Kinetic theory of gases and statistical mechanics

1860-1877 Energy and motion theory

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A description of a gas not as a continuous medium but as a population of molecules known only through statistics: the distribution of velocities, mean values, the probability of one state or another.

The Scotsman James Clerk Maxwell derived the law of molecular velocities in 1860. The Austrian Ludwig Boltzmann proved in 1872 that collisions drive any initial distribution towards Maxwell's, and in 1877 identified entropy with the probability of a state.

Out of this came the calculation of viscosity, heat conduction and diffusion, the first estimate of molecular size (Loschmidt, 1865), and the understanding of the second law as statistical rather than absolute. The state-counting method invented here was carried over to radiation by Planck in 1900, who thereby obtained the quantum of action. The reality of the molecules themselves stayed disputed until Perrin's experiments: the Austrian Ernst Mach and the German Wilhelm Ostwald held the atom to be a superfluous and unproven hypothesis.

Why this date

The axis year is 1860, the year the method was first published: Maxwell, Illustrations of the Dynamical Theory of Gases, Philosophical Magazine, 1860; the paper was read in 1859. The upper bound of the label, 1877, marks Boltzmann's paper identifying entropy with the probability of a state. The H-theorem was published between those dates, in 1872.

What it grew from

• Mechanical equivalent of heat (Joule)

  influenced · Confidence: high

  The mechanical equivalent of heat and the conservation of energy made the idea of heat as molecular motion admissible for serious consideration, and so opened the way for the kinetic theory in the 1850s.

  Link sources

  • Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

    The 'kinetic theory of heat' -- the old idea that heat is directly related to the kinetic energy of atomic motion -- had to be given serious consideration as soon as energy conservation and thermodynamics had been introduced in the middle of the 19th century.

  • Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

    The kinetic theory was not widely accepted in the 18th century; most scientists preferred the Newtonian repulsion theory, which was compatible with the idea that heat is a fluid, 'caloric,' rather than the energy of atomic motion.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• Chemical atomic theory

  influenced · Confidence: medium

  Chemistry gave the kinetic theory a discrete particle with a definite weight, and Maxwell treated the agreement between molecular masses computed from dynamics and the chemists' conclusions as an argument for the real existence of molecules.

  Dispute: No source links Dalton to the kinetic theory directly: Maxwell speaks of 'the deductions of chemists' in general, and the determination of molecular weights from vapour densities is Cannizzaro's subject, not Dalton's. Moreover Brush stresses that their pictures of a gas are opposites: Dalton's particles are almost motionless and surrounded by atmospheres of caloric, he adopted Newtonian repulsion and rejected Gay-Lussac's conclusions ('Dalton explicitly adopted Newton's hypothesis that gas pressure is due to short-range repulsive forces between particles, and envisioned these particles as being surrounded by caloric atmospheres in contact with those of their neighbors'). What the kinetic theory took from chemistry is the discrete particle with a weight, not Dalton's model of a gas.

  Link sources

  • J. C. Maxwell, Atom, Encyclopaedia Britannica, 9th ed., vol. 3 (1875) Verified 20 Sep 2026

    The consistency of this information with the deductions of chemists from the phenomena of combination, greatly strengthens the evidence in favour of the actual existence and motion of gaseous molecules.

  • Klodian Coko (rev. of A. Chalmers), Atomism from the 17th to the 20th Century, Stanford Encyclopedia of Philosophy (2026) Verified 20 Sep 2026

    This legitimated the use of vapour densities for the determination of relative molecular weights. This in turn led to definitive atomic weights and formulae that coincided with those that organic chemistry had yielded by the 1860s.

  • Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

    This model is radically different from the one proposed at the beginning of the 19th century by the English chemist John Dalton (1766-1844), founder of the modern chemical atomic theory.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• 'Sur l'homme': the astronomers' law of error applied to people

  influenced · Confidence: high

  In 1860 Maxwell obtained the distribution of molecular velocities by carrying into physics the normal law which had until then been applied to people: Brush calls this the crucial step and states outright that the formula was borrowed from the social sciences, where Quetelet had applied it extensively. It is the same branch that leads from Quetelet to Galton, only turned towards physics.

  Dispute: Historians differ over whether Maxwell read Quetelet himself or got the same idea from John Herschel's review of Quetelet's 'Letters on the theory of probabilities' (Edinburgh Review, July 1850). Porter (The Rise of Statistical Thinking, 1986), and several authors following him, trace Maxwell's acquaintance with the procedure to Herschel's review; Herschel is not mentioned in the chapter of Brush cited here. For the map this does not change the direction of the link, but it means the transmission went through an intermediary.

  Link sources

  • Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

    To describe this distribution he borrowed a mathematical formula from the social sciences. The crucial step was his translation of the 'normal distribution law' or 'law of errors,' discovered by Adrain and Gauss and extensively applied by Quetelet, into a distribution law for molecular velocities.

  • J. C. Maxwell, Illustrations of the Dynamical Theory of Gases, Philosophical Magazine 19-20 (1860) Verified 20 Sep 2026

    the velocities are distributed among the particles according to the same law as the errors are distributed among the observations in the theory of the 'method of least squares.'

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

  Further reading

  Not evidence: these carry no quotation and support nothing we wrote. They are books and scans to read next.

  • Porter T. M. «The Rise of Statistical Thinking, 1820-1900», Princeton University Press (1986)

What grew out of it

• Quantum of action

  prerequisite for · Confidence: high

  Planck derived the radiation formula by applying the combinatorial counting of states from Boltzmann's 1877 paper, and it was that counting which required energy to be divided into finite elements, whence the quantum of action.

  Dispute: Historians disagree about exactly what Planck took from Boltzmann. Klein and those following him hold that in 1900 Planck introduced something genuinely new into physics, finite elements of energy, and knew it. Kuhn in 1978 objects: for Planck energy remained continuous, the combinatorics was a technical device, and discontinuity appeared only in 1905-1906 with Einstein and Ehrenfest. Gearhart adds a third point: Planck adopted Boltzmann's formalism but not his probabilistic view of the second law, which Planck held to be absolute ('Planck thought it entirely impossible that there could be any exceptions…to the law of entropy increase').

  Link sources

  • Martin J. Klein, Max Planck and the Beginnings of the Quantum Theory, Archive for History of Exact Sciences 1 (1962) Verified 20 Sep 2026

    During those two months PLANCK had successfully changed the direction of all of his previous thought in recognizing and adopting BOLTZMANN'S insight into the relationship between entropy and probability.

  • Clayton A. Gearhart, Planck, the Quantum, and the Historians, Physics in Perspective 4 (2002) Verified 20 Sep 2026

    In December 1900, Planck cited Boltzmann, 'especially' his 1877 paper, and said that by introducing probability considerations, he could derive an expression for the entropy of a resonator.

  • M. Planck, quoted in Martin J. Klein, Max Planck and the Beginnings of the Quantum Theory, Archive for History of Exact Sciences 1 (1962) Verified 20 Sep 2026

    I was not only indifferent but to a certain extent even hostile to the atomic theory which was the foundation of his entire research.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• Energetics: physics without atoms

  influenced · Confidence: medium

  The kinetic theory became the main target of energetics: it refused to explain thermal processes through the mechanics of moving molecules, and in 1895 at Lübeck the argument came to an open clash between Ostwald and Boltzmann.

  Dispute: This concerns confrontation, not origin: the sources do not claim that energetics arose as a reaction to the kinetic theory, and they trace Ostwald's motives to the successes of thermodynamics and chemical energetics instead. Mach rejected atoms for different, positivist reasons, and his position did not coincide with Ostwald's energetics.

  Link sources

  • Jos Uffink, Boltzmann's Work in Statistical Physics, Stanford Encyclopedia of Philosophy (2024) Verified 20 Sep 2026

    It resisted attempts to comprehend energy, or these transformations in terms of mechanical pictures.

  • Jos Uffink, Boltzmann's Work in Statistical Physics, Stanford Encyclopedia of Philosophy (2024) Verified 20 Sep 2026

    In the case of Ostwald, and his 'energetics', Boltzmann did become involved in a more heated dispute at a meeting in Lübeck in 1895.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

• Brownian motion as a measurement of molecules

  prerequisite for · Confidence: high

  Einstein derived the law of Brownian motion directly from the molecular-kinetic theory of heat, as the very title of his 1905 paper states.

  Link sources

  • A. Einstein, Annalen der Physik 322 (8), 549-560 (1905)

    Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen

  • Jean Baptiste Perrin, Biographical, The Nobel Foundation (1926) Verified 20 Sep 2026

    His results in this field were able to confirm Einstein's theoretical studies in which it was shown that colloidal particles should obey the gas laws, and hence to calculate Avogadro's number N.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Other links

• Unified system of atomic weights (Cannizzaro, Karlsruhe)

  parallel to · Confidence: medium

  Avogadro's hypothesis was reached by two independent routes in 1859-1860: Cannizzaro made it the basis of the chemical scale of atomic weights, while Maxwell derived it from the mechanics of moving molecules.

  Dispute: The agreement did not come at once or by itself: according to Brush the early versions of the kinetic theory flatly contradicted Avogadro's hypothesis, and the agreement appeared only with Maxwell in 1859. There is no direct statement by a historian that Cannizzaro and Maxwell proceeded independently: that is an inference from the fact that both reached the same result in the same year by different means. The direction of influence is also the reverse of what one expects: chemistry did not supply physics, rather the kinetic theory retrospectively justified the chemical method of determining weights from vapour densities.

  Link sources

  • Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

    Avogadro's hypothesis is favorable to the kinetic theory of of gases insofar as it implies that the volume occupied by a certain number of molecules is independent of their size and shape, which suggests they are not ordinarily in contact. But it was not until 1859 that Maxwell showed that the hypothesis could be derived from the kinetic theory; the earlier version of kinetic theory we are now doing to consider is actually inconsistent with it.

  • Klodian Coko (rev. of A. Chalmers), Atomism from the 17th to the 20th Century, Stanford Encyclopedia of Philosophy (2026) Verified 20 Sep 2026

    It also predicted Avogadro's law that equal volumes of gases contain equal numbers of molecules and thus also explained Gay-Lussac's law.

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Links under review

• The two laws of thermodynamics (Clausius 1850, Thomson 1851)

  influenced · Confidence: medium

  Both quotations are Boltzmann's own words, and Uffink disputes the first; a historian's judgement on the link between the second law and kinetic theory is still lacking (Brush, The Kind of Motion We Call Heat).

  The second law and Clausius's entropy set the problem that the kinetic theory solved anew, restating the increase of entropy as an increase in the probability of a state.

  Dispute: Both phrases quoted are Boltzmann's own words and not a historian's conclusion, and Uffink rejects the first of them: on his reading the real result is a different one, a generalization of entropy to non-equilibrium states ('The true import of Boltzmann's results is rather that they provide a generalization of the entropy concept to non-equilibrium states'). The influence also runs both ways: Clausius himself built a kinetic theory in 1857 and introduced entropy only in 1865, after Maxwell's 1860 paper.

  Link sources

  • L. Boltzmann (1872), quoted in Jos Uffink, Boltzmann's Work in Statistical Physics, Stanford Encyclopedia of Philosophy (2024) Verified 20 Sep 2026

    This provides an analytical proof of the Second Law in a way completely different from those attempted so far.

  • L. Boltzmann (1877), quoted in Jos Uffink, Boltzmann's Work in Statistical Physics, Stanford Encyclopedia of Philosophy (2024) Verified 20 Sep 2026

    According to the present interpretation, [the Second Law] states nothing else but that the probability of the total state of a composite system always increases

  "Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

Node sources

• J. C. Maxwell, Illustrations of the Dynamical Theory of Gases, Philosophical Magazine 19-20 (1860) Verified 20 Sep 2026

  the velocities are distributed among the particles according to the same law as the errors are distributed among the observations in the theory of the 'method of least squares.'

• L. Boltzmann (1872), quoted in Jos Uffink, Boltzmann's Work in Statistical Physics, Stanford Encyclopedia of Philosophy (2024) Verified 20 Sep 2026

  It has thus been rigorously proved that whatever may have been the initial distribution of kinetic energy, in the course of time it must necessarily approach the form found by Maxwell.

• Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

  Clausius had used probability concepts in his derivation of the mean-free-path formula, but it was Maxwell who converted the kinetic theory of gases into a fully statistical doctrine.

• Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026

  As a result, the atom came to be regarded as no longer a merely hypothetical concept but a real physical entity, subject to quantitative measurement, even though it could not be 'seen.'

"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.

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