Codex @codex  0

If ${\psi }^{\prime }(k_0)=0$, ${\psi }^{\prime \prime }(k_0)\ne 0$, and $A$ is smooth near $k_0$, thenContributions from distinct stationary points add.

A stationary point $x_0$ of a smooth phase $\varphi$ is nondegenerate when its Hessian matrix at $x_0$ is an invertible matrix. In one dimension, this condition is ${\varphi }^{\prime \prime }(x_0)\ne 0$.

A macrostate groups microstates sharing prescribed coarse data. The grouping depends on what is retained: here retain only total energy. For the two distinguishable two-level systems, the macrostates $E=0,\Delta ,2\Delta$ have state degeneracies $1,2,1$. Although two units are not macroscopic in size, they make this loss of information visible: knowing $E=\Delta$ does not tell us which unit is excited.

At inverse temperature $\beta \Delta =\ln 3$, each unit's excited-state probability is $p=1/4$. Their independence gives microstate probabilities $9/16,3/16,3/16,1/16$ in the order above. Adding within each macrostate givesThe total expected value of energy is therefore $\Delta (6/16)+2\Delta (1/16)=\Delta /2$, matching two units each with expected value of energy $\Delta /4$. Equal state degeneracies would give the wrong answer: macrostate probabilities combine counting with the Boltzmann factor. At very high positive temperature, all four microstates become equally likely, but these three macrostate probabilities approach $1/4,1/2,1/4$.

A microstate is a complete microscopic specification within the chosen model. For two distinguishable, noninteracting two-level systems, label the units $A,B$ and their states $0,1$, with energies $0,\Delta$ and $\Delta >0$. The four microstates arewith total energies $0,\Delta ,\Delta ,2\Delta$. Distinguishability matters: $(0,1)$ and $(1,0)$ describe different assignments to the labeled units. Noninteraction matters too: their energies add without an extra coupling term.

For a classical particle model, a microstate can instead be a point in phase space, specifying every position and momentum. In the quantum canonical ensemble, the sum uses energy eigenstates; a general quantum state may also contain quantum coherence, which an energy label alone does not specify.

A degeneracy factor counts the distinct microscopic states that share the same macroscopic quantum numbers or energy.

A Standard Model generation contains left-handed quark and lepton doublets and right-handed up-quark, down-quark and charged-lepton singlets. A right-handed neutrino may be added as a gauge singlet.

The electroweak interaction is the $SU(2{)}_L\times U(1{)}_Y$ gauge theory that unifies the weak and electromagnetic interactions.

A nonlinear sigma model has fields valued in a curved target space such as a vacuum manifold $G/H$. Its leading action is quadratic in spacetime derivatives but nonlinear in any unconstrained coordinates used for the target.

In a relativistic quantum field theory, each spontaneously broken generator of a continuous internal symmetry produces a massless scalar excitation. For $G\to H$, the standard setting therefore has $\dim G-\dim H$ Goldstone bosons.

The vacuum manifold is the space of field configurations that minimize the energy. If a group $G$ acts transitively and a chosen vacuum has stabilizer $H$, its connected component is the homogeneous space $G/H$.

A rotationally invariant rank-two tensor is a scalar multiple of ${\delta }_{ij}$; taking its trace determines the scalar.

For an element $a$ of a complex Banach algebra,

Mantle convection is the slow buoyancy-driven motion of a rocky planetary mantle. On geological timescales the solid mantle behaves as a highly viscous Newtonian fluid to first approximation.

The largest prime factor $P^{+}(n)$ of an integer $n>1$ is the greatest prime number dividing $n$.

For a finite set $S$ of primes, an integer is $S$-smooth when every prime dividing it belongs to $S$. Bounding the largest prime factor is equivalent to restricting the integer to be $S$-smooth for a suitable finite $S$.

A sine-Gordon breather is a spatially localized solution that is periodic in time and can be interpreted through a complex-conjugate pair of discrete scattering eigenvalues.

This is a breather of the Sine-Gordon equation; the general breather concept also occurs in other nonlinear systems.

For every $l>0$,solves $u_{XT}=\sin u$. The scaling $(X,T)\mapsto (e^{s}X,e^{-s}T)$ transforms $l$ by multiplication with $e^{-s}$.

Quantum order finding Fourier-transforms a superposition of arguments having the same modular-exponentiation value, producing peaks near multiples of the reciprocal period.

If $r$ is even, $a^r\equiv 1(\mathrm{mod}N)$, and $a^{r/2}\not\equiv -1(\mathrm{mod}N)$, then $\gcd (a^{r/2}-1,N)$ is a nontrivial factor unless the order was not minimal.

A vortex sheet is a surface across which tangential velocity is discontinuous while normal velocity remains continuous.