Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical explanation of the second law of thermodynamics. In 1877, he provided the current definition of entropy, , where Ω is the number of microstates whose energy equals the system's energy, interpreted as a measure of the statistical disorder of a system. Max Planck named the constant the Boltzmann constant.
Statistical mechanics is one of the pillars of modern physics. It describes how macroscopic observations (such as temperature and pressure) are related to microscopic parameters that fluctuate around an average. It connects thermodynamic quantities (such as heat capacity) to microscopic behavior, whereas, in classical thermodynamics, the only available option would be to measure and tabulate such quantities for various materials.
Biography Childhood and education Boltzmann was born in Erdberg, a suburb of Vienna, into a Catholic family. His father, Ludwig Georg Boltzmann, was a revenue official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann's mother, Katharina Pauernfeind, was originally from Salzburg. Boltzmann was home-schooled until the age of ten, and then attended high school in Linz, Upper Austria. When Boltzmann was 15, his father died.
Starting in 1863, Boltzmann studied mathematics and physics at the University of Vienna. He received his doctorate in 1866 and his venia legendi in 1869. Boltzmann worked closely with Josef Stefan, director of the institute of physics. It was Stefan who introduced Boltzmann to James Clerk Maxwell's work.
Academic career In 1869, at age 25, thanks to a letter of recommendation written by Josef Stefan, Boltzmann was appointed full Professor of Mathematical Physics at the University of Graz in the province of Styria. In 1869, he spent several months in Heidelberg working with Robert Bunsen and Leo Königsberger and in 1871 with Gustav Kirchhoff and Hermann von Helmholtz in Berlin. In 1873, Boltzmann joined the University of Vienna as Professor of Mathematics and there he stayed until 1876.
In 1872, long before women were admitted to Austrian universities, he met Henriette von Aigentler, an aspiring teacher of mathematics and physics in Graz. Her nieces were Slovenian painters Avgusta Šantel and Henrika Šantel. Henriette was refused permission to audit lectures unofficially. Boltzmann supported her decision to appeal, which was successful. On 17 July 1876 Ludwig Boltzmann married Henriette; they had three daughters: Henriette (1880), Ida (1884) and Else (1891); and a son, Arthur Ludwig (1881). 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.
Boltzmann was appointed to the Chair of Theoretical Physics at the Ludwig-Maximilians-Universität München in Bavaria, Germany in 1890.
In 1894, Boltzmann succeeded his teacher Joseph Stefan as Professor of Theoretical Physics at the University of Vienna.
Final years and death
Boltzmann spent a great deal of effort in his final years defending his theories.Cercignani, Carlo (1998). Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press. He did not get along with some of his colleagues in Vienna, particularly Ernst Mach, who became a professor of philosophy and history of sciences in 1895. That same year Georg Helm and Wilhelm Ostwald presented their position on energetics at a meeting in Lübeck. They saw energy, and not matter, as the chief component of the universe. Boltzmann's position carried the day among other physicists who supported his atomic theories in the debate. In 1900, Boltzmann went to Leipzig University, on the invitation of Wilhelm Ostwald. Ostwald offered Boltzmann the professorial chair in physics, which became vacant when Gustav Heinrich Wiedemann died.
After Mach retired due to bad health, in 1902 Boltzmann returned to Vienna. In addition to his lectures on physics, in 1903 Boltzmann started teaching the philosophy course (which had previously been given by Mach). Boltzmann's lectures on natural philosophy were very popular and received considerable attention. His first lecture was an enormous success: people stood all the way down the staircase outside the largest available lecture hall, and the Emperor invited him to a reception.The Boltzmann Equation: Theory and Applications, E. G. D. Cohen, W. Thirring, ed., Springer Science & Business Media, 2012 Also in 1903, Boltzmann, together with Gustav von Escherich and Emil Müller, founded the Austrian Mathematical Society. His students included Karl Přibram, Paul Ehrenfest and Lise Meitner.
In 1905, he gave an invited course of lectures in the summer session at the University of California, Berkeley, which he described in a popular essay A German professor's trip to El Dorado.
In May 1906, Boltzmann's deteriorating mental condition (described in a letter by the Dean as "a serious form of neurasthenia") forced him to resign his position. His symptoms indicate he experienced what might today be diagnosed as bipolar disorder. Four months later he died by suicide on 5 September 1906, by hanging himself while on vacation with his wife and daughter in Duino, near Trieste (then Austria).Muir, Hazel, Eureka! Science's greatest thinkers and their key breakthroughs, p.152, Upon Boltzmann's death, Friedrich ("Fritz") Hasenöhrl became his successor in the professorial chair of physics at Vienna. He is buried in the Viennese Zentralfriedhof. His tombstone bears the inscription of Boltzmann's entropy formula: .
Philosophy Boltzmann's kinetic theory of gases seemed to presuppose the reality of atoms and molecules, but many German philosophers and scientists like Ernst Mach and the physical chemist Wilhelm Ostwald disbelieved their existence. Boltzmann had been exposed to molecular theory by James Clerk Maxwell’s paper, "Illustrations of the Dynamical Theory of Gases," which described temperature as dependent on the speed of the molecules. This inspired Boltzmann to embrace atomism, introducing statistics into physics and extending the theory.Nancy Forbes, Basil Mahon (2019). Faraday, Maxwell, and the Electromagnetic Field. Chapter 11. .
Boltzmann wrote treatises on philosophy such as "On the question of the objective existence of processes in inanimate nature" (1897). He was a realist. In his work "On Thesis of Schopenhauer's", Boltzmann refers to his philosophy as materialism and says further: "Idealism asserts that only the ego exists, the various ideas, and seeks to explain matter from them. Materialism starts from the existence of matter and seeks to explain sensations from it."
Physics Boltzmann's most important scientific contributions were in the kinetic theory of gases based upon the second law of thermodynamics. This was important because Newtonian mechanics did not differentiate between past and future motion, but Rudolf Clausius’ invention of entropy to describe the second law was based on disgregation or dispersion at the molecular level so that the future was one-directional. Boltzmann was twenty-five years of age when he came upon James Clerk Maxwell's work on the kinetic theory of gases which hypothesized that temperature was caused by collision of molecules. Maxwell used statistics to create a curve of molecular kinetic energy distribution from which Boltzmann clarified and developed the ideas of kinetic theory and entropy based upon statistical atomic theory, creating the Maxwell–Boltzmann distribution as a description of molecular speeds in a gas.Ludwig Boltzmann, Lectures on the Theory of Gases, translated by Stephen G. Brush, "Translator's Introduction", 1968. It was Boltzmann who derived the first equation to model the dynamic evolution of the probability distribution Maxwell and he had created.Penrose, Roger. "Foreword". In Cercignani, Carlo, Ludwig Boltzmann: The Man Who Trusted Atoms, . Boltzmann's key insight was that dispersion occurred due to the statistical probability of increased molecular "states". Boltzmann went beyond Maxwell by applying his distribution equation to not solely gases, but also liquids and solids. Boltzmann also extended his theory in his 1877 paper beyond Carnot, Rudolf Clausius, James Clerk Maxwell and Lord Kelvin by demonstrating that entropy is contributed to by heat, spatial separation, and radiation.Boltzmann, Ludwig (1877). Translated by Sharp, K.; Matschinsky, F. "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium". Sitzungberichte der Kaiserlichen Akademie der Wissenschaften. Mathematisch-Naturwissen Classe. Part II, LXXVI. 76:373–435. Vienna. Reprinted in Wissenschaftliche Abhandlungen, Vol. II, reprint 42, pp. 164–223, Barth, Leipzig, 1909. Entropy 2015, 17, 1971–2009. Maxwell–Boltzmann statistics and the Boltzmann distribution remain central in the foundations of classical statistical mechanics. They are also applicable to other phenomena that do not require quantum statistics and provide insight into the meaning of temperature.
He made multiple attempts to explain the second law of thermodynamics, with the attempts ranging over many areas. He tried Helmholtz's monocycle model, a pure ensemble approach like Gibbs, a pure mechanical approach like ergodic theory, the combinatorial argument, the Stoßzahlansatz, etc.
Most chemists, since the discoveries of John Dalton in 1808, and James Clerk Maxwell in Scotland and Josiah Willard Gibbs in the United States, shared Boltzmann's belief in atoms and molecules, but much of the physics establishment did not share this belief until decades later. Boltzmann 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), based on Einstein's theoretical studies of 1905, confirmed the values of the Avogadro constant and the Boltzmann constant, convincing the world that the tiny particles really exist.
To quote Planck, "The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases".Max Planck, p. 119. This famous formula for entropy S isThe concept of entropy was introduced by Rudolf Clausius in 1865. He was the first to enunciate the second law of thermodynamics by saying that "entropy always increases".
where is the Boltzmann constant, and ln is the natural logarithm. (for , a German word meaning "probability") is the probability of occurrence of a macrostate, p. 21 or, more precisely, the number of possible microstates corresponding to the macroscopic state of a system – the number of (unobservable) "ways" in the (observable) thermodynamic state of a system that 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 ranges over all possible molecular conditions, and where denotes factorial. The "correction" in the denominator account for indistinguishable particles in the same condition.