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Ludwig Boltzmann

Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics.

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

In 1869, at age 25, he 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 then in 1871 he was with Gustav Kirchhoff and Hermann von Helmholtz in Berlin. In 1873 Boltzmann joined the University of Vienna as Professor of Mathematics and where he stayed till 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. 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 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. However, Boltzmann did not get along with some of his colleagues there; particularly when Ernst Mach became professor of philosophy and history of sciences in 1895. Thus in 1900 Boltzmann went to the University of Leipzig, on the invitation of Wilhelm Ostwald. After the retirement of Mach due to bad health, Boltzmann came back to Vienna in 1902, where he stayed until his death. Among his students there were Karl Przibram, Paul Ehrenfest and Lise Meitner.

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'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.[1] This famous formula for entropy S is[2] [3]

where k = 1.3806505(24) × 10−23 J K−1 is Boltzmann's constant, and the logarithm is taken to the natural base e. W 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 N identical particles, of which Ni are in the i-th microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof.

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. W 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 S is engraved on Boltzmann's tombstone at the Vienna Zentralfriedhof — his second grave.

Boltzmann's bust in the courtyard arcade of the main building, University of Vienna.

The Boltzmann equation

The Boltzmann equation was developed to describe the dynamics of an ideal gas.

where f represents the distribution function of single-particle position and momentum at a given time (see the Maxwell-Boltzmann distribution), F is a force, m is the mass of a particle, t is the time and v 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 f 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.[4] 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.

Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.

Energetics of evolution

In 1922, Alfred J. Lotka [5] 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


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.

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 had, almost certainly, bipolar disorder. 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.

Boltzmann, 24 years old.

Boltzmann at age 31 with his wife, Henrietta, in 1875

Photo from 1887 a(standing, from the left) Nernst, Streintz, Arrhenius, Hiecke, (sitting, from the left) Aulinger, Ettingshausen, Boltzmann, Klemencic, Hausmanninger

Ludwig Boltzmann - The genius of disorder


  • ↑ 1. Max Planck, p. 119.
  • ↑ 2. The 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.
  • ↑ 3. An alternative is the information entropy definition introduced in 1948 by Claude Shannon.[6] It was intended for use in communication theory, but is applicable in all areas. It reduces to Boltzmann's expression when all the probabilities are equal, but can, of course, be used when they are not. Its virtue is that it yields immediate results without resorting to factorials or Stirling's approximation. Similar formulas are found, however, as far back as the work of Boltzmann, and explicitly in Gibbs (see reference).
  • ↑ 4. Wallace Carothers, who discovered neoprene and nylon and founded the science of long-chain polymers, finally drank his cocktail of cyanide-laced lemon juice in 1937, one year before nylon reached the market.
  • ↑ 5. See Tolman, Chapter VI, for an extensive discussion.
  • ↑ 6.A memorial activity and discussion meeting on Lattice Boltzmann methods may be held in September, 2006. The Lattice Boltzmann Method is a new method in Computational fluid dynamics which utilities the theories of Boltzmann.


  • Brush, Stephen G., ed. & tr., Boltzmann, Lectures on Gas Theory, Berkeley, CA: U. of California Press, 1964
  • Brush, Stephen G., ed., Kinetic Theory, NY: Pergamon Press, 1965
  • Brush, Stephen G., “Boltzmann” in Dictionary of Scientific Biography, NY: Scribner’s, 1970
  • Brush, Stephen G., The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases, Amsterdam: North-Holland, 1986
  • P. Ehrenfest & T. Ehrenfest (1911) Begriffliche Grundlagen der statistischen Auffassung in der Mechanik, in: Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen. Band IV, 2. Teil ( F. Klein and C. Müller (eds.). Leipzig: Teubner, pp. 3–90. Translated as The conceptual Foundations of the Statistical Approach in Mechanics. New York: Cornell University Press, 1959. ISBN 0-486-49504-3
  • Max Planck (1914). The Theory of Heat Radiation. P. Blakiston Son & Co. English translation by Morton Masius of the 2nd ed. of Waermestrahlung. Reprinted by Dover (1959) & (1991). ISBN 0-486-66811-8
  • Everdell, William R., “The Problem of Continuity and the Origins of Modernism: 1870-1913” History of European Ideas 9:5(1988), 531-552
  • Klein, Martin J. “The Development of Boltzmann’s Statistical Ideas” The Boltzmann Equation: Theory and Applications, ed., E. G. D. Cohen & W. Thirring, Acta Physica Austraica Suppl X(1973), 53-106, Vienna/NY: Springer, 1973
  • Richard C. Tolman (1938). The Principles of Statistical Mechanics. Oxford University Press. Reprinted: Dover (1979). ISBN 0-486-63896-0
  • J. Willard Gibbs (1901). Elementary Principles in Statistical Mechanics. Ox Bow Press (1981). ISBN 0-918024-19-6.
  • David Lindley (Physicist) Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics. ISBN 0-684-85186-5
  • A.J.Lotka (1922) 'Contribution to the energetics of evolution'. Proc Natl Acad Sci, 8: pp. 147–51.

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