Breakthroughs in Molecular Statistical Mechanics
=================================================


.. include:: /_generated/nav/statmech.rst

.. epigraph::

   "S = k . log W" -- inscription on Ludwig Boltzmann's tombstone, Vienna
   Zentralfriedhof (a formula Boltzmann himself never actually wrote down
   in this notation -- see 1877 -- 1901, below)

Statistical mechanics answers a deceptively simple chemistry question:
given the mechanical laws governing an individual molecule -- how it
translates, rotates, and vibrates -- what do those laws imply about a
mole of them together, at equilibrium, at a measurable temperature? The
translational, rotational, and vibrational partition functions in
:mod:`chemistrykit.statmech` are the direct answer, each one a
century-old piece of that program: Maxwell's and Boltzmann's kinetic
theory of gas speeds, the puzzle of *which* motions actually store
thermal energy classically and which do not, Gibbs's ensemble
formalism for turning a partition function into every macroscopic
thermodynamic quantity at once, and Sackur, Tetrode, Einstein, and
Langmuir's applications of the (at the time barely two-decades-old)
quantum of action to translational entropy, vibrational heat capacity,
and adsorption equilibrium. This chronology traces those breakthroughs,
with a pointer to the corresponding implementation in this package at
each stop.

.. contents:: Timeline
   :local:
   :depth: 1

1860 -- 1872 -- Maxwell, Boltzmann, and the Molecular Speed Distribution
-------------------------------------------------------------------------

James Clerk Maxwell's "Illustrations of the Dynamical Theory of Gases"
derived, from little more than the assumption that a gas's three
Cartesian velocity components are independent and that the distribution
of speeds cannot depend on the choice of axes, the specific functional
form every molecule's speed must be distributed as at equilibrium: not
uniform, not exponential, but the distribution now bearing his name,
peaked at a most-probable speed and falling off as a Gaussian in
:math:`v^2`. It was a strange derivation even by Maxwell's own later
admission -- an argument from symmetry and independence rather than from
any detailed accounting of molecular collisions -- and it left open
*why* a gas should relax to exactly this distribution rather than some
other one consistent with the same symmetry.

Ludwig Boltzmann closed that gap twelve years later. His 1872 paper
introduced the H-functional -- a single number computable from any
velocity distribution, however far from equilibrium -- and proved his
H-theorem: molecular collisions alone, with no further assumption, drive
H monotonically downward over time until it reaches its unique minimum,
attained precisely at Maxwell's distribution and no other. What Maxwell
had guessed from symmetry, Boltzmann showed was the inevitable long-time
outcome of ordinary molecular collisions -- the first serious mechanical
argument for irreversibility, and the reason the distribution is
properly named after both men.

.. math::

   f(v) = 4\pi\left(\frac{m}{2\pi k_BT}\right)^{3/2}v^2\,e^{-mv^2/(2k_BT)}

*Implementation:* :class:`chemistrykit.statmech.MaxwellBoltzmannSpeedDistribution`
implements exactly this pdf (and its cdf, via the error function), with
:func:`~chemistrykit.statmech.most_probable_speed`,
:func:`~chemistrykit.statmech.mean_speed`, and
:func:`~chemistrykit.statmech.rms_speed` giving the three characteristic
speeds that stand in a fixed, temperature-independent ratio -- a purely
geometric property of the distribution Maxwell's symmetry argument alone
already fixes.

*References:* J. C. Maxwell, "Illustrations of the Dynamical Theory of
Gases," Philos. Mag. 19, 19-32 (1860); L. Boltzmann, "Weitere Studien
über das Wärmegleichgewicht unter Gasmolekülen," Sitzungsberichte Akad.
Wiss. Wien 66, 275-370 (1872); McQuarrie, *Statistical Mechanics*, Ch.
27.

.. minigallery:: ../../examples/statmech/maxwell_boltzmann/plot_01_maxwell_boltzmann.py

1871 -- 1879 -- Boltzmann, Maxwell, and the Equipartition Theorem's Puzzle
-----------------------------------------------------------------------------

Ludwig Boltzmann's 1871 paper "Einige allgemeine Sätze über
Wärmegleichgewicht" generalized a special case already implicit in
Maxwell's kinetic theory into a sweeping general claim: at thermal
equilibrium, *every* independent quadratic term in a system's classical
energy -- one translational, one rotational, one vibrational degree of
freedom, it makes no difference which -- carries exactly the same average
energy, :math:`\frac12k_BT`. For a rigid diatomic molecule this predicts
:math:`\frac32k_BT` of translational energy, :math:`k_BT` of rotational
energy (two quadratic rotational degrees of freedom), and, once
vibration is admitted, a further :math:`k_BT` of vibrational energy (one
quadratic kinetic term, one quadratic potential term) -- a clean,
parameter-free prediction with no adjustable constants at all.

James Clerk Maxwell, extending the theorem himself in 1879 to systems
with arbitrarily many degrees of freedom, was also the one who stated
most sharply why this was a problem rather than a triumph: real
diatomic and polyatomic gases' measured heat capacities fell well short
of the equipartition prediction at ordinary temperatures, as though
entire vibrational (and sometimes rotational) degrees of freedom were
simply absent. Maxwell called it "the greatest difficulty which the
molecular theory has yet encountered," and the puzzle -- which classical
mechanics had no resource left to resolve -- stood unexplained for nearly
three decades, until Einstein's 1907 quantum theory of specific heats
(below) supplied the missing ingredient: a mode's energy spacing, not
just its existence, governs whether it can absorb thermal energy at a
given temperature at all.

*Implementation:* :class:`chemistrykit.statmech.RotationalPartitionFunctionLinear`
implements only the classical high-temperature limit equipartition
predicts (:meth:`~chemistrykit.statmech.RotationalPartitionFunctionLinear.heat_capacity_v`
returns exactly :math:`Nk_B` at *every* temperature, as its own
docstring flags), while
:class:`~chemistrykit.statmech.VibrationalPartitionFunctionHarmonic` is
an exact quantum treatment that only approaches its own equipartition
value :math:`Nk_B` as :math:`T\to\infty` -- so
:class:`~chemistrykit.statmech.IdealGasMolecule`, which sums both, is a
direct working demonstration of exactly the gap between the classical
prediction and the quantum reality that puzzled Maxwell.

*References:* L. Boltzmann, "Einige allgemeine Sätze über
Wärmegleichgewicht," Sitzungsberichte Akad. Wiss. Wien 63, 679-711
(1871); J. C. Maxwell, "On Boltzmann's Theorem on the Average
Distribution of Energy in a System of Material Points," Trans. Camb.
Philos. Soc. 12, 547-570 (1879) (the "greatest difficulty" remark is
widely quoted from this paper in secondary literature).

.. minigallery:: ../../examples/statmech/partition_functions/plot_02_equipartition_ladder.py

1877 -- 1901 -- Boltzmann, Planck, and the Statistical Entropy S = k ln W
----------------------------------------------------------------------------

Ludwig Boltzmann's 1877 paper "Über die Beziehung zwischen dem zweiten
Hauptsatze der mechanischen Wärmetheorie und der
Wahrscheinlichkeitsrechnung" made the case that entropy is not a
mysterious separate quantity but a direct measure of a macroscopic
state's probability: the number of microscopic arrangements `W`
("Wahrscheinlichkeit," though closer in meaning to multiplicity than
probability in the modern sense) consistent with the observed macroscopic
description. A gas spontaneously spreads to fill its container not
because some new force pushes it there, but because the "spread out"
macrostate is compatible with astronomically more microstates than any
compressed one -- the same purely combinatorial logic behind why shuffled
playing cards almost never come back out sorted.

The tombstone inscription "S = k . log W" is, however, a later
retrospective attribution rather than a formula Boltzmann himself ever
wrote in that compact notation, with a *named* constant `k`: his own
papers work with proportionalities and unnamed constants of
proportionality, not the explicit equation with Boltzmann's constant
picked out as a fundamental quantity in its own right. That step belongs
to Max Planck, who -- deriving the blackbody radiation law in 1900-1901 --
was the first to write the entropy-probability relation explicitly with a
named constant, and who named that constant after Boltzmann in
recognition of the statistical idea it was built on. The formula
immortalized on Boltzmann's grave is therefore, in the precise historical
sense, Planck's formula for Boltzmann's idea.

*Implementation:* :meth:`chemistrykit.statmech.LatticeGasAdsorption.canonical_entropy`
computes exactly this combinatorial entropy,
:math:`S=k_B\ln\binom{M}{N}`, for `N` indistinguishable adsorbed
molecules arranged on `M` lattice sites -- via the numerically exact
:func:`chemistrykit.statmech.ln_binomial` rather than Stirling's
approximation -- and shows it maximized at half filling, exactly where
the number of microscopic arrangements `W` is largest. The named
constant itself, ``chemistrykit.constants.K_B``, is the quantity
every entropy and heat-capacity formula throughout this package is built
from.

*References:* L. Boltzmann, "Über die Beziehung zwischen dem zweiten
Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung,
respektive den Sätzen über das Wärmegleichgewicht," Sitzungsberichte Akad.
Wiss. Wien 76, 373-435 (1877); M. Planck, "Über das Gesetz der
Energieverteilung im Normalspektrum," Ann. Phys. 4, 553-563 (1901).

.. minigallery:: ../../examples/statmech/lattice_gas/plot_02_boltzmann_entropy_counting.py

1902 -- Gibbs's Elementary Principles in Statistical Mechanics
-----------------------------------------------------------------

Josiah Willard Gibbs's *Elementary Principles in Statistical Mechanics*
-- his last major work, published the year before his death -- gave
statistical mechanics its modern, systematic form: rather than reason
about one physical system's time-averaged behavior directly, imagine an
*ensemble* of an enormous number of imagined identical copies of the
system, distributed over every microstate consistent with the
macroscopic constraints, and compute averages over that ensemble
instead. For a system held at fixed temperature (the canonical
ensemble), Gibbs showed that essentially every thermodynamic quantity of
interest -- internal energy, entropy, free energy, heat capacity -- follows
directly from a single generating function, the canonical partition
function :math:`Q=\sum_i e^{-E_i/k_BT}`, via
:math:`A=-k_BT\ln Q`. It is the reason a single object, the partition
function, sits at the center of every subsequent quantum-statistical
result below.

*Implementation:* :class:`chemistrykit.statmech.PartitionFunction`
declares exactly this common interface -- ``value``, ``internal_energy``,
``entropy``, ``heat_capacity_v`` -- and its
:meth:`~chemistrykit.statmech.PartitionFunction.helmholtz_free_energy`
computes :math:`A=U-TS`, Gibbs's own free-energy relation, generically
from any subclass's energy and entropy;
:class:`~chemistrykit.statmech.IdealGasMolecule` composes independent
translational, rotational, and vibrational partition functions into one
combined molecular partition function exactly as Gibbs's ensemble theory
licenses (the total partition function of independent modes factorizes,
:math:`q=q_{\text{trans}}q_{\text{rot}}q_{\text{vib}}`), and its
:meth:`~chemistrykit.statmech.IdealGasMolecule.thermodynamic_functions`
bundles the resulting `U`, `S`, `Cv`, and `A` together.

*References:* J. W. Gibbs, *Elementary Principles in Statistical
Mechanics, Developed with Especial Reference to the Rational Foundation
of Thermodynamics* (New Haven: Yale University Press, 1902).

.. minigallery:: ../../examples/statmech/partition_functions/plot_03_gibbs_canonical_partition_function.py

1907 -- Einstein's Quantum Theory of Specific Heats
-------------------------------------------------------

Albert Einstein resolved the equipartition puzzle above by applying
Planck's quantum hypothesis -- until then invoked only for the specific
case of blackbody radiation -- to something entirely different: the
vibrations of atoms in a solid. Modeling a solid as a collection of
independent quantum oscillators each vibrating at the same frequency
:math:`\nu`, Einstein showed that a mode's average energy is not the
classical, temperature-independent :math:`k_BT` but a temperature-
*dependent* quantity that only approaches :math:`k_BT` once :math:`k_BT`
comfortably exceeds the mode's own quantum :math:`h\nu`; well below that
threshold the mode's heat capacity vanishes exponentially, "frozen out"
of thermal equilibrium entirely. This was the first time the quantum of
action was shown to matter for an everyday material property completely
unrelated to radiation, and it explained at a stroke why diamond's heat
capacity (very high vibrational frequency, hence a very high freeze-out
temperature) behaved so differently from softer, lower-frequency solids
at the same temperature -- exactly the vibrational side of the
equipartition puzzle Maxwell had flagged decades earlier.

.. math::

   C_V = Nk_B\left(\frac{\Theta_{\text{vib}}}{T}\right)^2
         \frac{e^{-\Theta_{\text{vib}}/T}}{\left(1-e^{-\Theta_{\text{vib}}/T}\right)^2},
   \qquad \Theta_{\text{vib}} = \frac{h\nu}{k_B}

*Implementation:* :meth:`chemistrykit.statmech.VibrationalPartitionFunctionHarmonic.heat_capacity_v`
implements exactly this formula (reused here for a single molecular
vibrational mode rather than Einstein's original 3D lattice of solid-state
oscillators), vanishing as :math:`T\to0` and approaching the classical
equipartition value :math:`k_B` per mode as :math:`T\to\infty`; its
:attr:`~chemistrykit.statmech.VibrationalPartitionFunctionHarmonic.vibrational_temperature`
property gives :math:`\Theta_{\text{vib}}` directly, and
:meth:`~chemistrykit.statmech.VibrationalPartitionFunctionHarmonic.from_wavenumber`
builds a mode straight from an experimental IR wavenumber.

*References:* A. Einstein, "Die Plancksche Theorie der Strahlung und die
Theorie der spezifischen Wärme," Ann. Phys. 22, 180-190 (1907).

.. minigallery:: ../../examples/statmech/partition_functions/plot_01_vibrational_heat_capacity.py

1911 -- 1912 -- Sackur and Tetrode's Translational Entropy
---------------------------------------------------------------

Otto Sackur and, independently, Hugo Tetrode set out to compute
something that sounds elementary -- the absolute translational entropy of
an ideal monatomic gas -- and found they could only get a formula that
agreed with experiment (vapor-pressure and chemical-equilibrium data for
mercury and other monatomic gases) by inserting Planck's constant into a
classical statistical-mechanical calculation, years before anyone had a
coherent theory of *why* a classical phase-space calculation should
involve a quantum of action at all. Sackur (1911) and Tetrode (1912),
working independently and using slightly different routes to essentially
the same result, effectively used `h` to set the size of one
quantum-mechanical cell in classical phase space -- an ad hoc but
strikingly successful anticipation, a full fifteen years before
Schrodinger's wave mechanics, of the idea that phase space is not
infinitely subdividable.

.. math::

   S = Nk_B\left[\ln\left(\frac{1}{N}\left(\frac{2\pi mk_BT}{h^2}\right)^{3/2}V\right)+\frac52\right]

*Implementation:* :meth:`chemistrykit.statmech.TranslationalPartitionFunction.entropy`
implements exactly this formula, and the standalone
:func:`chemistrykit.statmech.sackur_tetrode_entropy` convenience wrapper
takes pressure and temperature directly, reproducing argon's tabulated
standard molar entropy (Atkins & de Paula, Table 13.1) to three
significant figures from nothing but its molecular mass.

*References:* O. Sackur, "Die Anwendung der kinetischen Theorie der Gase
auf chemische Probleme," Ann. Phys. 36, 958-980 (1911); H. Tetrode, "Die
chemische Konstante der Gase und das elementare Wirkungsquantum," Ann.
Phys. 38, 434-442 (1912).

.. minigallery:: ../../examples/statmech/partition_functions/plot_04_sackur_tetrode_entropy.py

1912 -- Debye's T-Cubed Law for the Heat Capacity of Solids
----------------------------------------------------------------

Einstein's 1907 theory (above) explained why a solid's heat capacity
falls below the Dulong-Petit value :math:`3R` on cooling, but it predicted
an *exponential* fall, far faster than the measurements Walther Nernst's
laboratory was then making on solids near liquid-hydrogen temperatures.
Peter Debye saw the reason: the atoms of a crystal do not vibrate
independently at one frequency, but collectively, as sound waves with a
whole spectrum of frequencies. Treating the solid as an elastic continuum
gives a density of vibrational states growing as :math:`\nu^2`, which
Debye cut off at a maximum frequency :math:`\nu_D` so that the crystal
has exactly :math:`3N` modes in total. However low the temperature, the
longest-wavelength sound waves still have energy spacings smaller than
:math:`k_BT` and stay thermally active, so the heat capacity falls only as
a power law, :math:`C_V\propto T^3`. The single parameter
:math:`\Theta_D=h\nu_D/k_B`, the Debye temperature, fits the heat
capacities of many simple solids over the whole temperature range, and
the :math:`T^3` law remains the standard way to extrapolate
calorimetric data to absolute zero when determining third-law entropies.

.. math::

   C_V = 9Nk_B\left(\frac{T}{\Theta_D}\right)^3
         \int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}\,dx
   \;\xrightarrow{T\ll\Theta_D}\;
   \frac{12\pi^4}{5}Nk_B\left(\frac{T}{\Theta_D}\right)^3

*Implementation:* :class:`chemistrykit.statmech.DebyeSolid` evaluates
this integral numerically in
:meth:`~chemistrykit.statmech.DebyeSolid.heat_capacity_v` (with the
matching thermal energy in
:meth:`~chemistrykit.statmech.DebyeSolid.internal_energy`), and
:meth:`~chemistrykit.statmech.DebyeSolid.low_temperature_heat_capacity`
gives the limiting :math:`T^3` law it approaches as :math:`T\to0`.

*References:* P. Debye, "Zur Theorie der spezifischen Wärmen," Ann.
Phys. 39, 789-839 (1912); McQuarrie, *Statistical Mechanics*, Ch. 11.

.. minigallery:: ../../examples/statmech/solids/plot_01_debye_t_cubed_law.py

1918 -- Langmuir's Statistical Theory of Adsorption
---------------------------------------------------------

Irving Langmuir, studying gas adsorption onto tungsten filaments and
other clean metal surfaces at General Electric, proposed a simple
microscopic picture that -- unlike the empirical Freundlich isotherm
already in use -- could be derived from first principles: a fixed number
of independent, non-interacting adsorption sites, each either empty or
occupied by exactly one adsorbed molecule, in dynamic equilibrium with
the surrounding gas. Treating the sites as an open system in equilibrium
with an ideal-gas reservoir gives a coverage that rises linearly with
pressure at low pressure and saturates smoothly toward complete monolayer
coverage at high pressure -- the Langmuir isotherm -- with a single
physically meaningful parameter, the adsorption energy, controlling where
the transition happens.

.. math::

   \theta(P) = \frac{P}{P+P_0}, \qquad
   P_0 = \frac{k_BT}{\Lambda^3}e^{-\epsilon/k_BT}

*Implementation:* :class:`chemistrykit.statmech.LatticeGasAdsorption`
implements exactly this grand-canonical lattice-gas derivation --
:meth:`~chemistrykit.statmech.LatticeGasAdsorption.coverage` gives
:math:`\theta(P)`, and
:meth:`~chemistrykit.statmech.LatticeGasAdsorption.p_half` gives the
half-coverage pressure :math:`P_0` directly from the adsorption energy,
molecular mass, and temperature via the gas-phase chemical potential.

*References:* I. Langmuir, "The Adsorption of Gases on Plane Surfaces of
Glass, Mica and Platinum," J. Am. Chem. Soc. 40, 1361-1403 (1918).

.. minigallery:: ../../examples/statmech/lattice_gas/plot_01_langmuir_adsorption.py

1920 -- Stern's Molecular-Beam Verification of the Speed Distribution
--------------------------------------------------------------------------

Every result above through 1918 was a *theoretical* derivation of the
Maxwell-Boltzmann distribution's consequences (entropy, adsorption,
heat capacity); none of it directly measured a molecule's speed. Otto
Stern's 1920 rotating-drum molecular-beam apparatus -- a silver-coated
platinum wire vaporized at the center of an evacuated, spinning cylinder --
gave the first direct experimental access to actual molecular speeds,
inferred from where the deposited silver landed on the rotating drum's
inner wall relative to where it would land from an infinitely fast
beam. The measured mean speed matched the kinetic-theory prediction to
within Stern's admittedly generous experimental uncertainty; more
demanding confirmations of the full distribution's *shape*, not just its
mean, came over a decade later from Leonard Zartman's and, with
improved velocity selectors, I. Estermann and Otto Stern's own further
refinements of the technique through the 1930s.

*Implementation:* :meth:`chemistrykit.statmech.MaxwellBoltzmannSpeedDistribution.sample`
draws a synthetic "molecular beam" of speeds by sampling three
independent Gaussian velocity components and taking their norm -- exactly
the underlying statistical model a real molecular-beam measurement
infers back from -- letting a finite-sample histogram be compared
directly against the exact analytic pdf, the same kind of
theory-versus-measurement check Stern's apparatus first made possible.

*References:* O. Stern, "Eine direkte Messung der thermischen
Molekulargeschwindigkeit," Z. Phys. 2, 49-56 (1920); L. Zartman, "A
Direct Measurement of Molecular Velocities," Phys. Rev. 37, 383-389
(1931).

.. minigallery:: ../../examples/statmech/maxwell_boltzmann/plot_02_stern_experiment.py

1924 -- 1925 -- de Broglie's Matter Waves and the Thermal Wavelength
--------------------------------------------------------------------------

Louis de Broglie's doctoral thesis proposed, by analogy with the
already-established wave-particle duality of light, that every material
particle of momentum `p` has an associated wavelength
:math:`\lambda=h/p` -- a conjecture with no experimental support at the
time it was made, confirmed only three years later by Davisson and
Germer's electron-diffraction experiment. Applied not to a single
particle's momentum but to the whole thermal spread of momenta a gas
molecule carries at temperature `T`, de Broglie's relation gives a
natural length scale, the thermal de Broglie wavelength -- roughly, how
"smeared out" a typical molecule's position is by its own quantum
wave-like nature. Whenever that wavelength is small compared to the
average distance between molecules, as it is for essentially every
ordinary gas well above its condensation point, treating translational
motion as classical (position and momentum both sharply defined) is an
excellent approximation; only when the thermal wavelength grows
comparable to the interparticle spacing -- liquid helium, a dilute
ultracold atomic gas, the conduction electrons in a metal -- does quantum
statistics (below) become unavoidable.

.. math::

   \Lambda = \frac{h}{\sqrt{2\pi mk_BT}}

*Implementation:* :func:`chemistrykit.statmech.thermal_de_broglie_wavelength`
implements exactly this formula, confirmed in its own docstring to be a
small fraction of an angstrom for room-temperature argon -- far shorter
than the mean interparticle spacing in a gas at atmospheric pressure, the
formal justification for treating translational motion classically
throughout this package's
:class:`~chemistrykit.statmech.TranslationalPartitionFunction`. It is
also, less obviously, the length scale entering the gas-phase chemical
potential :class:`~chemistrykit.statmech.LatticeGasAdsorption` uses to
convert a gas pressure into the Langmuir isotherm's half-coverage
pressure :math:`P_0` (see 1918, above).

*References:* L. de Broglie, "Recherches sur la théorie des quanta," PhD
thesis, University of Paris (1924), published as Ann. Phys. 10, 22-128
(1925).

.. minigallery:: ../../examples/statmech/quantum_statistics/plot_01_thermal_de_broglie_wavelength.py

1924 -- 1926 -- Bose, Einstein, Fermi, Dirac, and the Classical Limit of Quantum Statistics
-----------------------------------------------------------------------------------------------

Satyendra Nath Bose's 1924 derivation of Planck's radiation law by
treating light quanta as genuinely indistinguishable particles obeying a
new counting rule -- refined by Einstein into a full quantum statistical
mechanics of an ideal gas of indistinguishable bosons -- and Enrico
Fermi's and Paul Dirac's independent 1926 derivations of the
complementary statistics obeyed by indistinguishable fermions (subject to
the Pauli exclusion principle, itself only a year old), together
completed the picture the classical, Maxwell-Boltzmann statistics used
everywhere above are a *limit* of, not a universally exact description.
Both quantum statistics reduce smoothly to the classical Maxwell-Boltzmann
distribution whenever the thermal de Broglie wavelength (1924-25, above)
is small compared to the interparticle spacing -- precisely the
"dilute, high-temperature" regime every partition function elsewhere in
this package assumes without further comment, and the reason
distinguishing bosons from fermions never actually arises in a package
built entirely on classical molecular statistics.

*Connection:* this package implements no Bose-Einstein or Fermi-Dirac
distribution -- every partition function in
``chemistrykit.statmech.systems.partition_functions`` is built on the
classical Maxwell-Boltzmann counting these results show is only ever a
limiting case -- but
:func:`chemistrykit.statmech.thermal_de_broglie_wavelength`'s own
confirmation that :math:`\Lambda` is minuscule for ordinary molecular
gases (see 1924-25, above) is exactly the quantitative condition under
which that omission is a genuine physical approximation rather than an
oversight, and the Maxwell-Boltzmann speed distribution plotted throughout
this package is precisely what both quantum statistics degenerate into in
that limit.

*References:* S. N. Bose, "Plancks Gesetz und Lichtquantenhypothese," Z.
Phys. 26, 178-181 (1924); A. Einstein, "Quantentheorie des einatomigen
idealen Gases," Sitzungsber. Preuss. Akad. Wiss. (1924), 261-267, and
(1925), 3-14; E. Fermi, "Sulla quantizzazione del gas perfetto
monoatomico," Rend. Lincei 3, 145-149 (1926); P. A. M. Dirac, "On the
Theory of Quantum Mechanics," Proc. R. Soc. Lond. A 112, 661-677 (1926).

.. minigallery:: ../../examples/statmech/quantum_statistics/plot_02_bose_einstein_fermi_dirac_limit.py

1925 -- Ising's One-Dimensional Model of a Ferromagnet
-----------------------------------------------------------

Wilhelm Lenz proposed, and his student Ernst Ising solved for his 1924
Hamburg dissertation, the simplest possible model of cooperative
behavior: a row of "spins" :math:`s_i=\pm1`, each interacting only with
its nearest neighbors through an energy :math:`-Js_is_{i+1}`, in a field
`h`. The same model, read as occupied and empty sites, is a lattice gas
with nearest-neighbor attraction, the natural interacting extension of
Langmuir's independent sites (1918, above). Ising found that the
one-dimensional chain never orders at any temperature above absolute
zero: a single broken bond costs a finite energy but gains an entropy
that grows with the chain's length, so thermal fluctuations always win.
He concluded, wrongly as it turned out, that the model could not describe
a phase transition in any dimension. The exact solution is most easily
written with the transfer matrix introduced later by Kramers and Wannier
(1941, below): the partition function of a ring of `N` spins is
:math:`Z_N=\lambda_+^N+\lambda_-^N`, where :math:`\lambda_\pm` are the
eigenvalues of a 2x2 matrix.

.. math::

   \lambda_\pm = e^{K}\left[\cosh b\pm\sqrt{\sinh^2b+e^{-4K}}\right],
   \qquad K=\frac{J}{k_BT},\quad b=\frac{h}{k_BT}

*Implementation:* :class:`chemistrykit.statmech.Ising1D` implements the
transfer-matrix solution:
:meth:`~chemistrykit.statmech.Ising1D.partition_function`,
:meth:`~chemistrykit.statmech.Ising1D.free_energy_per_spin`,
:meth:`~chemistrykit.statmech.Ising1D.magnetization` (zero in zero field
at every :math:`T>0`), the finite
:meth:`~chemistrykit.statmech.Ising1D.correlation_length`, and the smooth
:meth:`~chemistrykit.statmech.Ising1D.heat_capacity_per_spin`.

*References:* E. Ising, "Beitrag zur Theorie des Ferromagnetismus," Z.
Phys. 31, 253-258 (1925); S. G. Brush, "History of the Lenz-Ising Model,"
Rev. Mod. Phys. 39, 883-893 (1967).

.. minigallery:: ../../examples/statmech/ising/plot_01_ising_chain_transfer_matrix.py

1937 -- Mayer's Cluster Expansion and the Second Virial Coefficient
------------------------------------------------------------------------

Every result above treats molecules as non-interacting apart from the
occasional collision. Joseph E. Mayer, building on H. D. Ursell's 1927
expansion of the configurational integral, showed how to put
intermolecular forces into the partition function systematically. The
Boltzmann factor of each pair is written as :math:`1+f(r)`, where the
Mayer function :math:`f(r)=e^{-u(r)/k_BT}-1` is non-zero only when two
molecules are close together. Expanding the product over all pairs and
grouping the terms into connected "clusters" gives the virial equation of
state :math:`PV_m/RT=1+B_2(T)/V_m+B_3(T)/V_m^2+\cdots`, in which every
coefficient is an explicit integral over the pair potential. This turned
measured gas non-ideality into a direct probe of intermolecular forces.
Fitting :math:`B_2(T)` data was for decades the main way the parameters
of potentials such as the Lennard-Jones potential were determined.

.. math::

   B_2(T) = -2\pi N_A\int_0^\infty\left(e^{-u(r)/k_BT}-1\right)r^2\,dr

*Implementation:* :func:`chemistrykit.statmech.second_virial_coefficient`
evaluates this integral by quadrature for any spherical pair potential,
including hard-core potentials, and reproduces the hard-sphere value
:math:`\frac{2\pi}{3}N_A\sigma^3`, the closed-form square-well result, and
the Lennard-Jones Boyle temperature :math:`T_B\approx3.418\,\epsilon/k_B`
(tested in ``chemistrykit/statmech/tests/test_virial.py``).

*References:* H. D. Ursell, "The Evaluation of Gibbs' Phase-Integral for
Imperfect Gases," Proc. Cambridge Philos. Soc. 23, 685-697 (1927); J. E.
Mayer, "The Statistical Mechanics of Condensing Systems. I," J. Chem.
Phys. 5, 67-73 (1937); J. E. Mayer and M. G. Mayer, *Statistical
Mechanics* (New York: Wiley, 1940); McQuarrie, *Statistical Mechanics*,
Ch. 12.

.. minigallery:: ../../examples/statmech/virial/plot_01_mayer_second_virial_coefficient.py

1941 -- Kramers and Wannier's Duality and the Exact Critical Temperature
-----------------------------------------------------------------------------

Hendrik Kramers and Gregory Wannier reopened the question Ising had
closed (1925, above) by studying the two-dimensional square lattice. They
introduced the transfer-matrix method and found a hidden symmetry: the
high-temperature expansion of the partition function (a sum over closed
loops of bonds) and its low-temperature expansion (a sum over domain
walls around flipped spins) have exactly the same form, with the
coupling :math:`K=J/k_BT` in one replaced by a dual coupling :math:`K^*`
in the other. The duality maps high temperature onto low temperature.
If the model has a single phase transition, it must therefore sit at the
self-dual point :math:`K_c=K_c^*`, which fixes the critical temperature
exactly, three years before anyone could solve the model. It was the
first exact critical temperature for any lattice model, and the first
firm evidence that the Ising model does order in two dimensions.

.. math::

   \sinh 2K\,\sinh 2K^* = 1
   \quad\Longrightarrow\quad
   \sinh 2K_c = 1,\qquad
   T_c = \frac{2J}{k_B\ln\left(1+\sqrt2\right)}\approx 2.269\,\frac{J}{k_B}

*Implementation:* :func:`chemistrykit.statmech.kramers_wannier_dual_coupling`
computes :math:`K^*=-\tfrac12\ln\tanh K` (an involution whose fixed point is
:math:`K_c`), and :func:`chemistrykit.statmech.ising_2d_critical_temperature`
returns the exact :math:`T_c` for a given coupling `J`.

*References:* H. A. Kramers and G. H. Wannier, "Statistics of the
Two-Dimensional Ferromagnet. Part I," Phys. Rev. 60, 252-262 (1941).

.. minigallery:: ../../examples/statmech/ising/plot_02_kramers_wannier_duality.py

1944 -- 1952 -- Onsager's Exact Solution of the Two-Dimensional Ising Model
--------------------------------------------------------------------------------

Lars Onsager calculated the partition function of the zero-field square
lattice Ising model exactly, by diagonalizing Kramers and Wannier's
transfer matrix with an algebraic method of his own. It was the first
exact solution of any model with a genuine phase transition, and it
settled the long debate over whether the partition function, a sum of
smooth exponentials, could produce a singularity at all: in the limit of
an infinite lattice, it does. The solution also showed that the
approximate mean-field (Bragg-Williams) theory, the standard treatment
of the time, is qualitatively wrong near :math:`T_c`: the heat capacity
does not jump but diverges logarithmically. Onsager announced the
spontaneous magnetization, :math:`m=[1-\sinh^{-4}2K]^{1/8}`, at a 1949
conference without publishing a derivation. C. N. Yang supplied the proof
in 1952, and its exponent :math:`\beta=1/8` (against the mean-field
value 1/2) became a benchmark for the theory of critical phenomena.

.. math::

   -\frac{f}{k_BT} = \ln(2\cosh 2K)+\frac1\pi\int_0^{\pi/2}
   \ln\frac{1+\sqrt{1-\kappa^2\sin^2\phi}}{2}\,d\phi,
   \qquad \kappa=\frac{2\sinh 2K}{\cosh^2 2K}

*Implementation:* :class:`chemistrykit.statmech.Ising2DOnsager` evaluates
Onsager's free energy
(:meth:`~chemistrykit.statmech.Ising2DOnsager.free_energy_per_spin`), and
the internal energy and heat capacity in closed form via complete
elliptic integrals
(:meth:`~chemistrykit.statmech.Ising2DOnsager.internal_energy_per_spin`,
:meth:`~chemistrykit.statmech.Ising2DOnsager.heat_capacity_per_spin`), as
well as Yang's
:meth:`~chemistrykit.statmech.Ising2DOnsager.spontaneous_magnetization`.
The tests check each quantity against numerical derivatives of the free
energy.

*References:* L. Onsager, "Crystal Statistics. I. A Two-Dimensional Model
with an Order-Disorder Transition," Phys. Rev. 65, 117-149 (1944); C. N.
Yang, "The Spontaneous Magnetization of a Two-Dimensional Ising Model,"
Phys. Rev. 85, 808-816 (1952); K. Huang, *Statistical Mechanics*, 2nd ed.
(New York: Wiley, 1987), Ch. 15.

.. minigallery:: ../../examples/statmech/ising/plot_03_onsager_exact_solution.py

See Also
--------

- :doc:`/api/statmech`
- :doc:`/history/quantum_breakthroughs`