History of Kinetic Theory

The text in the links has been kindly supplied by Stephen G. Brush. It too is a work in progress.

A Chronology of Kinetic Theory

• 1660, Robert Boyle:

• 1738, Daniel Bernoulli: The first kinetic theory.

• 1807, Louis Joseph Gay-Lussac:

Heat Dynamics

• 1807, J.B.J. Fourier: Fourier submits his first manuscript on the dynamics of heat to the Institut de France . It is rejected.

• 1819, J.B.J. Fourier: Fourier applies his theory to the cooling of the earth.

• 1820, John Herapath:

• 1822, J.B.J. Fourier: Fourier's great The Analytic Theory of Heat appears.

• 1824, Sadi Carnot: The Carnot cycle is introduced.

• 1845, John James Waterston: His "lost manuscript" is submitted to the Royal Society, rejected, and buried in the archives.

The Conservation of Energy

• 1847, Hermann Helmholtz: Helmholtz asserts the law of conservation of energy and that thermal energy is a form of mechanical energy.

The Second Law of Thermodynamics Begins to Emerge

• 1850, Rudolf Clausius:

• 1852, William Thomson (Lord Kelvin): Thomson asserts that there is ``a universal tendency in nature to the dissipation of mechanical energy''.

• 1854, Hermann Helmholtz: Helmholtz asserts that all energy will become heat. This is the famous statement of the "heat death" of the universe.

• 1855, Rudolf Clausius: Clausius states that heat can never pass from a colder to a warmer body without intervention.

• 1857, Rudolf Clausius: In his first paper on kinetic theory, Clausius shows molecules can move with speeds much greater than the magnitude of the bulk fluid velocity.

• 1858, Rudolf Clausius: The concept of mean-free-path is introduced.

• 1860, James Clerk Maxwell: This is the first of Maxwell's three great papers the subject. It is the bridge from the era that went before to the era that follows. In it Maxwell introduces the velocity distribution (kinetic density), and hence probability into the subject. He then gives his first derivation of what we now call Maxwellian densities. He uses his densities to compute viscosities and thermal conductivities for a gas of hard spheres. These transport coefficients are seen to depend only on the temperature of the gas, and hence independent of the density. This result surprised many, but were confirmed by experiments carried out by Maxwell himself and published in 1865. This prediction was the first great success of kinetic theory.

• 1865, Rudolf Clausius: Entropy is introduced into thermodynamics.

• 1865, Josef Loschmidt: The size of atoms is estimated using Maxwell's 1860 theory and what is known about the viscosity of gases. The estimate has the correct order of magnitude and is much smaller than anything observable at the time.

• 1867, James Clerk Maxwell: This is the second and greatest of Maxwell's three great papers the subject. In it Maxwell introduces his ``general equation of continuity'', a kinetic equation which is formally equivalent to the Boltzmann equation. Maxwell derives the collision kernel for monatomic molecules with a repulsive power-law intermolecular potential. He gives his second derivation of the Maxwellian densities and shows them to be equilibria of his kinetic equation. He gives an argument that they are the only equilibria, but it is unconvincing. He observes that the collisional terms of his kinetic equation simplify for the case of what are now called Maxwell molecules. He computes viscosities and thermal conductivities for a gas of Maxwell molecules through a balance argument in a regime in which mean free paths are small compared to macroscopic length scales.

Irreversibility

• 1867, James Clerk Maxwell: In a private letter to Tait, Maxwell invents a ``demon'' to illustrate that irreversibility is statistical in nature. The demon appear in many letters between British physicists as they struggle to understand how microscopic dynamics that is reversible can give rise to macroscopic dynamics that is not -- namely, the compressible Navier-Stokes equations. The demon finally make a public appearance in a 1871 paper by Maxwell.

• 1868, Ludwig Boltzmann: The Maxwellians are extended to cases where there is an external potential.

• 1872, Ludwig Boltzmann: This is the first of Boltzmann's four great papers the subject. The Boltzmann equation is introduced. The H-Theorem is ``proved''. In particular, Boltzmann shows his equation is dissipative and proves that Maxwellians are the only equilibria of the collision operator. Mistakenly, Boltzmann claims to have established how irreversible dynamics arise from reversible microscopic dynamics. He extends the kinetic equation to polyatomic molecules.

• 1873, Josiah Willard Gibbs: Gibbs publishes his first two papers on thermodynamics. He introduces a novel graphical approach that revolutionized the subject. You can find out more about this at "http://www.public.iastate.edu/~jolls/".

• 1874, William Thomson (Lord Kelvin): Thomson raises the question of how a reversible microscopic picture of the world can be consistent with irreversible macroscopic theories such as (for example) thermodynamics, which by then were widely accepted. He gives a probabilistic analysis of the diffusion of gases in the limit of the number of molecules goes to infinity. He states that irreversibility can arise from such a limit process. In other words, irreversibility holds in a limit.

• 1876, Josef Loschmidt: Loschmidt raises the objection that the Boltzmann equation itself is not reversible, while the microscopic dynamics that it allegedly arises from are.

• 1876, Josiah Willard Gibbs: Gibbs publishes the first part of his revolutionary third paper on thermodynamics. In it he exploits his graphical approach of 1873 to enlarge the domain of phenomena covered by thermodynamics.

• 1877, Ludwig Boltzmann: Boltzmann publishes two papers. The first addresses the 1876 objection of Loschmidt. The second is also the second and perhaps deepest of Boltzmann's four great papers the subject. In it he follows up on a remark he makes in his response to Loschmidt and establishes the connection between entropy and the number of microscopic states available to a system in a given macroscopic state.

• 1878, Josiah Willard Gibbs: Gibbs publishes the second part of his revolutionary third paper on thermodynamics. In it he shows that his techniques apply to mixtures. You can find more about this at "http://www.public.iastate.edu/~jolls/".

• 1878, James Clerk Maxwell: Maxwell critiques and extends some of Boltzmann's eariler work; in particular, the 1868 paper.

• 1879, James Clerk Maxwell: This is the third and least well-known of Maxwell's three great papers the subject. It includes the first treatment of boundary conditions in kinetic theory. It also identifies a regime in which mean free paths are small compared to macroscopic length scales, yet the compressible Navier-Stokes are not valid. For a gas of Maxwell molecules he derives corrections to fluid dynamics that anticipate results of Burnett over fifty years later. This paper showed that kinetic theory could explain a number of experimental observations that had been unexplained by traditional fluid mechanics.

• 1884, Ludwig Boltzmann: This is the third of Boltzmann's four great papers the subject. In it he applies his ideas to give a thermodynamic explanation of the energy and pressure of electromagnetic that had been suggested experimentally by Stefan. This is the famous Stefan-Boltzmann law.

• 1884, Ludwig Boltzmann: This is the last and least well-known of Boltzmann's four great papers the subject. In it he develops a theory of equilibrium statistical mechanics. He introduces the notions that Gibbs later dubbed canonical and grand canonical ensembles.

Detailed and Semi-Detailed Balance

• 1887, Hendrik Antoon Lorentz: Lorentz finds a mistake in Boltzmann's analysis of the collisions of polyatomic molecules. This mistake renders Boltzmann's "proof" of the H-theorem for that case wrong even on a formal level. Specifically, Lorentz points out that Boltzmann assumed the collision kernel satisfies a property we now call "detailed balance", which need not be satified by collision kernels for polyatomic molecules.

• 1887, Ludwig Boltzmann: Boltzmann responds to Lorentz by pointing out that the H-theorem can be proved assuming only that the collision kernel satisfies a property we now call "semi-detailed balance", which is weak than detailed balance. He goes on to argue that this property holds for collision kernels for polyatomic molecules. This last argument rests on some approximations in the calculation of the kernels that make it not totally convincing. However, much later it is shown that this property is easy to establish for quantum mechanically derived collision kernels.

• 1893, Henri Poincare: In a short note, Poincare raises the objection that kinetic theory contradicts his recurrence theorem. He had asserted his theorem in 1890 but in order to simplify his presentation only published a proof for the case of system with a three dimensional phase space. Of course, this case does not apply to kinetic theory. This note did not receive much attention until Zermelo took up the objection three years later.

Molecular Chaos and Determinism

• 1894, S.H. Burbury: During a meeting of British physicists in London at which Boltzmann was present, Burbury observed that any two particles that collide would henceforth be correlated, a fact which would seem to eventually render invalid the assumption made by Maxwell and Boltzmann that particles are uncorrelated before collisions.

• 1894, Ludwig Boltzmann: Responding to Burbury, Boltzmann asserts that, while two particles that collide would henceforth be correlated, as they each undergo other collisions they would become less and less correlated. Hence, he asserts, the assumption that are uncorrelated before collisions is justified. The physicists in London at which Boltzmann was present, Burbury observed that any two particles that collide would henceforth be correlated, a fact which would seem to eventually render invalid the assumption made by Maxwell and Boltzmann that particles are uncorrelated before collisions.

• 1896, Ludwig Boltzmann: The first volume of Boltzmann's classic text, Lectures on Gas Theory, is published in German.

• 1896, Ernest Zermelo: Zermelo publishes a proof of Poincare's recurrence theorem in a setting general enough to apply to kinetic theory. He goes on to amplify Poincare's 1893 objection, stating that both thermodynamics and kinetic theory contradict Poincare's recurrence theorem.

• 1896, Ludwig Boltzmann: Boltzmann refutes Poincare and Zermelo. He correctly argues that the recurrence time is too large to be physically meaningful on the time scale over which his equation is valid.

• 1896, Ernest Zermelo: Zermelo replies to Boltzmann's refutation, and reasserts that kinetic theory contradicts Poincare's recurrence theorem. Basically Zermelo correctly asserts that Boltzmann's argument is not on firm mathematical while his argument is, but goes on to incorrectly assert that it follows that Boltzmann is wrong.

• 1897, Ludwig Boltzmann: Boltzmann refutes Zermelo again. He points out the obvious fallacy in Zermelo's logic and goes on to reassert that his argument is correct.

• 1898, Ludwig Boltzmann: The second volume of Boltzmann's classic text, Lectures on Gas Theory, is published in German.

Mathematical Development Begins

• 1900, David Hilbert: Hilbert explicitly mentions the validity of the Boltzmann equation and its connections to fluid dynamics as part of his sixth problem in his famous address to the International Congress of Mathematicians.

• 1902, Josiah Willard Gibbs: Gibbs publishes his classic treatise Elementary Principles in Statistical Mechanics, which becomes the standard reference on equilibrium statistical mechanics.

• 1906, Henri Poincare: Poincare challenges mathematicians to make sense of kinetic theory.

• 1909, David Hilbert: The theory of integral operators is developed, in part with an eye toward making a contribution to the solution of the problem mentioned above.

• 1912, David Hilbert: Employing the theory of integral operators, Hilbert introduces a formal expansion for the kinetic density. With this expansion he computes transport coefficients for a gas of hard spheres, which is the only microscopically derived collision kernel for which the Boltzmann equation makes sense.

Formal Development

• 1917, Sydney Chapman: Chapman shows how to adapt Maxwell's technique to compute transport coefficients for more general intermolecular potentials.

• 1917, David Enskog: Inspired by Hilbert's expansion, Enskog proposes an alternative expansion that has subsequently been dubbed the ``Chapman-Enskog'' expansion.

• 1922, David Enskog: The Enskog equation is introduced.

• 1933, Torsten Carleman: Carleman analyzes the spatially homogeneous Boltzmann equation.

• 1935, David Burnett:

• 1937, David Burnett:

• 1949, Harold Grad:

Modern Mathematical Developments

• 1962, Harold Grad:

• 1972, Leif Arkeryd:

• 1974, Ellis and Mark Pinsky:

• 1974, S. Ukai:

• 1975, Oscar E. Landford III: Landford proves the validity for at least small times of the Boltzmann equation for a gas of hard spheres in what he dubs the Boltzmann-Grad limit.

• 1980, Russel Caflisch:

• 1986, Reinhard Illner and Mario Pulverenti: Illner and Pulverenti improve Landford's result in two dimensions.

• 1989, Reinhard Illner and Mario Pulverenti: Illner and Pulverenti improve Landford's result in three dimensions.

Finally, Global Mathematical Theories for Large Data

• 1989, Ron DiPerna and Pierre-Louis Lions: DiPerna and Lions prove the first theorem that asserts global existence for all physical initial data for the Boltzmann equation with a cut-off collision kernel. Their theory readily applies to spatial domains without boundary. They worked out the theory for the whole space. The extension to the spatially periodic case is direct. This paper ushers in the current era of rapid mathematical development in the area.

• 1993, Claude Bardos, Francois Golse, and David Levermore:

• 1994, Pierre-Louis Lions: Regularity of the gain term.

• 1998, Claude Bardos, Francois Golse, and David Levermore:

• 1999, Laure Saint Raymond:

• 2000, Pierre-Louis Lions and Nader Masmoudi:

• 2000, Francois Golse and David Levermore:

• 2001, Francois Golse and Laure Saint Raymond:

• 2001, David Levermore and Nader Masmoudi: