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For $N$ independent spins with $s_z\in \{-1,0,1\}$ and one-spin Hamiltonian $-\mu B{s}_z$,Writing $x=\beta \mu B$, its heat capacity is

The partition function $Z(\beta )={\sum }_s{e}^{-\beta E_s}$ is the sum of the Boltzmann weights over individual microstates. Dividing by it makes their probabilities sum to one. When summing distinct energies instead, include the state degeneracy: $Z={\sum }_{E}g(E)e^{-\beta E}$. For two independent distinguishable two-level systems,The middle coefficient counts two microstates, exactly as in the preceding figure. For $N$ such units, $Z_N=Z_1^N$; this factorization requires additive energy and independence, and generally fails with interactions.

The usefulness goes beyond normalization. For fixed energies, derivatives of the natural logarithm of the partition function yield measurable quantities:Here $U$ is internal energy, the second expression is the energy variance, and $S$ is Gibbs entropy. For one two-level system at fixed $\Delta$, its heat capacity $C=\partial U/\partial T$ and entropy areAt $\beta \Delta =\ln 3$, these give $U/\Delta =0.25$, $C/k_B\simeq 0.2263$ and $S/k_B\simeq 0.5623$. For independent units, all three quantities add.

The figure shows why heat capacity peaks. At low temperature, the excited energy is rarely occupied; at high temperature, the probabilities are almost equal and further warming barely changes internal energy. The largest response lies between these limits, at $k_{B}T/\Delta \simeq 0.4168$, where $C/k_B\simeq 0.4392$. Meanwhile entropy rises from zero toward $k_B\ln 2$, reflecting two equally likely alternatives. This high-temperature limit belongs to the stipulated two-level system; extra accessible energy eigenstates would change it.

The second variation is the quadratic term in a functional's expansion along $u+\epsilon \eta$; positivity on admissible variations is a local-minimum test.

Noether’s theorem associates a conserved quantity to each continuous variational symmetry.

Fermat-type optical paths in cylindrical coordinates extremize refractive index times Euclidean arclength.

Among functions with fixed boundary values, the solution of $\nabla \cdot (\kappa \nabla \varphi )=0$ with $\kappa >0$ uniquely minimizes the weighted energy

A point is stationary for a differentiable function or functional when its first derivative or first variation vanishes in every admissible direction.

A functional is a function whose inputs are themselves functions, curves, or other elements of a function space.

The direct method takes a minimizing sequence, extracts a subsequence using compactness, and uses lower semicontinuity to show that its limit attains the infimum.

A holomorphic or meromorphic integrand has a single-valued path integral on a domain whenever its integral around every closed curve in that domain vanishes.

If a cut joins two poles with opposite residues, every loop avoiding the cut has equal winding number about the endpoints, so the two residue contributions cancel.

Analytic continuation of an integral of a meromorphic function around a closed loop changes its value by $2\pi i$ times the winding-number-weighted sum of enclosed residues.

For steady incompressible flow with imposed outer pressure gradient,The normal momentum equation makes pressure approximately uniform across the layer.

Balancing $U^2/x$ with $\nu U/{\delta }^2$ gives $\delta \sim \sqrt{\nu x/U}$.

Magnetization is magnetic dipole moment per unit volume. It produces bound volume and surface currentsAcross an interface, the net bound sheet current is $({\mathbf{M}}_1-{\mathbf{M}}_2)\times \mathbf{n}$.

The critical dimension is the target-spacetime dimension in which quantum anomalies obstructing Lorentz or Weyl symmetry cancel. It is 26 for the bosonic string.

Here critical means cancellation of the string’s quantum anomalies; it does not mean the upper or lower critical dimension of a phase transition.

The Fourier modes ${\alpha }_n^i$ and ${\tilde{\alpha }}_n^i$ of the transverse closed-string coordinates obey harmonic-oscillator commutation relations. Modes with negative index create string excitations.

The Polyakov action introduces an independent worldsheet metric $g_{\alpha \beta }$:Its metric equation makes $g_{\alpha \beta }$ conformal to the induced worldsheet metric, reducing it classically to the Nambu–Goto action.

String tension is energy per unit length. The Regge-slope parameter ${\alpha }^{\prime }$ is conventionally defined by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$.

For a product state, probabilities of local measurement outcomes multiply.