Statistical Mechanics

Thermodynamics, developed in Part VI — Classical Thermodynamics, states relations between measured macroscopic quantities without ever asking what matter is made of; statistical mechanics derives those same relations from the mechanics of the constituents, and in doing so predicts things thermodynamics cannot — the magnitude of fluctuations, the absolute value of the entropy, and the point at which the classical description must fail. This chapter opens Part XII — Statistical Mechanics by building that bridge: phase space and Liouville's theorem inherited from Hamiltonian Mechanics, the Boltzmann and Gibbs definitions of entropy, the microcanonical, canonical and grand canonical ensembles with their partition functions, the thermodynamic limit in which they agree, and the classical ideal gas as the case where every step can be carried through in closed form. It states the ergodic hypothesis explicitly and reports its status honestly — proved for a small class of systems, assumed for the rest — rather than passing over the gap in the foundations.

The chapter is deliberately classical: the counting is corrected for indistinguishability where the measurements demand it (the Gibbs paradox and the Sackur–Tetrode entropy), but the quantum statistics that explain why are deferred to Quantum Statistics, the kinetic and transport theory to Kinetic Theory of Gases and Nonequilibrium Thermodynamics and Transport, and phase transitions to Phase Transitions and Critical Phenomena. Its direct experimental sequel is Experiment: Brownian Motion and Avogadro's Number, where the fluctuations predicted here are measured and Avogadro's constant extracted from them. Founding sources are Boltzmann [Boltzmann:1877] and Gibbs [Gibbs:1902]; the modern treatments used throughout are [Landau:1980] [Pathria:2011], with the thermodynamic side in [Callen:1985].

From mechanics to thermodynamics

Why a statistical description

[Reserved: the impossibility in practice and the irrelevance in principle of following \(\sim\!10^{23}\) trajectories; Maxwell's introduction of a distribution of molecular velocities as the object of study rather than the trajectories themselves [Maxwell:1860], and Boltzmann's transport equation for its evolution [Boltzmann:1872]; the division of labour with Kinetic Theory of Gases (dynamics of the distribution) and Classical Thermodynamics (relations among the averages); what “equilibrium” means operationally.]

Phase space and Liouville's theorem

[Reserved: the \(6N\)-dimensional phase space of Hamiltonian Mechanics, the ensemble density \(\rho(q,p,t)\), and Liouville's theorem \(\pp_{t}\rho + \pb{\rho}{\Ham} = 0\) as the statement that the flow preserves phase-space volume, with the symplectic geometry of Symplectic Geometry of Phase Space behind it; stationarity of any \(\rho\) that depends on the phase point only through conserved quantities; Gibbs's formulation of the whole subject in these terms [Gibbs:1902].]

Macrostates, microstates and coarse graining

[Reserved: the distinction between a microstate and the macrostate specified by a few extensive variables; the multiplicity \(W\) of a macrostate; the coarse-grained density and why the fine-grained Gibbs entropy is constant under Liouville evolution while the observed entropy is not — the Ehrenfests' analysis of the issue [Ehrenfest:1911b]; the cell size and its cancellation from all entropy differences, with the absolute value fixed only by \(h\) (Quantum Statistics).]

Entropy and the Boltzmann principle

The combinatorial definition

[Reserved: Boltzmann's 1877 counting of complexions — the distribution of a fixed energy among discrete cells, the multiplicity of each distribution, and the identification of the most probable distribution with equilibrium [Boltzmann:1877]; the Stirling limit; the logarithm as the only function turning multiplicative multiplicities into an additive quantity; the recovery of the Maxwell–Boltzmann distribution as the maximiser.]

The relation between entropy and multiplicity

[Reserved: \(S = k\log W\), written in that form by Planck when he introduced the constant \(k\) in the radiation analysis [Planck:1901]; the constant as a conversion factor between energy and temperature, now exact by the 2019 redefinition of the kelvin [BIPM:2019] with the value quoted in Measurement, SI Units, and the Theory of Errors and [Tiesinga:2021]; the sense in which the relation is a definition and the sense in which it is a testable claim.]

The H-theorem and the reversibility objections

[Reserved: the functional \(H=\int f\log f\,\dd^{3}v\) and Boltzmann's proof that it decreases under the collision term [Boltzmann:1872]; Loschmidt's reversibility objection — the reversed microstate has increasing \(H\) [Loschmidt:1876]; the recurrence objection built by Zermelo [Zermelo:1896] on Poincaré's recurrence theorem [Poincare:1890], and the estimated recurrence time; where the arrow enters — the molecular-chaos assumption — stated as an assumption and not concealed; the initial-condition question left open and referred to Evidence-Based Cosmology.]

The Gibbs entropy

[Reserved: \(S = -k\int \rho \log\rho \,\dd\Gamma\) and its discrete form \(-k\sum_{i} p_{i}\log p_{i}\) [Gibbs:1902]; agreement with the Boltzmann expression in the thermodynamic limit and disagreement away from it; the concavity and additivity properties; its identification with the Shannon functional of Section 118.8.1.]

Ensembles

The microcanonical ensemble

[Reserved: the postulate of equal a priori probabilities on the energy shell, the density of states \(\Omega(E,V,N)\), and \(S=k\log\Omega\) [Gibbs:1902]; temperature, pressure and chemical potential as derivatives of \(S\); the shell thickness and its irrelevance; the two-subsystem derivation of the zeroth law and of the equality of temperatures at equilibrium.]

The canonical ensemble

[Reserved: a system in contact with a reservoir, the Boltzmann factor \(\ee^{-\beta E_{i}}\), and the partition function \(Z=\sum_{i}\ee^{-\beta E_{i}}\) [Gibbs:1902]; the Helmholtz free energy \(F=-kT\log Z\) as the generating function of every thermodynamic quantity; the energy distribution and its sharpness; the classical phase-space integral with the \(1/(N!h^{3N})\) prefactor justified in Section 118.4.3; standard development in [Landau:1980].]

The grand canonical ensemble

[Reserved: exchange of particles as well as energy, the fugacity \(z=\ee^{\beta\mu}\), and the grand partition function \(\Xi=\sum_{N} z^{N} Z_{N}\) [Gibbs:1902]; the grand potential \(\Phi = -kT\log\Xi = -pV\); particle-number fluctuations and their relation to the isothermal compressibility; why this is the natural ensemble for the quantum gases of Quantum Statistics.]

Equivalence of ensembles

[Reserved: the coincidence of the three ensembles for all intensive quantities as \(N\to\infty\), and the \(O(N^{-1/2})\) relative size of the differences; the Legendre-transform structure relating \(S\), \(F\) and \(\Phi\); the conditions under which equivalence fails — long-range interactions (Section 118.5.3) and at a first-order transition; rigorous statements in [Ruelle:1969].]

Partition functions in practice

[Reserved: factorisation for non-interacting degrees of freedom; translational, rotational and vibrational partition functions of a molecule and the resulting heat capacities, compared with measured values for diatomic gases; the classical virial expansion for weak interactions; the systematic use of \(\log Z\) as a cumulant generator, linking to Probability and Statistics; thermodynamic identities collected in [Callen:1985].]

The classical ideal gas

The Maxwell–Boltzmann distribution

[Reserved: the speed distribution \(f(v)\propto v^{2}\ee^{-mv^{2}/2kT}\) [Maxwell:1860] and Boltzmann's generalisation to an arbitrary external potential [Boltzmann:1868]; the mean, root mean square and most probable speeds; the direct measurement of the distribution in a potassium and thallium atomic beam by Miller and Kusch, agreeing with the predicted form to about \(1\,\mathrm{\%}\) [Miller:1955]; the barometric formula as the same law in gravity, which is the observable used in Experiment: Brownian Motion and Avogadro's Number.]

Equipartition and its failure

[Reserved: \(\tfrac{1}{2}kT\) per quadratic degree of freedom [Boltzmann:1868]; the Dulong–Petit law for solids [Dulong:1819] as its most successful instance; the failures Maxwell singled out as the greatest difficulty facing the molecular theory — the missing vibrational and rotational contributions to diatomic heat capacities, and the fall of solid heat capacities at low temperature [Maxwell:1875]; the resolution by level quantisation, carried out in Black-Body Radiation and Planck's Hypothesis and Phonons and Lattice Dynamics and Quantum Statistics.]

The Gibbs paradox

[Reserved: the spurious entropy of mixing between two samples of the same gas, and Gibbs's own statement and discussion of it [Gibbs:1876]; the \(1/N!\) correction and the resulting extensivity of \(S\); the fact that classical mechanics gives no reason for the factor, so that the paradox is evidence for indistinguishability before quantum mechanics supplies it (Identical Particles and Quantum Statistics); the operational reading in terms of which mixings can be undone.]

The Sackur–Tetrode entropy

[Reserved: the absolute entropy of a monatomic ideal gas, \(S=Nk[\log(V/N\lambda^{3})+\tfrac{5}{2}]\) with the thermal wavelength \(\lambda = h/\sqrt{2\pi mkT}\), obtained independently by Sackur [Sackur:1911] and Tetrode [Tetrode:1912]; the appearance of Planck's constant in a classical result and what that signals; the vapour-pressure test — Tetrode's comparison with mercury vapour data, and the chemical-constant measurements that followed — as the first quantitative confirmation; the Sackur–Tetrode constant among the CODATA quantities [Tiesinga:2021].]

The thermodynamic limit

Existence and extensivity

[Reserved: the limit \(N\to\infty\), \(V\to\infty\) at fixed \(N/V\); van Hove's proof that the free energy per particle converges for interactions with a hard core and a sufficiently rapid decay [vanHove:1949], and the systematic treatment in [Ruelle:1969]; stability and tempering as the hypotheses; why the limit is what makes thermodynamics sharp rather than approximate.]

Analyticity and its failure

[Reserved: the partition function of a finite system as an analytic function of \(\beta\), so that no finite system has a phase transition; the Yang–Lee account of how non-analyticity appears in the limit through the pinching of the real axis by zeros of the partition function [Yang:1952]; the transfer of this to Phase Transitions and Critical Phenomena.]

Long-range forces and non-extensive systems

[Reserved: systems in which the energy is not extensive — self-gravitating matter above all; negative heat capacity and the gravothermal catastrophe [LyndenBell:1968]; the inequivalence of ensembles that follows, and why a self-gravitating system has no maximum-entropy equilibrium; the treatment of stellar and compact matter in Stellar Structure and Nucleosynthesis and Compact Stars and Relativistic Astrophysics, where these caveats are load-bearing.]

The ergodic hypothesis: status

What the hypothesis is asked to do

[Reserved: the identification of time averages along a trajectory with ensemble averages over the energy shell; the Ehrenfests' distinction between the original ergodic hypothesis (the trajectory passes through every point), which is false for dimensional reasons, and the quasi-ergodic hypothesis [Ehrenfest:1911b]; the role the assumption actually plays — justifying the microcanonical postulate of Section 118.3.1 — and the alternatives to it.]

The ergodic theorems

[Reserved: Birkhoff's pointwise ergodic theorem, giving almost everywhere convergence of the time average [Birkhoff:1931], and von Neumann's mean ergodic theorem in \(L^{2}\) [vonNeumann:1932a]; metric transitivity as the condition under which the limit is the ensemble average; Khinchin's argument that only sum functions of many degrees of freedom need behave well, which weakens what must be proved [Khinchin:1949].]

Systems proved ergodic

[Reserved: Sinai's proof of ergodicity and mixing for a gas of hard elastic spheres in a box [Sinai:1970]; hyperbolicity and sensitive dependence as the mechanism (Nonlinear Dynamics and Chaos); the smallness of the class — no realistic interatomic potential has been shown ergodic — stated plainly.]

Honest status of the foundations

[Reserved: the Fermi–Pasta–Ulam–Tsingou numerical experiment, in which a weakly nonlinear chain failed to equipartition and instead recurred [Fermi:1955], and the KAM tori that explain it; the resulting position — statistical mechanics works far better than its proofs justify — recorded as an open problem rather than argued away; the pragmatic defence by timescales; forward reference to What We Observe but Do Not Understand.]

Thermalisation in isolated quantum systems

[Reserved: the eigenstate thermalisation hypothesis of Deutsch [Deutsch:1991] and Srednicki [Srednicki:1994], in which individual energy eigenstates already encode thermal expectation values; the measured approach to thermal equilibrium of an isolated lattice of ultracold bosons through the growth of entanglement entropy [Kaufman:2016]; the relation to decoherence and open systems in Open Quantum Systems and Decoherence.]

Fluctuations

Energy and particle-number fluctuations

[Reserved: \(\langle \Delta E^{2}\rangle = kT^{2}C_{V}\) from the canonical ensemble and the analogous compressibility relation in the grand canonical ensemble [Gibbs:1902]; the relative size \(O(N^{-1/2})\), which is why thermodynamics appears deterministic; the inversion of the argument — a measured fluctuation determines \(k\) and hence Avogadro's constant — which is the strategy of Experiment: Brownian Motion and Avogadro's Number; Einstein's general fluctuation formula [Einstein:1910].]

Critical opalescence

[Reserved: the divergence of the compressibility near a critical point, the resulting long-wavelength density fluctuations, and the strong scattering of light that makes a critical fluid turn milky; Smoluchowski's kinetic account [Smoluchowski:1908] and Einstein's quantitative theory of the scattered intensity [Einstein:1910]; the observation as a direct visual demonstration that fluctuations are real; critical exponents deferred to Phase Transitions and Critical Phenomena.]

Thermal noise

[Reserved: Johnson's measurement of a voltage noise across a resistor proportional to resistance and to absolute temperature and independent of material [Johnson:1928], and Nyquist's derivation of \(\langle V^{2}\rangle = 4kTR\,\Delta f\) from equilibrium statistical mechanics alone [Nyquist:1928b]; the measurement as a second independent determination of \(k\); the general fluctuation–dissipation statement developed in Nonequilibrium Thermodynamics and Transport.]

Entropy, information and inference

The Shannon functional

[Reserved: Shannon's uniqueness derivation of \(-\sum p_{i}\log p_{i}\) from continuity, monotonicity and the composition rule [Shannon:1948]; its formal identity with the Gibbs entropy of Section 118.2.4; the care needed in reading that identity physically — the physical entropy carries units of \(\mathrm{J}/\mathrm{K}\) and refers to a specified macroscopic description.]

Maximum entropy as inference

[Reserved: Jaynes's derivation of the canonical and grand canonical distributions by maximising the Shannon entropy subject to the known expectation values, with the Lagrange multipliers becoming \(\beta\) and \(\beta\mu\) [Jaynes:1957]; what this reformulation buys — a uniform derivation and a clear account of which assumptions are doing the work — and what it does not settle, namely why the resulting distributions describe the system after a finite time; Lagrange multipliers as in Calculus of Variations.]

Information and thermodynamic cost

[Reserved: Szilard's one-molecule engine and the entropy cost of a measurement [Szilard:1929]; Landauer's principle, that erasing one bit dissipates at least \(kT\log 2\) [Landauer:1961]; the experimental verification with a colloidal particle in a double-well optical trap, confirming the bound at the level of a few percent of \(kT\) [Berut:2012]; the exorcism of Maxwell's demon that follows, and its relation to the second law as stated in Classical Thermodynamics.]

Negative absolute temperature

[Reserved: a bounded-energy spectrum admitting population inversion and hence \(\pp S/\pp E<0\); the nuclear spin system of lithium fluoride prepared by Purcell and Pound, which relaxed through an inverted state with a measurable negative spin temperature [Purcell:1951]; Ramsey's demonstration that such states are consistent thermodynamic equilibria hotter than any positive temperature [Ramsey:1956]; the motional degrees of freedom of an ultracold lattice gas prepared at negative temperature [Braun:2013]; why this does not violate the second law.]

Beyond equilibrium

[Reserved: the Jarzynski equality relating the free-energy difference to an exponential average of the work done in arbitrarily fast processes [Jarzynski:1997], and its verification by mechanically unfolding a single RNA molecule [Liphardt:2002]; the second law recovered as an inequality from an equality, with rare trajectories of negative entropy production observed; the systematic development in Nonequilibrium Thermodynamics and Transport.]