Portrait of Ludwig Boltzmann.
At a time when the classical continuum model of matter and energy reigned supreme, one physicist and philosopher was able to see beyond it: Ludwig Eduard Boltzmann. An important advocate of the atomic theory, Boltzmann made fundamental contributions to statistical mechanics and thermodynamics based on his atomicmolecular kinetic conception of matter.
Boltzmann interpreted entropy as a mathematical measure of a system’s disorder and derived a now famous equation (known as the Boltzmann equation) that describes the time evolution of the statistical properties of a gas consisting of molecules. It was the first equation to describe the progression of a probability in mathematical physics, presenting proof of the irreversibility of macroscopic phenomena.
Boltzmann was honored nationally and internationally during his lifetime, but he was also a figure of some controversy during his day because the scientific community had not yet embraced the atomic theory he advocated. His prescient work helped to lay the groundwork for the field of quantum mechanics.
Early life, education and academic career
Boltzmann was born the eldest son in a middleclass family in Vienna on 20 February 1844. His grandfather had moved to Vienna from Berlin to manufacture clocks and music boxes. Boltzmann’s father, Ludwig Georg, was a commercial tax collector, and his mother, Maria Pauernfeind, was a devoted caregiver and the daughter of a Salzburg businessman. Sadly, Boltzmann’s father passed away from tuberculosis when his son was just 15.
Boltzmann was a curious child with a keen interest in the natural world. He began his elementary education with a private teacher in his parents’ home. The piano lessons he took from noted Austrian composer Anton Bruckner sparked a lifelong interest in music. Boltzmann continued to play piano throughout his life, often with his son Arthur Ludwig, who accompanied him on the violin.
His formal education began in Linz, Austria, at the Akademisches Gymnasium, after the family had moved there for his father’s work. After passing the Gymnasium’s final examination in 1863, he studied mathematics and physics at the University of Vienna. The Physical Institute at the University had been founded 14 years earlier by Christian Doppler.
In order to study physics and mathematics at the University at that time, all students were required to enroll in the philosophy faculty. Boltzmann benefited from this prerequisite. He took 10 courses in philosophy, and his interest in this area remained strong throughout his life. His works derived from his joint interest in mathematical physics and philosophy.
During his last year of university study, Boltzmann worked with Josef Stefan, director of the Physical Institute starting in 1866. From their collaboration, Boltzmann developed a strong interest in the theory of gases. Stefan introduced Boltzmann to the publications of James Maxwell.
Also at the Physical Institute, Boltzmann met Johann Josef Loschmidt, who became his mentor and close friend. In 1865, Loschmidt, using the kinetic theory of Rudolf Clausius and James Clerk Maxwell, and Maxwell’s definition of the mean free path—the average distance a molecule moves between successive collisions—published the first accurate estimate of molecular dimensions. He estimated the diameter of a molecule of air (nitrogen or oxygen) to be 10^{–7} cm. The number of particles per unit volume of an ideal gas at standard temperature and pressure was given the name Loschmidt’s number in his honor, with a value of 2.6867774 x 10^{25} m^{–3}.
Boltzmann received his Ph.D. in 1866. Two years later, he became a lecturer (privatdozent) of mathematical physics at the University of Vienna. Then, in 1869, Boltzmann was awarded the venia legend, or the right to lecture, at a university. A year later, he became the chair of mathematical physics at the University of Graz.
Boltzmann’s “H”theorem and distribution law
Boltzmann was part of a movement in physics toward more statistical techniques, which began in the late 1850s. Interestingly, the statistical approach first gained traction in the social sciences, after statistical methods were applied to census information; Maxwell—one of the pioneers of statistical physics—acknowledged this in his 1873 lecture, in which he talked about how statistics were used to calculate the average macroscopic properties of gases.
During Boltzmann’s time at Graz, he published his “H”theorem, in which H indicates the negative of entropy, which is denoted by the symbol S. The theorem predicts an increase in the entropy of an irreversible process. In 1867, Maxwell published a paper deriving the probability distribution of the velocities of constituents of a gas at equilibrium.
Boltzmann’s seminal paper, “Further researches on the thermal equilibrium of gas molecules,” came in 1872. It contained his equation describing how the probability distribution of the velocities of constituents of a gas at equilibrium evolves over time, as well as a proof of the irreversibility of macroscopic phenomena.
The Boltzmann distribution is widely used in laser physics to calculate the population of states. In a system of atoms at temperature T, it gives the probability distribution function f_{i} that an atom has the discrete energy E_{i} asf_{i} (E_{i})=cg_{i}e^{–(Ei/kT)}. Here c is a normalization factor, g_{i} is the statistical weight for the level i, and k is Boltzmann’s constant. Particles that have this energy distribution follow MaxwellBoltzmann statistics.
Family life
Boltzmann returned to Vienna and, shortly after his paper was published, he became a professor of mathematics at the University of Vienna. A few years later, in 1875, he returned to the University of Graz. The following year, he married Henriette von Aigentler, who shared his passion for mathematics and physics. When Henriette decided to study mathematics at the university, she was initially denied access to classes. Boltzmann helped her to overrule that decision, and she became the first female student at the University of Graz.
Boltzmann and his wife had three daughters and two sons together. Boltzmann was a caring father who was extremely fond of his children. He maintained a strong love for nature and often took long walks with his wife and children and taught them botany. While living in Graz, he purchased a farm and chose to live in the country with his family. He was well known for his wonderful sense of humor. He remained in Graz for the next 14 years—a happy and productive time that sharply contrasted with the later years of his life.
Peak scientific years
It was at Graz where Boltzmann made what may be his greatest achievement: His monumental statistical interpretation of the second law of thermodynamics, which was published in 1877. Interestingly, this work resulted directly from a strong criticism of his earlier Htheorem. Boltzmann’s colleague Loschmidt had condemned that work, stating that it should not be possible to deduce an irreversible process from timesymmetric dynamics.
Boltzmann’s tombstone features a bust of the scientist with his famous formula S = k log W.
In addressing this criticism, he made a statistical interpretation of the second law of thermodynamics, in which he showed that the concept of entropy is a logarithmic function of the number of microstates that corresponds to a thermodynamic state of a system. Einstein labeled this as the Boltzmann Principle. This is an important example of how the criticism of one’s work can result in the generation of new theoretical developments.
To comprehend it, one must first understand the concept of macrostates and microstates. A macrostate of a system relates to its macroscopic properties, such as pressure and temperature; those are the extensive properties that we measure. A microstate, on the other hand, is a specific configuration of a thermodynamic system that undergoes thermal fluctuations. During these changes, the system has a probability to exist in a given microstate—in other words, each microstate is associated with a given probability.
Boltzmann showed that, for any system that exists in an improbable state, it will evolve through more and more probable states until it finally comes to be in the state described by the Maxwell distribution. The Boltzmann Principle can be summarized as follows: The entropy S of a macrostate is proportional to the logarithm of the number of microstates W, or as expressed in the famous Boltzmann formula: S = k log W. The constant k, known as the Boltzmann constant, was evaluated in 1900 by Max Planck.
In 1884, Boltzmann derived theoretically what his mentor Stefan had previously empirically proposed: The total energy radiated by a black body, which by definition is an object that absorbs all the incident electromagnetic radiation and is an ideal radiator of thermal energy, is proportional to the fourth power of its absolute temperature.
At this time Boltzmann’s reputation was at its peak. He attracted brilliant foreign students to Graz, including the future Nobel laureates Svante Arrhenius and Walther Nernst. He rose to the position of rector of the University in 1887. He received an honorary doctorate from Oxford University and was made a member of many scientific academies.
Theoretical physics
Boltzmann decided to leave Graz, and in 1891, he was appointed to the position of chair of theoretical physics at the University of Munich in Germany. There, he lectured on theoretical physics and mathematics, specifically the theory of numbers. At that time his sight had begun to deteriorate to the point that his wife had to read aloud scientific papers for him.
Discouraged by health problems and missing his native Austria, Boltzmann returned to his beloved Vienna after 18 years in the spring of 1895 in the position of professor of theoretical physics at the University of Vienna. While Boltzmann is largely known as a theoretical physicist, he was also a talented experimentalist. For example, he provided experimental verification for Maxwell’s electromagnetic theory before Heinrich Hertz’s experiments in 18871888 and he investigated the dielectric constants of solid insulators and gases between 1873 and 1874.
The main problems Boltzmann worked on included nonequilibrium systems, the kinetic interpretation of irreversible processes and statistical mechanics, which explain the bulk properties of matter in terms of their microscopic constituents. He integrated statistical, mechanical and atomic concepts into his physics.
Around this time, Boltzmann entered into some difficult years. Always prone to highs and lows in his mood, Boltzmann was formally diagnosed in 1888 with neurasthenia—a term used to describe a sad, nervous or depressed state of mind. He is believed to have suffered from the rapid alternation between depression and elevated mood that is today referred to as bipolar disorder.
In 1889, Boltzmann was dealt another crushing blow: His first son Ludwig died from appendicitis at the age of 11. Boltzmann felt responsible for the death, since he had failed to question the general physician’s diagnosis, which incorrectly suggested that the illness was not serious.
The existence of atoms and molecules
Boltzmann struggled in his final years to defend his atomicmolecular theories. Most Germanspeaking physicists and philosophers opposed the concepts of atoms and molecules. Boltzmann conflicted with some of his colleagues in Vienna, particularly Ernst Mach, who became a professor of philosophy and the history of science in 1885. Mach demanded experimental evidence for scientific concepts, and at the time there was a paucity of data to support the existence of atoms.
The physical chemist Wilhelm Ostwald expressed even stronger opposition to Boltzmann’s theories. Ostwald claimed that our senses only detect energy, and that atoms are merely hypothetical constructs. Both Mach and Ostwald were proponents of the philosophy of positivism, which held that valid knowledge is based on sensory experience and positive verification. (It was later refuted by the Russian philosopher Vladimir Illyich Lenin, who noted that matter existed long before there were humans who could experience any sensations.)
Starting in the late 1880s, Boltzmann attempted to formulate a middle position that would be acceptable to both the atomists and the antiatomists and to end the fierce acrimony of the debates. Boltzmann pointed to the philosophical concept introduced by Heinrich Hertz in which atoms were considered to be useful models, pictures or representation of matter. Building on that idea, he promoted the modern use of models to help elucidate our understanding of the world.
Boltzmann’s bold and prescient support of the existence of atoms was enhanced in 1905 by the works of Einstein. Indeed, Einstein used Boltzmann’s ideas in his 1905 papers on molecular dimensions and another paper on Brownian motion. Those works gave great support to the atomic hypothesis.
Boltzmann was interested in communicating beyond the scientific community: His book Populäre Schriften (Popular Writings) contains essays on his personal views of science and his life experiences. His sense of humor pervades this work. In one essay, “Reise einses deutschen professors in Eldorado,” he humorously compares California, a state he visited in 1905, to his native Austria.
Josiah Willard Gibbs: Building on Boltzmann’s Work
Boltzmann’s 1884 paper developed the fundamentals of statistical mechanics. But it was Josiah Willard Gibbs who reconstructed Boltzmann’s works and developed them further in order to form our modern approach to statistical mechanics.
Born in 1839, Gibbs was a theoretical physicist, mathematician and a chemist who spent most of his career at Yale University. He entered Yale as an undergraduate at age 15 and graduated in 1858. From there, he went on to receive from Yale the first American Ph.D. in engineering in 1863. Between 1866 and 1869, he studied in Paris, Berlin and Heidelberg, where his professors included Kirchhoff and Helmholtz. Back at Yale, he was appointed professor of mathematical physics in 1871.
From 1874 to 1878, Gibbs published papers on chemical thermodynamics in The Transactions of the Connecticut Academy of Sciences. He derived the concepts that form the foundation of the field of physical chemistry, including chemical potential, Gibbs free energy and the Gibbs phase rule and its threedimensional models. These papers were eventually published as his monograph, On the Equilibrium of Heterogeneous Substances.
One of Gibbs’ aims was to popularize the use of vector analysis in modern physics. This included scalar and vector products, divergence and curl. Gibbs also sought to demonstrate the notional and computational advantages of vector analysis over quaternions, which were previously widely used in mathematical physics.
To this end, between 1881 and 1884, Gibbs circulated a short pamphlet called Elements of Vector Analysis. In 1901, Yale University Press published a book, titled Vector Analysis: A TextBook for the Use of Students of Mathematics and Physics, that is based on the lectures of Gibbs on the topic of vector analysis; it was written by Gibbs’ student Edwin Bidwell Wilson. Oliver Heaviside, an English engineer, physicist and mathematician, independently developed many of the same concepts of vector analysis and published them in his Electromagnetic Theory in 1893.
After 1889, Gibbs developed his concept of statistical mechanics. The modern student of statistical mechanics will be familiar with terms that Gibbs defined in his seminal book, Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics, which was published in 1902 by Yale University Press. For example, Gibbs’ work led to the modern terms ensemble, canonical ensemble, grand canonical ensemble or macroconnonical ensemble, and microcanonical ensemble.
Because Gibbs published in the Transactions of the Connecticut Academy of Sciences, his work was not distributed to a widespread audience. He circumvented this problem by sending copies of his papers to approximately 100 scientists; the list included James Clerk Maxwell, Friedrich Wilhelm Ostwald, Henri Louis le Chatelier and Ludwig Boltzmann. After Oswald translated Gibbs’ papers into German in 1892, and le Chatelier translated them into French in 1899, his concepts spread rapidly throughout Europe.
Gibbs’ works influenced such disparate authors and educators as the physical chemists Gilbert N. Lewis and Merle Randall, the economist Paul Samuelson, and the mathematician Norbert Wiener. Boltzmann was aware of Gibbs’ contributions, but the two never met.


Late life and legacy
By 1903, Boltzmann’s health had further deteriorated. He suffered from asthma attacks at night as well as angina pectoris and intense headaches. His manicdepressive disorder contributed to his suffering, often leading to a lack of sleep.
Despite these circumstances, Boltzmann agreed to participate in a scientific meeting held at the 1904 World’s Fair in St. Louis, Mo., U.S.A. Accompanied by his son Arthur Ludwig, he lectured at the World’s Fair and then visited Detroit and Chicago. This was his second visit to the United States; he and his wife had also journeyed there in 1899, visiting Massachusetts, New York, Montreal, Buffalo, Washington, Baltimore and Philadelphia. Boltzmann returned to the United States in 1905 to present 30 lectures in a summer school at the University of California at Berkeley.
But sadly, Boltzmann was a haunted man. On 5 September 1906, during his summer vacation in Duino, Italy, near Trieste, he committed suicide by hanging. We may never understand the reasons that led Boltzmann to take his own life. His longterm battles with the famous physicists and philosophers of the times over his concept of atoms and molecules, his deteriorating health and his mental struggles are cited as possible contributing factors. Boltzmann is buried in the Central Cemetery in Vienna; his tombstone contains a bust and behind it is his famous formula S = k log W.
Within years of Boltzmann’s death, the reality of atoms and molecules was validated and became a part of mainstream physics. In 1908, when Jean Baptiste Perrin investigated colloidal suspensions, he based his analysis on Einstein’s theoretical papers of 1905 and validated the values of both Avogadro’s number and the Boltzmann constant. In 1909, Perrin published a paper under the title, “Brownian movement and molecular reality,” in which he experimentally confirmed the theoretical prior predictions of both Einstein and Marian Smoluchowski. He clearly shows the power of the kineticmolecular approach to problems in both physics and chemistry. In 1926, Perrin received the Nobel Prize for his research on colloids.
Perrin wrote a prescient thought in his 1912 book entitled Atoms: “A time will perhaps come in the future when atoms can be seen directly and will become as easy to observe as microbes are today.” Much later atoms did indeed become visible. In 1955, Erwin Wilhelm Müller used his fieldion microscope that he invented in 1951, to observe individual tungsten atoms on the tip of a tungsten needle at 78 K.
One possible reason that Boltzmann’s ideas did not achieve widespread acceptance during his lifetime was that his writing style was difficult to comprehend. Following his death, Paul Ehrenfest, a student of Boltzmann’s, and his wife Tatiana spent five years reworking an article on Boltzmann’s statistical mechanics, which was published in 1912 in the German Encyclopedia of Mathematical Sciences, and then in their book, The Conceptual Foundations of the Statistical Approach in Mechanics. Their clear, logical exposition of Boltzmann’s theoretical developments of statistical mechanics helped to popularize his work. Further credit should be given to Josiah Willard Gibbs, who reformulated the theoretical works of Boltzmann and expanded upon them to form our modern approach to statistical mechanics.
Boltzmann’s legacy is large: He bridged the gap between the science of Maxwell and that of Einstein. His work was also used in 1900, when Max Planck used his combinatorial mathematics in his pioneering work on blackbody radiation and his discovery of energy quanta, a source of the quantum theory. The bold vision of this exceptionally versatile scientist paved the way for a true revolution in our understanding of the physical universe.
Barry R. Masters is a Fellow of AAAS, OSA and SPIE. He is a visiting scientist with the department of biological engineering, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.
References and Resources
>> J.W. Gibbs. Vector Analysis, Founded upon the lectures of J.W. Gibbs, by E.B. Wilson, New Haven, Yale University Press (1901).
>> J.W. Gibbs, Elementary Principles in Statistical Mechanics, Developed with Especial Reference to the Rational Foundation of Thermodynamics, unabridged republication of 1902 edition, New York, Dover Publications (1960).
>> S.G. Brush. Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager. Princeton, Princeton University Press (1983).
>> S.G. Brush. The Kind of Motion We Call Heat, 2 volumes, Amsterdam, NorthHolland (1986).
>> H. Hörz and A. Laaß. Ludwig Boltzmann’s Wege nach Berlin, Berlin, AkademieVerlag (1989).
>> L. Boltzmann. Lectures on Gas Theory, translated by S.G. Brush. New York, Dover Publications, Inc. (1995).
>> C. Cercignani. Ludwig Boltzmann, The Man who Trusted Atoms. Oxford, Oxford University Press (1998).
>> L. Boltzmann. Entropie und Wahrscheinlichkeit, Ostwalds Klassiker der Exakten Wissenschaften, Band 286, Frankfurt am Main, Verlag Harri Deutsch (2002).
>> S.G. Brush. The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, London, Imperial College Press (2003).
>> W.L. Reiter. “In memoriam, Ludwig Boltzmann: A life of passion,” Phys. Perspect, 9, 357 (2007).