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A critical phenomenon is behavior near a continuous phase transition where fluctuations extend over a large correlation length and thermodynamic responses can become singular. Near a liquid-gas critical point, for example, the distinction between liquid and vapor diminishes while density fluctuations become unusually large.

Correlation length and response functions often obey power laws in the distance from the transition, characterized by critical exponents. Different microscopic materials can share those exponents when their long-distance behavior belongs to the same universality class. The renormalization group explains this by following effective interactions as the observation scale changes. A finite sample, finite observation time or imperfect tuning cuts off the ideal divergences, so experimental power laws need a stated range of scales.

The ideal gas free energy in the canonical ensemble follows from $F=-k_{B}T\log Z$. For a dilute classical monatomic ideal gas, with the usual indistinguishable-particle counting and no internal excitations,where ${\lambda }_T$ is the thermal de Broglie wavelength and the expression uses the large-$N$ approximation to $N!$. The classical regime requires $n{\lambda }_T^3\ll 1$ for a single internal state.

At fixed particle number and temperature, its volume dependence is $-N{k}_{B}T\log V$. Differentiating gives pressure $p=-(\partial F/\partial V{)}_{T,N}=N{k}_{B}T/V$. The same function supplies the chemical potential and entropy by other derivatives, connecting state counting to several measurable quantities at once. Omitting the temperature-dependent terms is harmless for that pressure derivative but not for entropy or internal energy.

An ideal-gas atmosphere in uniform gravity combines thermal particle statistics with a spatial potential energy $mgz$. At fixed temperature $T$, the Boltzmann factor giveswhere $m$ is the particle mass and $H$ is the scale height. The same result follows from hydrostatic equilibrium $dp/dz=-mng$ and the ideal gas law $p=n{k}_{B}T$.

The two derivations connect a microscopic probability weight to a macroscopic force balance. A height increase $H$ reduces both pressure and number density by a factor $e$. In a noninteracting mixture held at thermal equilibrium, different molecular masses have different scale heights; efficient mixing changes the relevant atmospheric model. The result assumes a uniform temperature and gravitational field, so it is a controlled idealization rather than a complete description of a real atmosphere.

The canonical ensemble is the probability distribution for a system in thermal equilibrium with a heat bath at fixed temperature $T$, with fixed particle number and external parameters. It allows energy exchange, so individual energy measurements fluctuate. With Boltzmann constant $k_B$ and inverse temperature $\beta =1/(k_{B}T)$,The heat bath is idealized as large enough that exchanges barely change its temperature; coupling must permit equilibration while contributing negligible interaction energy. A finite heat bath approximates this description only when its temperature change is negligible over the relevant exchanges. Strong coupling or a small heat bath can require a different description.

For one two-level system at positive temperature, the Boltzmann weights are $1,e^{-\beta \Delta }$, henceAt $\beta \Delta =\ln 3$, $Z_1=4/3$ and $p=1/4$. This is a probabilistic thermal mixed state. For quantum levels, its density matrix is $\rho =(1-p)|0⟩⟨0|+p|1⟩⟨1|$. The pure state $\sqrt{1-p}|0⟩+\sqrt{p}|1⟩$ has the same energy measurement probabilities, but also has quantum coherence between the energy eigenstates. Matching those probabilities does not make the states equivalent.

For an additive system with short-range interactions and a regular extensive entropy, microcanonical, canonical, and grand canonical predictions agree for bulk observables in the thermodynamic limit because relative fluctuations vanish. Equivalence can fail for finite systems, long-range interactions, or at singular phase-coexistence points.

The grand canonical ensemble describes equilibrium with a reservoir that fixes temperature $T$ and chemical potential $\mu$, while the system's volume is fixed and both energy and particle number can fluctuate. A microstate with energy $E$ and particle number $N$ has probability proportional to $e^{-\beta (E-\mu N)}$, with $\beta =1/(k_{B}T)$. The normalization is the grand canonical partition function $\Xi$.

A small region of gas exchanging particles and heat with a much larger gas reservoir illustrates the ensemble. Differentiating $\log \Xi$ gives $⟨N⟩=k_{B}T{\partial }_{\mu }\log \Xi$, and a second derivative gives the number variance. Thus a response to chemical potential and spontaneous particle-number fluctuations encode related information. The ensemble does not mean that total particle number in system plus reservoir is unconstrained; fluctuations describe the chosen subsystem.

The microcanonical ensemble describes an isolated equilibrium system with fixed volume, particle number and energy within a narrow shell. It assigns equal probability to the accessible microstates inside that shell. For a discrete count $\Omega (E)$, the entropy is $S=k_B\log \Omega$; classical phase-space counting also requires a specified measure and shell width.

An idealized insulated rigid container motivates the constraints. Its particles exchange energy internally, but the whole container exchanges neither heat nor particles with its surroundings. A small subsystem can still have fluctuating energy because the rest acts as a reservoir. In the thermodynamic regime where that reservoir has a well-defined temperature, the subsystem approaches a canonical ensemble. Equal shell probabilities are an ensemble prescription; whether a particular dynamics explores the accessible shell on an experimental timescale is a separate equilibration question.

For a discrete probability distribution over microstates, the Gibbs entropy is

Thermodynamics studies energy transfers and the constraints on macroscopic changes. Its vocabulary connects internal energy, entropy, temperature, pressure and volume to heat and work. The first law of thermodynamics balances energy; the Second law of thermodynamics decides which changes can occur spontaneously and limits how much work a heat engine can deliver. These principles let us bound performance before choosing a material or a detailed machine design. Applications include power generation, propulsion and refrigerators. ^[ref]

Start by choosing a system boundary. A closed system can exchange energy but no matter; an isolated system exchanges neither. A state function, such as internal energy or entropy, depends on the equilibrium state. A process specifies how that state changes. Thus a body's temperature is a property it has, whereas heat records energy crossing its boundary during a process. A body contains internal energy, rather than a stored quantity of heat.

Two insulated bodies initially at different positive temperatures approach thermal equilibrium when brought into thermal contact. Their combined energy is conserved, but their combined entropy increases. They do not spontaneously separate again into a hotter body and a colder one. This familiar asymmetry makes thermodynamics interesting: conserving energy leaves many imaginable changes available, and entropy supplies a further constraint. The same distinction underlies both entropy maximization under thermal contact and the thermal efficiency limit in Carnot's theorem.

For photons in thermal equilibrium at temperature $T$, Planck's law gives the photon number density per frequency interval. In terms of the Planck function, it is $dn/d\nu =4\pi B_{\nu }(T)/(ch\nu )$, hence

Quantum ideal-gas statistics describes noninteracting identical particles while retaining the exchange rules of bosons or fermions. In thermal equilibrium, a one-particle state of energy $E$ has mean occupationThe minus sign is the Bose-Einstein distribution; the plus sign is the Fermi-Dirac distribution. A bosonic state permits arbitrary occupation, whereas Pauli exclusion principle restricts each fermionic state to zero or one particle. ^[ref]

The mean particle number and internal energy follow by summing $\overline{N}={\sum }_r{\overline{n}}_r$ and $U={\sum }_r{E}_r{\overline{n}}_r$. When every occupation is small, both denominators reduce to the same exponential and reproduce Maxwell–Boltzmann statistics. Thus absence of interactions does not make a gas classical.

This counts microstates sharing a macrostate; when the shared quantity is energy, it gives the multiplicity of degenerate energy levels. The degeneracy $g(l)$ of a macrostate labelled by $l$ is the number of distinct microstates having that value of $l$.

This is the one-dimensional version of a freely jointed chain, an ideal chain. A chain of $n$ links of length $a$, each pointing independently right or left, has extensionand multiplicity $n!/(n_{+}!n_{-}!)$.

The Boltzmann equation describes a dilute gas through its phase-space distribution function $f(\mathbf{x},\mathbf{v},t)$. With $f$ normalized so that $\int f{d}^{3}v$ is number density, a velocity-independent external force $\mathbf{F}$ givesThe left side transports molecules through position and velocity space; the collision term $C[f]$ accounts for redistribution by molecular encounters.

A small gas sample with an anisotropic velocity distribution illustrates the two effects. Molecules carry the distribution across the container while collisions tend to redistribute their velocities. For elastic binary collisions, the collision term conserves particle number, momentum and energy, although it changes the distribution's shape. Deriving a closed one-particle collision term requires assumptions such as molecular chaos for the Boltzmann equation. Taking velocity moments connects this microscopic description to fluid conservation laws and transport coefficients.

Relativistic causality requires physical influences and controllable information to propagate only within or on the future light cone of their source.

The four-acceleration is the covariant derivative of four-velocity along a massive observer's worldline: $A^{\mu }=d{U}^{\mu }/d\tau +{\Gamma }_{\nu \rho }^{\mu }{U}^{\nu }{U}^{\rho }$. In Cartesian Minkowski spacetime coordinates the Christoffel symbols vanish, leaving the ordinary proper time derivative. Differentiating the constant velocity norm gives $U\cdot A=0$; in the instantaneous rest frame, $A=(0,\mathbf{a})$, so $A\cdot A=-a^2$ with $+---$ metric signature. The magnitude $a$ is the proper acceleration.

A massive observer's four-velocity is the tangent $U^{\mu }=d{X}^{\mu }/d\tau$ to its worldline, parametrized by proper time. Its invariant squared Minkowski norm is $c^2$ in the $+---$ metric signature, and $-c^2$ in the opposite convention. Unlike the coordinate values $X^{\mu }$, the tangent transforms as a vector under a general coordinate change.

A massive particle has four-momentumFor a massless particle, $P^{\mu }=(E/c,\mathbf{p})$ with $E=c|\mathbf{p}|$ and $P^2=0$.

The sum of two future-pointing timelike four-momenta is timelike. It therefore cannot equal a photon's null four-momentum, which forbids spontaneous photon decay into a massive particle pair in vacuum.

A subluminal speed has magnitude strictly below the speed of light: $|v|<c$. Massive particles follow timelike worldlines and therefore move subluminally in every inertial frame.