# Boltzmann, Ludwig Eduard

# BOLTZMANN, LUDWIG EDUARD

(*b*. Vienna, Austria, 20 February 1844; *d*. Duino, near Trieste, 5 September **1906** )

*physics*,*thermodynamics*, ** statistical mechanics** . For the original article on Boltzmann see

*DSB*, vol. 2.

After the original ** DSB** article, historical research focused on the study of specific aspects of Boltzmann’s work, particularly the application of mechanics to thermodynamic systems, the role of the ergodic hypothesis, the analysis of the statistical arguments, and his contribution to the philosophical debate at the end of the nineteenth century. Furthermore, the publication of a large section of Boltzmann’s scientific and personal correspondence has contributed to the shedding of light upon the historical and philosophical context (see Fasol-Boltzmann, 1990; Höflechner, 1994; Blackmore, 1995). This postscript provides a picture of the most interesting elements of this critical work.

**Mechanics and Thermodynamics.** Throughout the 1850s and early 1860s, Rudolf Clausius’s and James Clerk Maxwell’s works made clear that the study of thermodynamic properties was closely related to the mechanical analysis of systems with many degrees of freedom. On the basis of this relation, during the **1860–1885** period a number of attempts were made to obtain mechanical analogies of thermodynamic laws, that is, expressions which were formally identical to the latter, but which contained mechanical quantities only (see Klein, 1974; Bier-halter, 1992). The most important among these analogies concerned the second law of thermodynamics.

Ludwig Boltzmann was the first, as early as 1866, to use tools of the Lagrangian analytical mechanics to investigate this issue. In his paper “Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie,” he showed that if the quantity of heat provided to the system is interpreted as mechanical energy, and the temperature is interpreted as the average kinetic energy of the particles, the principle of least action leads to an equation that is formally identical with the second law. In 1871 Clausius (who was especially concerned with finding a mechanical interpretation for the disgregation function) and the Hungarian physicist Kálmán Szily arrived independently at a similar result. However, Boltzmann considered the mechanical analogy as the starting point of a more ambitious program: to specify the conditions that a mechanical system has to satisfy in order to represent thermodynamic behaviors.

Indeed, to obtain the mechanical analogy of the second law, the action integral for a system trajectory must be solved. In general, the solution depends on the initial and the final states, which are not knowable because of the extreme complexity of the system. However, if some

special conditions are met, this requirement can be neglected. Boltzmann realized that the kinetic theory needed an additional particular assumption about the long-term behavior of the system. This assumption was as of 2007 known as the ergodic hypothesis: a mechanical system tends to pass through all the states allowed by its general constraints; in other words, its trajectory fills all the allowed phase space.

From Boltzmann’s point of view, the ergodic hypothesis is closely connected with thermodynamic behaviors such as the dependence on general constraints, irreversible evolution towards equilibrium, and so on. Hence, Boltzmann devoted some sections of his papers of 1871 (“Einige allgemeine Sätze über Wärmegleichgewicht”) and of 1877 (“Bemerkungen über einige Probleme der mechanischen Wärmetheorie”) to the study of mechanical trajectories in order to characterize the features of the ergodic kind of motion. In 1879, Maxwell also recognized the importance of the ergodic hypothesis, although his version of this assumption differs significatively from Boltzmann’s (see von Plato, 1991).

**Role of Statistical Arguments.** The ergodic hypothesis was important also for the statistical understanding of thermodynamic phenomena and the application of statistical arguments. In fact, if the trajectory of the system covers the entire phase space, then a description of such a trajectory can be obtained by means of a combinatorial calculation of the possible configurations in the phase space itself. Accordingly, the statistical arguments of which Boltzmann made use after his 1868 paper “Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten” are not simply a generalization of Maxwell's; on the contrary, they played a constructive part in Boltzmann’s general approach to thermodynamics.

Research in the 1990s revealed interesting differences between the arguments used in 1868 and in the 1877 paper “Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht” (see Bach, 1990; Costantini and Garibaldi, 1997; Costantini and Garibaldi, 1998). In 1868 Boltzmann calculated the marginal distribution of energy for a molecule, that is, the probability that a molecule has a certain energy independent of the energy distribution of the other molecules. To perform his calculation, Boltzmann assumed that the energy was divided in many intervals of equal dimension and defined probability in a classical way as the ratio between favorable cases and total cases. It is worth noting that Boltzmann’s marginal argument is probabilistically exact, because it uses classical probability rather than relative frequencies. In 1927 a similar procedure was used by Louis Brillouin in his important paper on quantum statistics. Furthermore, a deep analysis shows that Boltzmann was able to derive and to apply statistical distributions that would be studied by professional statisticians only many years later, notably the Polya bivariate distribution, the Beta distribution, and the Gamma distribution (see Costantini and Garibaldi, 1997).

In 1877, Boltzmann had to resort again to statistical arguments to answer the paradox of reversibility raised by Josef Loschmidt, but the procedure he adopted is remarkably different from that of 1868. First, Boltzmann considered *packets* of energy instead of intervals in order to apply his famous urn model. While an interval is defined by two values of energy, a packet is defined by one value only; this fact changes the number of statistical predicates and the form of the normalization factor. Second, he calculated the equilibrium distribution by means of the maximization of the state probability. Third, he understood the probability as relative frequency and made use of the Stirling approximation.

These developments notwithstanding, the role of statistical arguments in Boltzmann’s approach to thermodynamics remained not completely clear. The difficulty was partly due to ambiguities and inconsistencies in Boltzmann’s papers. For instance, the issue of what kind of statistics Boltzmann actually used is not settled. In the 1990s, interpreters have hypothesized that Boltzmann assumed the equiprobability not for the individual configurations, but for the distributions of the occupation numbers (how many molecules were in each energy cell). This assumption would imply that Boltzmann used the Bose-Einstein statistics rather than the classical Maxwell-Boltzmann statistics (see Bach, 1990).

Boltzmann’s popular and philosophical writings were also the subject of detailed analysis from the 1970s through the 1990s. Scholars agree that Boltzmann’s methodology and epistemology evolved over time, but there was wide disagreement in interpreting the direction of this evolution. Lakatosian writers claimed that Boltzmann started from realism and in his late years moved to the positivistic side (see Elkana, 1974; Clark, 1976). According to other interpretations, Boltzmann’s late position is, rather, a more tolerant form of realism coupled with a pluralistic methodology of evolutionistic flavor (see Hiebert, 1981; de Regt, 1996). However, this issue, as well as the relation between Boltzmann’s philosophical view and his scientific work, remained as of 2007 an open question.

## SUPPLEMENTARY BIBLIOGRAPHY

### WORKS BY BOLTZMANN

Fasol-Boltzmann, Ilse M., ed., *Principien der Naturfilosofi: Lectures on Natural Philosophy, 1903–1906*. Berlin-Heidelberg: Springer Verlag, 1990. Boltzmann’s philosophical lectures at the University of Vienna.

McGuinness, Brian, ed. *Theoretical Physics and PhilosophicalProblems*:*Selected Writings/Ludwig Boltzmann*. Dordrecht, Holland and Boston: Reidel, 1974. An English translation of Boltzmann’s popular papers.

### OTHER SOURCES

Bach, Alexander. “Boltzmann’s Probability Distribution of 1877.”*Archive for History of Exact Sciences* 41 (1990): 1–40. An interpretation of Boltzmann’s statistical arguments in terms of Bose-Einstein statistics.

Battimelli, Gianni, Maria Grazia Ianniello, and Otto Kresten, eds.*Proceedings of the International Symposium on Ludwig Boltzmann*. Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1993.

Bierhalter, Günther. “Von L. Boltzmann bis J. J. Thomson: die Versuche einer mechanischen Grundlegung der Thermodynamik (1866–1890).” *Archive for History of Exact Sciences* 44 (1992): 25–75. A survey on the problem of the mechanical analogy of the second law.

Blackmore, John, ed. *Ludwig Boltzmann, His Later Life andPhilosophy, 1900–1906*. Dordrecht, Holland and Boston: Kluwer, 1995.

Brush, Stephen. *Statistical Physics and the Atomic Theory ofMatter*. Princeton, NJ: Princeton University Press, 1983.

Clark, Peter. “Atomismus versus Thermodynamics.” In *Method and Appraisal in the Physical Sciences*:*The Critical Background to Modern Science, 1800–1905*, edited by Colin Howson. Cambridge, U.K.: Cambridge University Press, 1976. A Lakatosian interpretation of Boltzmann’s scientific work. Costantini, Domenico, and Ublado Garibaldi. “A Probabilistic Foundation of Elementary Particle Statistics. Part I.” *Studies in History and Philosophy of Modern Physics* 28 (1997): 483–506.

———. “A Probabilistic Foundation of Elementary Particle Statistics. Part II.” *Studies in History and Philosophy of Modern Physics* 29 1 (1998): 37–59. A deep analysis of Boltzmann’s statistical argument in modern terminology. De Regt, Hank W. “Philosophy and the Kinetic Theory of Gases.” *British Journal for the Philosophy of Science* 47 (1996): 31–62.

Elkana, Yehuda. “Boltzmann’s Scientific Research Programme and Its Alternatives.” In *The Interaction between Science and Philosophy*, edited by Yehuda Elkana. Atlantic Highlands, NJ: Humanities Press, 1974.

Hiebert, Erwin N. “Boltzmann’s Conception of Theory Construction: The Promotion of Pluralism, Provisionalism and Pragmatic Realism.” In *Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science*. Vol. II, edited by Jakko Hintikka, David Gruender, and Evandro Agazzi. Dordrecht, Holland and Boston: Reidel, 1981.

Höflechner, Walter, ed. *Ludwig Boltzmann. Leben und Briefe*. Graz: Akademische Druck und Verlaganstalt, 1994. Boltzmann’s scientific correspondence with comments and a biography.

Klein Martin J. “Boltzmann, Monocycles and Mechanical Explanation.” In *Philosophical Foundations of Science: Proceedings of Section L, 1969, American Association for the Advancement of Science*, edited by Raymond J. Seeger and Robert S. Cohen. Dordrecht, Holland and Boston: Reidel, 1974.

Von Plato, Jan. “Boltzmann’s Ergodic Hypothesis.” *Archive forHistory of Exact Sciences* 42 (1991): 71–89.

De Regt, Hank W. “Philosophy and the Kinetic Theory of Gases.” *British Journal for the Philosophy of Science* 47 (1996): 31–62.

*Massimiliano Badino*

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