m physical modeling / IICSM

CONNECTED PHYSICS / STATES OF MATTER

From quantum atoms to plasma

Connect quantum mechanics and quantum fields to atoms, solids, liquids, gases, and plasmas, with the assumptions at each change of description made explicit.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

CHAPTER 01

Quantum mechanics: atoms to matter

Build from quantum states and atomic binding to electronic bands, statistics, and the classical limits used across solids, liquids, gases, and plasmas.

Read derivations →CHAPTER 02

Quantum field theory: fields to matter

Connect quantized fields and many-body excitations to atoms, phonons, quantum fluids, photons, and plasma approximations.

Read derivations →CHAPTER 03

Atomic physics: bound states, spectra and interactions

Use quantum mechanics to derive atomic structure and spectra, then connect interacting atoms to solids, liquids, gases and plasmas.

Read derivations →CHAPTER 04

Solid-state physics

Ordered reference positions, small displacements, and quantum electronic states. Start with the microscopic Hamiltonian and keep the structure that makes a crystal a crystal.

Read derivations →CHAPTER 05

Liquid-state physics

Keep strong local correlations while releasing the assumption of permanent lattice sites. Thermodynamic structure and molecular motion now require distinct but related descriptions.

Read derivations →CHAPTER 06

Gaseous-state physics

Use diluteness to separate free flight from binary collisions. From a velocity distribution, derive pressure and fluid equations, then introduce ionization when the species composition changes.

Read derivations →CHAPTER 07

Plasma physics

Retain mobile charge and self-consistent fields. Choose collisional kinetics, collisionless kinetics, or a fluid approximation by comparing physical length and time scales.

Read derivations →

How the quantum foundations connect to each state

Read the atom-to-plasma route below as a sequence of modeling choices. Cooling, heating, compression, and chemical changes can follow different physical paths.

Quantum description → retained variables → material model
StageConnection and assumptionsContinue
AtomBound electron states and photon transitions; the nonrelativistic Coulomb model is a low-energy limit of electromagnetic field theory.Atomic derivation · QED connection
SolidInteracting orbitals form bonds and bands; expansion about a stable lattice yields vibrations and quantized phonons.Atom → solid · Solid chapter
LiquidElectronic energy surfaces supply interactions; compare phase free energies and sample configurations. Classical nuclei are an approximation; quantum liquids need quantum many-body models.Melting bridge · Liquid chapter
GasLow-density quantum scattering and Bose/Fermi statistics approach classical kinetics only when the relevant scale and degeneracy tests pass.Classical limit · Gas chapter
PlasmaIonization introduces mobile charges. Mean fields and coarse-grained kinetics describe many plasmas; dense, degenerate, or relativistic regimes need quantum or relativistic extensions.Field → kinetic limit · Plasma chapter

What changes between descriptions?

Approximations, retained physics, and validity tests
RegimeRetained descriptionMajor assumptionCheck before use
CrystalLattice displacements, phonons, electronic bandsSmall displacements / periodic referenceStability, anharmonicity, disorder, electronic correlations
LiquidPair structure and temporal correlationsSpecified potential and correlation closureDensity, thermodynamic consistency, sampling
Dilute gasOne-particle distribution and binary collisionsMolecular chaos / classical statisticsDensity correction, degeneracy, Knudsen number
PlasmaCharged distributions and collective fieldsChosen collision, screening, and fluid orderingDebye length, coupling, collisions, gyro and inertial scales

1. A shared microscopic starting point

H=∑ipi22mi+U(r1,…,rN)H=\sum_i\frac{\mathbf p_i^2}{2m_i}+U(\mathbf r_1,\ldots,\mathbf r_N)r˙i=∂H∂pi,p˙i=−∂H∂ri\dot{\mathbf r}_i=\frac{\partial H}{\partial\mathbf p_i},\quad\dot{\mathbf p}_i=-\frac{\partial H}{\partial\mathbf r_i}iℏ∂tρ^=[H^,ρ^]i\hbar\partial_t\hat\rho=[\hat H,\hat\rho]
  1. Specify the species, masses, charges, interaction energy, and boundary conditions. An effective interatomic potential may already average over electronic motion.
  2. Hamilton equations determine classical trajectories. If quantum statistics or electronic structure matters, evolve a quantum density operator instead.
  3. The same microscopic ingredients can support different equilibrium phases. Temperature, pressure, composition, and competing free energies select the stable phase.

Connection / worked consequence. Changing phase does not require inventing a new conservation law; practical models change the variables and correlations they retain.

↑ Return to the state pathway

2. From microscopic states to thermodynamics

Z=Tr⁡e−βH^,F=−kBTln⁡Z,β=(kBT)−1Z=\operatorname{Tr}e^{-\beta\hat H},\quad F=-k_{\mathrm B}T\ln Z,\quad\beta=(k_{\mathrm B}T)^{-1}Zcl=1N!h3N∫e−βH d3Nr d3NpZ_{\mathrm{cl}}=\frac{1}{N!h^{3N}}\int e^{-\beta H}\,d^{3N}r\,d^{3N}pp=−(∂F∂V)T,N,μ=(∂F∂N)T,Vp=-\left(\frac{\partial F}{\partial V}\right)_{T,N},\quad\mu=\left(\frac{\partial F}{\partial N}\right)_{T,V}
  1. Weight accessible states by their Boltzmann factor in the canonical ensemble. The quantum trace counts states with the appropriate particle statistics.
  2. For identical nondegenerate classical particles with no internal degeneracy, replace the trace by a phase-space integral and divide by N! and h^(3N).
  3. Differentiate the resulting free energy to obtain pressure and chemical potential. Structural approximations change the partition function and therefore the predicted thermodynamics.

Connection / worked consequence. Solid and liquid free energies must be compared on the same thermodynamic basis. A phonon-only model is not a complete melting model.

↑ Return to the state pathway

3. Melting and vaporization: equality of chemical potentials

μα(T,p)=μβ(T,p)\mu_\alpha(T,p)=\mu_\beta(T,p)dμ=−s dT+v dpd\mu=-s\,dT+v\,dpdpcoexdT=ΔsΔv=LTΔv\frac{dp_{\mathrm{coex}}}{dT}=\frac{\Delta s}{\Delta v}=\frac{L}{T\Delta v}
  1. For a pure substance, transfer a small number of particles between two phases at common temperature and pressure. Stationary total Gibbs energy requires equal chemical potentials.
  2. Differentiate that equality along the coexistence curve and substitute dμ=−s dT+v dp for each phase.
  3. Rearrange to obtain the Clapeyron slope. Here s, v, and latent heat L=TΔs are all per particle; molar quantities also work if used consistently.

Connection / worked consequence. A continuous chain of explanations does not mean a continuous change of order or density. Melting and boiling can be discontinuous; pressure paths can also bypass a liquid through sublimation or a critical point.

↑ Return to the state pathway

4. One hierarchy, different dynamical closures

∂tfN+{fN,H}=0\partial_t f_N+\{f_N,H\}=0(∂t+v1⋅∇1+Fext,1m⋅∇v1)f1=−1m∇v1⋅∫F12f2 d3r2 d3v2\left(\partial_t+\mathbf v_1\cdot\nabla_1+\frac{\mathbf F_{\mathrm{ext},1}}m\cdot\nabla_{v_1}\right)f_1=-\frac1m\nabla_{v_1}\cdot\int\mathbf F_{12}f_2\,d^3r_2\,d^3v_2
  1. Conservation of probability along Hamiltonian trajectories gives Liouville’s equation; the braces denote a Poisson bracket.
  2. Integrate out all but one particle to obtain the first BBGKY equation. Here f1 and f2 are number-normalized reduced distributions for identical particles.
  3. Its force term depends on pair statistics. Equations for pairs depend on triples, producing a hierarchy. Each usable kinetic theory must specify how it truncates or approximates that hierarchy.

Connection / worked consequence. Dense liquids retain pair structure; dilute gases approximate incoming binary encounters; weakly coupled collisionless plasmas retain a collective field and neglect collisional correlations. These are different limits, not automatic successive algebraic substitutions.

↑ Return to the state pathway

Follow the three transitions

  1. Solid → liquid: compare free energies; replace fixed lattice positions by configurational statistics. A harmonic expansion alone does not predict melting.
  2. Liquid → gas: compare chemical potentials for vaporization, or take a separate dilute-gas limit to derive a virial expansion.
  3. Gas → plasma: introduce ionization and charge conservation, then test whether collective field behavior matters. Equilibrium Saha and nonequilibrium rate kinetics are different routes.

Notation used throughout

n: number density (m⁻³); N: particle count; ρ: mass density (kg m⁻³); ρc: charge density; p: pressure; T: temperature (K); kB: Boltzmann constant; h, ℏ: Planck constants; β=1/(kBT); μ: chemical potential; f: phase-space distribution; g(r): pair distribution. In solid displacements u is a displacement; in the liquid closure u(r) is pair energy; in fluid equations u is bulk velocity. Subscripts identify phase or species. Every approximation must use consistent SI units or explicitly stated reduced units.