzmann's science was being threatened by another philosophical objection. Some phyicists, including Mach's student, Gustav Jaumann, interpreted Hertz to mean that all electromagnetic behavior was continuous as if there were no atoms and molecules and as if all physical behavior was ultimately electromagnetic. This movement around 1900 deeply depressed Boltzmann since it could mean the end of his kinetic theory and statistical interpretation of the second law of thermodynamics. After Mach's resignation in Vienna in 1901, Boltzmann returned there and decided to become a philosopher himself to refute philosophical objections to his physics, but he soon became discouraged again. In 1904 at a physics conference in St. Louis where most physicists seemed to reject atoms and he was not even invited to the physics section but was stuck in a section called "applied mathematics" he violently attacked philosophy, especially on allegedly Darwinian grounds but actually in terms of Lamarck's theory of the inheritance of acquired characteristics that people inherited bad philosophy from the past and that it was hard for scientists to overcome such inheritance. In 1905 Boltzmann corresponded extensively with the Austro-German philosopher Franz Brentano in hope of mastering philosophy better apparently so that he could refute its presence in science better, but he became discouraged about this approach as well. In the following year 1906 his mental condition became so bad that he had to resign his position. He committed suicide in September of that same year.
Much of the physics establishment did not share his belief in the reality of atoms and molecules — a belief shared, however, by Maxwell in Scotland and Gibbs in the United States; and by most chemists since the discoveries of John Dalton in 1808. He had a long-running dispute with the editor of the preeminent German physics journal of his day, who refused to let Boltzmann refer to atoms and molecules as anything other than convenient theoretical constructs. Only a couple of years after Boltzmann's death, Perrin's studies of colloidal suspensions (1908-1909) confirmed the values of Avogadro's number and Boltzmann's constant, and convinced the world that the tiny particles really exist.
where = 1.3806505(24) × 10−23 J K−1 is Boltzmann's constant, and the logarithm is taken to the natural base . is the Wahrscheinlichkeit, the frequency of occurrence of a macrostate or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system — number of (unobservable) "ways" the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann’s paradigm was an ideal gas of identical particles, of which are in the -th microscopic condition (range) of position and momentum. can be counted using the formula for permutations
where i ranges over all possible molecular conditions. ( denotes factorial.) The "correction" in the denominator is because identical particles in the same condition are indistinguishable. is called the "thermodynamic probability" since it is an integer greater than one, while mathematical probabilities are always numbers between zero and one.
The Boltzmann equation was developed to describe the dynamics of an ideal gas.
where represents the distribution function of single-particle position and momentum at a given time (see the Maxwell-Boltzmann distribution), is a force, is the mass of a particle, is the time and is an average velocity of particles.
This equation describes the temporal and spatial variation of the probability distribution for the position and momentum of a density distribution of a cloud of points in single-particle phase space. (See Hamiltonian mechanics.) The first term on the left-hand side represents the explicit time variation of the distribution function, while the second term gives the spatial variation, and the third term describes the effect of any force acting on the particles. The right-hand side of the equation represents the effect of collisions.
In principle, the above equation completely describes the dynamics of an ensemble of gas particles, given appropriate boundary conditions. This first-order differential equation has a deceptively simple appearance, since can represent an arbitrary single-particle distribution function. Also, the force acting on the particles depends directly on the velocity distribution function f. The Boltzmann equation is notoriously difficult to integrate. David Hilbert spent years trying to solve it without any real success.
The form of the collision term assumed by Boltzmann was approximate. However for an ideal gas the standard Chapman-Enskog solution of the Boltzmann equation is highly accurate. It is expected to lead to incorrect results for an ideal gas only under shock wave conditions.
Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation — his famous H-theorem. However the key assumption he made in formulating the collision term was "molecular chaos", an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law. It was from the probabilistic assumption alone that Boltzmann's apparent success emanated, so his long dispute with Loschmidt and others over Loschmidt's paradox ultimately ended in his failure.
Finally, in the 1970s E.G.D. Cohen and J.R. Dorfman proved that a systematic (power series) extension of the Boltzmann equation to high densities is mathematically impossible. Consequently nonequilibrium statistical mechanics for dense gases and liquids focuses on the Green-Kubo relations, the fluctuation theorem, and other approaches instead.
Boltzmann had a tremendous admiration for Darwin and he wished to extend Darwinism from biological to cultural evolution. In fact he considered biological and cultural evolution as one and the same things. ... In short, cultural evolution was a physical process taking place in the brain. Boltzmann included ethics in the ideas which developed in this fashion ...
Howard T. Odum later sought to develop these views when looking at the evolution of ecological systems, and suggested that the maximum power principle was an example of Darwin's law of natural selection.