History of Kinetic Theory
This webpage is a work in progress. There are many important
contributions missing. The dates given may be wrong. Names may be
incomplete or misspelled. The description of the content of a work
may be missing or misrepresent the work. There is much work to be
done before anyone should use this page as either a reference, or
consider that it accurately reflects its creator's sum knowledge of
the subject. I hope that this condition is relatively temporary. I
further hope that people will freely offer input about the page.
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
Early Theories of Gases
- 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.
Neglected Pioneers
- 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.
Revival of Kinetic Theory by Clausius
- 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.
New Foundations Laid by Maxwell and Boltzmann
- 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.
- 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.
Poincare Recurrence Theorem
- 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.
- 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:
Updated 25 August 2001
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