Ludwig Boltzmann

Boltzmann's kinetic theory of gases seemed to presuppose the reality of atoms and molecules, but almost all German philosophers and many scientists like Ernst Mach and the physical chemist Wilhelm Ostwald disbelieved their existence.[19] Boltzmann was exposed to molecular theory by the paper of atomist James Clerk Maxwell entitled "Illustrations of the Dynamical Theory of Gases" which described temperature as dependent on the speed of the molecules thereby introducing statistics into physics. This inspired Boltzmann to embrace atomism and extend the theory.[20]

Boltzmann wrote treatises on philosophy such as "On the question of the objective existence of processes in inanimate nature" (1897). He was a realist.[21]

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.[22] It was Boltzmann who derived the first equation to model the dynamic evolution of the probability distribution Maxwell and he had created.[23] 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.[24] 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.

Boltzmann's 1898 I₂ molecule diagram showing atomic "sensitive region" (α, β) overlap

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".[25] This famous formula for entropy S is[26] An alternative is the information entropy definition introduced in 1948 by Claude Shannon.[27] 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).

$${\displaystyle S=k_{\mathrm {B} }\ln W}$$

where k_B is the Boltzmann constant, and ln is the natural logarithm. W (for Wahrscheinlichkeit, a German word meaning "probability") is the probability of occurrence of a macrostate[28] 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 N identical particles, of which N_i are in the ith microscopic condition (range) of position and momentum. W can be counted using the formula for permutations

$${\displaystyle W=N!\prod _{i}{\frac {1}{N_{i}!}}}$$

where i ranges over all possible molecular conditions, and where ${\displaystyle$ !} denotes factorial. The "correction" in the denominator account for indistinguishable particles in the same condition.

Boltzmann could also be considered one of the forerunners of quantum mechanics due to his suggestion in 1877 that the energy levels of a physical system could be discrete, although Boltzmann used this as a mathematical device with no physical meaning.[29]

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

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

$${\displaystyle {\frac {\partial f}{\partial t}}+v{\frac {\partial f}{\partial x}}+{\frac {F}{m}}{\frac {\partial f}{\partial v}}={\frac {\partial f}{\partial t}}\left.{\!\!{\frac {}{}}}\right|_{\mathrm {collision} }}$$

where ƒ 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 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's grave in the Zentralfriedhof, Vienna, with bust and entropy formula

The idea that the second law of thermodynamics or "entropy law" is a law of disorder (or that dynamically ordered states are "infinitely improbable") is due to Boltzmann's view of the second law of thermodynamics.

In particular, it was Boltzmann's attempt to reduce it to a stochastic collision function, or law of probability following from the random collisions of mechanical particles. Following Maxwell,[30] Boltzmann modeled gas molecules as colliding billiard balls in a box, noting that with each collision nonequilibrium velocity distributions (groups of molecules moving at the same speed and in the same direction) would become increasingly disordered leading to a final state of macroscopic uniformity and maximum microscopic disorder or the state of maximum entropy (where the macroscopic uniformity corresponds to the obliteration of all field potentials or gradients).[31] The second law, he argued, was thus simply the result of the fact that in a world of mechanically colliding particles disordered states are the most probable. Because there are so many more possible disordered states than ordered ones, a system will almost always be found either in the state of maximum disorder – the macrostate with the greatest number of accessible microstates such as a gas in a box at equilibrium – or moving towards it. A dynamically ordered state, one with molecules moving "at the same speed and in the same direction", Boltzmann concluded, is thus "the most improbable case conceivable...an infinitely improbable configuration of energy."[32]

Boltzmann accomplished the feat of showing that the second law of thermodynamics is only a statistical fact. The gradual disordering of energy is analogous to the disordering of an initially ordered pack of cards under repeated shuffling, and just as the cards will finally return to their original order if shuffled a gigantic number of times, so the entire universe must some-day regain, by pure chance, the state from which it first set out. (This optimistic coda to the idea of the dying universe becomes somewhat muted when one attempts to estimate the timeline which will probably elapse before it spontaneously occurs.)[33] The tendency for entropy increase seems to cause difficulty to beginners in thermodynamics, but is easy to understand from the standpoint of the theory of probability. Consider two ordinary dice, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small (1 in 36); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system must move to one of the more probable states.[34]

Ludwig Boltzmann's contributions to physics and philosophy have left a lasting impact on modern science. His pioneering work in statistical mechanics and thermodynamics laid the foundation for some of the most fundamental concepts in physics. For instance, Max Planck in quantizing resonators in his Black Body theory of radiation used Boltzmann’s constant to describe the entropy of the system to arrive at his formula in 1900.[35] However, Boltzmann's work was not always readily accepted during his lifetime, and he faced opposition from some of his contemporaries, particularly in regards to the existence of atoms and molecules. Nevertheless, the validity and importance of his ideas were eventually recognized, and they have since become cornerstones of modern physics. Here, we delve into some aspects of Boltzmann's legacy and his influence on various areas of science.

Boltzmann's work in statistical mechanics laid the groundwork for understanding the statistical behavior of particles in systems with a large number of degrees of freedom. In his 1877 paper he used discrete energy levels of physical systems as a mathematical device and went on to show that the same could apply to continuous systems which might be seen as a forerunner to the development of quantum mechanics.[39] One biographer of Boltzmann says that Boltzmann’s approach “pav[ed] the way for Planck.”[40]

The concept of quantization of energy levels became a fundamental postulate in quantum mechanics, leading to groundbreaking theories like quantum electrodynamics and quantum field theory. Thus, Boltzmann's early insights into the quantization of energy levels had a profound influence on the development of quantum physics.

• Verhältniss zur Fernwirkungstheorie, Specielle Fälle der Elektrostatik, stationären Strömung und Induction (in German). Vol. 2. Leipzig: Johann Ambrosius Barth. 1893.
• Theorie van der Waals, Gase mit zusammengesetzten Molekülen, Gasdissociation, Schlussbemerkungen (in German). Vol. 2. Leipzig: Johann Ambrosius Barth. 1896.
• Theorie der Gase mit einatomigen Molekülen, deren Dimensionen gegen die mittlere Weglänge verschwinden (in German). Vol. 1. Leipzig: Johann Ambrosius Barth. 1896.
• Abteilung der Grundgleichungen für ruhende, homogene, isotrope Körper (in German). Vol. 1. Leipzig: Johann Ambrosius Barth. 1908.
• Vorlesungen über Gastheorie (in French). Paris: Gauthier-Villars. 1922.

• 
  Volumes I and II of Vorlesungen über Gastheorie (1896-1898)
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  Title page to volumes I and II of Vorlesungen über Gastheorie (1896-1898)
• 
  Table of contents to volumes I and II of Vorlesungen über Gastheorie (1896-1898)
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  Introduction to volumes I and II of Vorlesungen über Gastheorie (1896-1898)

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. He was elected a member of the Royal Swedish Academy of Sciences in 1888 and a Foreign Member of the Royal Society (ForMemRS) in 1899.[1] Numerous things are named in his honour.

• Thermodynamics
• Statistical Mechanics

• Roman Sexl & John Blackmore (eds.), "Ludwig Boltzmann – Ausgewahlte Abhandlungen", (Ludwig Boltzmann Gesamtausgabe, Band 8), Vieweg, Braunschweig, 1982.
• John Blackmore (ed.), "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book One: A Documentary History", Kluwer, 1995. ISBN 978-0-7923-3231-2
• John Blackmore, "Ludwig Boltzmann – His Later Life and Philosophy, 1900–1906, Book Two: The Philosopher", Kluwer, Dordrecht, Netherlands, 1995. ISBN 978-0-7923-3464-4
• John Blackmore (ed.), "Ludwig Boltzmann – Troubled Genius as Philosopher", in Synthese, Volume 119, Nos. 1 & 2, 1999, pp. 1–232.
• Blundell, Stephen; Blundell, Katherine M. (2006). Concepts in Thermal Physics. Oxford University Press. p. 29. ISBN 978-0-19-856769-1.
• Boltzmann, Ludwig Boltzmann – Leben und Briefe, ed., Walter Hoeflechner, Akademische Druck- u. Verlagsanstalt. Graz, Oesterreich, 1994
• Brush, Stephen G. (ed. & tr.), Boltzmann, Lectures on Gas Theory, Berkeley, California: U. of California Press, 1964
• Brush, Stephen G. (ed.), Kinetic Theory, New York: Pergamon Press, 1965
• Brush, Stephen G. (1970). "Boltzmann". In Charles Coulston Gillispie (ed.). Dictionary of Scientific Biography. New York: Scribner. ISBN 978-0-684-16962-0.
• Brush, Stephen G. (1986). The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases. Amsterdam: North-Holland. ISBN 978-0-7204-0370-1.
• Cercignani, Carlo (1998). Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press. ISBN 978-0-19-850154-1.
• Darrigol, Olivier (2018). Atoms, Mechanics, and Probability: Ludwig Boltzmann's Statistico-Mechanical. Oxford University Press. ISBN 978-0-19-881617-1.
• Ehrenfest, P. & Ehrenfest, T. (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
• Everdell, William R (1988). "The Problem of Continuity and the Origins of Modernism: 1870–1913". History of European Ideas. 9 (5): 531–552. doi:10.1016/0191-6599(88)90001-0.
• Everdell, William R (1997). The First Moderns. Chicago: University of Chicago Press. ISBN 9780226224800.
• Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics. New York: Charles Scribner's Sons.
• Johnson, Eric (2018). Anxiety and the Equation: Understanding Boltzmann's Entropy. The MIT Press. ISBN 978-0-262-03861-4.
• Klein, Martin J. (1973). "The Development of Boltzmann's Statistical Ideas". In E.G.D. Cohen; W. Thirring (eds.). The Boltzmann Equation: Theory and Applications. Acta physica Austriaca Suppl. 10. Wien: Springer. pp. 53–106. ISBN 978-0-387-81137-6.
• Lindley, David (2001). Boltzmann's Atom: The Great Debate That Launched A Revolution In Physics. New York: Free Press. ISBN 978-0-684-85186-0.
• Lotka, A. J. (1922). "Contribution to the Energetics of Evolution". Proc. Natl. Acad. Sci. U.S.A. 8 (6): 147–51. Bibcode:1922PNAS....8..147L. doi:10.1073/pnas.8.6.147. PMC 1085052. PMID 16576642.
• Meyer, Stefan (1904). Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage 20. Februar 1904 (in German). J. A. Barth.
• Planck, Max (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
• Sharp, Kim (2019). Entropy and the Tao of Counting: A Brief Introduction to Statistical Mechanics and the Second Law of Thermodynamics (SpringerBriefs in Physics). Springer Nature. ISBN 978-3030354596
• Tolman, Richard C. (1938). The Principles of Statistical Mechanics. Oxford University Press. Reprinted: Dover (1979). ISBN 0-486-63896-0

• Uffink, Jos (2004). "Boltzmann's Work in Statistical Physics". Stanford Encyclopedia of Philosophy. Retrieved 11 June 2007.
• O'Connor, John J.; Robertson, Edmund F., "Ludwig Boltzmann", MacTutor History of Mathematics Archive, University of St Andrews
• Ruth Lewin Sime, Lise Meitner: A Life in Physics Chapter One: Girlhood in Vienna gives Lise Meitner's account of Boltzmann's teaching and career.
• Eftekhari, Ali, "Ludwig Boltzmann (1844–1906)." Discusses Boltzmann's philosophical opinions, with numerous quotes.
• Rajasekar, S.; Athavan, N. (7 September 2006). "Ludwig Edward Boltzmann". arXiv:physics/0609047.
• Ludwig Boltzmann at the Mathematics Genealogy Project
• Weisstein, Eric Wolfgang (ed.). "Boltzmann, Ludwig (1844–1906)". ScienceWorld.