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Childhood and Education Ludwig Edward Boltzmann was born on February 20, 1844 in Vienna. His father, Ludwig Georg Boltzmann was a tax official. His grandfather, who had moved to Vienna from Berlin, was a clock manufacturer, and Boltzmann’s mother, Katharina Pauernfeind, was originally from Salzburg. He received his primary education from a private tutor at the home of his parents. Boltzmann attended high school in Linz, Upper Austria. At age 15, Boltzmann lost his father. 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. Academic Career
Overview Boltzmann's most important scientific contributions were in kinetic theory, including the Maxwell-Boltzmann distribution for molecular speeds in a gas. In addition, Maxwell-Boltzmann statistics and the Boltzmann distribution over energies remain the foundations of classical statistical mechanics. They are applicable to the many phenomena that do not require quantum statistics and provide a remarkable insight into the meaning of temperature. Much of the physics establishment rejected his thesis about 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. To quote Planck, The logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases. This famous formula for entropy is 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, or 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 due to the fact that 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 equation for is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof — his second grave. The Boltzmann equation The Boltzmann equation was developed to describe the dynamics of an ideal gas. ight|_mathrm 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 only expected to lead to incorrect results for an ideal gas 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. For higher density fluids, the Enskog equation was proposed. For moderately dense gases this equation, which reduces to the Boltzmann equation at low densities, is fairly accurate. However the Enskog equation is basically an heuristic approximation without any rigorous mathematical basis for extrapolating from low to higher densities. 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. Energetics of evolution In 1922, Alfred J. Lotka * referred to Boltzmann as one of the first proponents of the proposition that available energy (also called exergy) can be understood as the fundamental object under contention in the biological, or life-struggle and therefore also in the evolution of the organic world. Lotka interpreted Boltzmann's view to imply that available energy could be the central concept that unified physics and biology as a quantitative physical principle of evolution. Howard T. Odum later developed this view as the maximum power principle. Significant contributions 1872 - Boltzmann equation; H-theorem 1877 - Boltzmann distribution; relationship between thermodynamic entropy and probability. 1884 - Derivation of the Stefan-Boltzmann law Evaluations Closely associated with a particular interpretation of the second law of thermodynamics, he is also credited in some quarters with anticipating quantum mechanics. For detailed and technically informed account of Boltzmann's contributions to statistical mechanics consult the article by E.G.D. Cohen. See also: Philosophy of thermal and statistical physics. Final years Boltzmann was subject to rapid alternation of depressed moods with elevated, expansive or irritable moods. 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; he was, almost certainly, manic-depressive. 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 he committed suicide during an attack of depression. Notes See also | ||||||||||||
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