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At fixed temperature, volume, and chemical potential, the grand canonical ensemble permits both energy and particle number to fluctuate and weights a state by $e^{-\beta (E-\mu N)}$.

The microcanonical ensemble assigns equal probability to accessible states in a narrow energy shell at fixed energy, volume, and particle number.

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

An ideal quantum gas has mean occupationwith one sign for bosons and the other for fermions.

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 evolves a phase-space distribution by balancing free streaming, external forces, and collisions.

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

Four-acceleration is the proper-time derivative of four-velocity and is orthogonal to it.

Four-velocity is the proper-time derivative of position and has invariant squared norm c^2.

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.

Minkowski spacetime is the flat four-dimensional spacetime of special relativity: one time coordinate and three spatial coordinates, equipped with a Minkowski metric. More precisely, it is an affine space, so no spacetime event is a preferred origin; differences between spacetime events are four-vectors. Inertial frames give different coordinates to the same spacetime events, but agree on their Minkowski interval.

This geometry answers two different measurement questions with one structure. Which spacetime events can be connected by a light signal or a massive observer? The light cone classifies them. How much time does a moving clock record? Its world line and the Minkowski metric determine its proper time. The minus sign in the Minkowski metric is what allows these answers to differ from ordinary Euclidean distance.

For a concrete one-space-dimensional slice, start at the origin $O$ and measure $t$ in seconds and $x$ in light-seconds. The spacetime event $A$ at $(t,x)=(4\mathrm{s},3\mathrm{ls})$ can be reached by an observer moving at $0.75c$, whereas $B=(2\mathrm{s},3\mathrm{ls})$ would require $1.5c$ to be reached directly from $O$. The Minkowski interval distinguishes them even though both are three light-seconds away in that inertial frame.

Hermann Minkowski presented this geometric viewpoint in his Cologne lecture of 21 September 1908. His account treats particle histories as world lines, changes of inertial frame as choices of time axis, and clock readings as proper time. These are connected geometric ideas, rather than separate corrections to ordinary mechanics.^[ref]

Causality requires an effect at an event to depend only on sources in its past light cone. Relativistic field equations implement this choice through retarded Green functions.

A four-vector is a vector whose four vector components transform under a Lorentz transformation as $U^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{U}^{\nu }$. Its individual vector components generally change between inertial frames; the Minkowski metric combines them into invariant scalar products. Listing four quantities is not enough to make a four-vector: their transformation law matters.

The most direct example is the displacement between two spacetime events, written $(c\Delta t,\Delta x,\Delta y,\Delta z)$. Multiplying the time difference by the speed of light gives all four vector components the same length units. A position four-vector describes an event relative to a chosen origin, while a displacement does not depend on a common translation of the origin. Four-momentum and four-velocity use the same transformation rule for different physical quantities.

A Lorentz transformation preserves the Minkowski metric. For a Lorentz boost, let $S^{\prime }$ move at constant velocity $+v$ along the $x$-axis of $S$, with parallel spatial axes and coincident origins at $t=t^{\prime }=0$. With $\beta =v/c$ and the Lorentz factor $\gamma$, the coordinates of the same spacetime event obey

The inverse Lorentz transformation replaces $v$ by $-v$. In particular, $x^{\prime }=0$ traces the moving origin $x=vt$, fixing the sign convention. The time formula is $t^{\prime }=\gamma (t-vx/c^2)$; dropping its position term would lose the relativity of simultaneity. ^[ref].

For the moving-clock example in time dilation, the second tick has $(t,x)=(5\mu \mathrm{s},3c\mu \mathrm{s})$. At $v=0.6c$, substitution gives $t^{\prime }=1.25(5-0.6\times 3)\mu \mathrm{s}=4\mu \mathrm{s}$ and $x^{\prime }=1.25(3-0.6\times 5)c\mu \mathrm{s}=0$. The clock stays at its own origin, and its coordinate time there equals its proper time.

For inertial frames with relative velocity $u$ along the $x$-axis, the Galilean transformation is