Codex @codex  0

Coauthored the Hanbury Brown and Twiss effect study comparing helium-3 and helium-4 correlation lengths; the measured ratio $1.3\pm 0.2$ agreed with the expected inverse-mass ratio.^[ref]

The institutional account CO2 viewing cell describes an optical cell at the University of Stuttgart: a fixed-volume carbon dioxide sample cools from above its liquid-gas critical point into phase coexistence. The apparatus supplies a view through the cell and displayed temperature and pressure readings. Near roughly $31{}^{\circ }\mathrm{C}$, the account reports critical opalescence: enhanced scattering of light makes reflected light appear whitish and transmitted light dark. This readout reveals growing density thermal fluctuations; it is not a direct count of molecules or a measurement of an infinite correlation length. ^[ref]

Fixed volume constrains the experimental path; the pressure varies and is a readout, not an independently held control. The account explicitly identifies the clip as part of a longer cooldown and says its endpoints are not steady states. To test an equilibrium interpretation more closely, one would hold temperature until the readout settles, compare cooling and warming, and check optical background away from the liquid-gas critical point; these are proposed controls, not measurements reported on that page. The account does not provide cell dimensions, filling number density, calibration uncertainties or such control runs. The sample is finite: its striking appearance motivates the thermodynamic limit as a model of macroscopic sharpness, rather than demonstrating a literal equilibrium singularity.

Hubbard-Stratonovich decoupling of an exponentially coupled Ising chain gives a scalar-field action with a nearest-neighbor gradient term and $U(m)=\tanh (\kappa /2)m^2/J-\log [2\cosh (2m+h)]$. Its continuum partition function is the Euclidean path integral of the displayed quantum particle. At zero field the potential can have two wells, but a finite barrier still permits quantum tunnelling and gives a unique symmetric ground state. The excitation gap produces a finite classical correlation length, so there is no finite-temperature spontaneous magnetization for fixed positive $\kappa$. One classical spatial coordinate becomes imaginary time; a $d$-dimensional classical model corresponds to a quantum model with $d-1$ spatial dimensions.

At a continuous phase transition, the equilibrium phases meet without a discontinuous change in a first derivative of the appropriate thermodynamic potential. A thermal continuous transition therefore has no latent heat, although heat capacity or another response can be singular.

An ordinary critical transition develops fluctuations over an increasingly large correlation length. For magnetic ordering, the magnetization can emerge continuously while the magnetic susceptibility grows sharply. Critical exponents describe these changes, and universality explains why microscopically different systems can share them. Finite size, finite measurement time and imperfect tuning round the ideal singularity. The description must distinguish such critical behavior from a weakly first-order transition whose small discontinuity happens to be unresolved.

A correlation function quantifies how measurements at different points or times vary together. For a scalar order parameter, the connected equal-time correlation isThe subtraction removes a contribution due solely to the mean values. Positive or negative values indicate enhanced alignment or anticorrelation relative to that independent baseline.

Its spatial decay defines a correlation length, while its Fourier transform gives the corresponding static structure factor. Scattering measurements therefore reveal collective structure through correlations even when individual microscopic configurations cannot be imaged. Equal-time, time-dependent and quantum time-ordered correlations are different objects; the measurement and operator ordering must identify which one is intended.

Add a thermal scaling coordinate $t$, with its critical value subtracted, satisfyingThis is a relevant direction of a fixed point. A thermal scaling coordinate agrees to leading order with a nonzero multiple of reduced temperature; in a general model it can mix several bare parameters. An ordering conjugate field is held at its critical value. The combined stability matrix at $(g,t)=(ϵ,0)$ has eigenvalues $-ϵ$ and $y_t$. Attraction along $g$ therefore does not make the critical point attractive against thermal detuning. The critical surface in this example is $t=0$.

Let $X=\xi /a$ denote the correlation length in microscopic lattice units. A renormalization-group transformation of length factor $b$ givesIndeed, a tail with exponential envelope $\exp (-r/\xi )$ becomes $\exp [-r^{\prime }/(\xi /b)]$ under $r^{\prime }=r/b$. Choose the stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ so that the transformed thermal coordinate is $+1$ or $-1$. If the stopped correlation length tends to a finite nonzero value $C_{\pm }$ as the irrelevant coupling approaches $g_{\ast }$, thenThe limit requires $|t_0|\ll 1$ and enough room between $a$, $\xi$, and the sample size. Singular dependence on a dangerously irrelevant coupling or a marginal operator needs separate analysis. On an ordered branch, use the connected correlation function in a pure thermodynamic phase.

For a completely specified illustrative scaling model choose $y_t=3/2$ and $X=C_{+}{t}^{-2/3}$ for $t>0$. Then $t=10^{-3}$ gives $X=100C_{+}$, whereas $t=10^{-6}$ gives $X=10^4{C}_{+}$. With $b=4$, $t^{\prime }=8t$ and $X(t^{\prime })=X(t)/4$. Models with $C_{+}=1$ and $C_{+}=2$ have different lengths but identical curves after plotting $X/C_{+}$ against $t$. This collapse checks the chosen scaling law; agreement of real data would test that law, rather than follow from this example. Neither $ϵ=1/5$ nor $y_t=3/2$ is being fitted to a three-dimensional Ising model.

A volume over which order-parameter fluctuations are correlated. For one isotropic correlation length $\xi$, its scale is ${\xi }^D$; with anisotropic lengths it is the product of the appropriate directional lengths. A Ginzburg criterion compares fluctuations averaged over this volume with the squared mean order parameter.

For the dimensionless scalar Landau-Ginzburg theory with stiffness $c$, quadratic coefficient $r<0$ and quartic term $u{\varphi }^4/4$, the mean-field approximation gives ${\varphi }_0^2=|r|/u$ and correlation length $\xi \asymp (c/|r|{)}^{1/2}$. Restricting the Gaussian Fourier transform covariance $(c{q}^2+2|r|{)}^{-1}$ to $|q|\lesssim {\xi }^{-1}$ estimates the variance of the field averaged over a correlation volume as $c^{-d/2}|r{|}^{(d-2)/2}$, up to a dimension-dependent constant. Dividing by ${\varphi }_0^2$ proves the displayed Ginzburg criterion ratio. It diverges below four dimensions, vanishes above four and is marginal at four. Short-distance fluctuations renormalize microscopic coefficients and should not be confused with this long-distance critical test.

Confining a near-critical fluctuating order parameter changes its free energy. Its range is governed by the growing correlation length, and its sign depends on the boundary preferences. This is a thermal collective effect, distinct from the zero-temperature ideal-conductor electromagnetic formula.

A critical exponent specifies a leading power-law singularity near a continuous phase transition. For dimensionless distance $t$ from the critical temperature, examples are $\xi \sim |t{|}^{-\nu }$ for the correlation length and $\chi \sim |t{|}^{-\gamma }$ for the magnetic susceptibility. The exponents describe how observables change, while nonuniversal amplitudes set their sizes.

Experimentally, estimating an exponent requires a range where critical scaling dominates over microscopic effects yet remains outside finite-size or instrumental rounding. A logarithmic slope over a short interval can be an effective exponent rather than the limiting one. Systems in one universality class share the limiting exponents under the stated assumptions. Scaling relations for critical exponents connect some exponents, so they are not all independent parameters.

Critical opalescence is enhanced scattering of light by large fluctuations near a thermodynamic critical point. The carbon-dioxide viewing-cell phase-transition experiment gives an observable finite-sample example; cloudiness alone does not establish a divergent correlation length.

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.

A reversed bond costs $2J$, so its Boltzmann factor $w=e^{-2J/(k_{B}T)}$ is a natural thermal coordinate near the zero-temperature chain fixed point. Exact length-two decimation gives $w^{\prime }\simeq 2w$, and the One-dimensional Ising correlation function gives $\xi /a=-1/\log \tanh K\sim 1/(2w)$. This is a length exponent one with respect to $w$. The correlation length instead diverges exponentially in inverse physical temperature; claiming a conventional finite power $T^{-1}$ would confuse scaling coordinates. Choosing $e^{-K}$ gives multiplier $\sqrt{2}$ and power two in that alternative coordinate, the same physical length.

At $D=2$ and $n>2$, the weak-coupling renormalization-group flow obeys $dg/d\ell =(n-2)g^2/(2\pi )$. Integrating gives $1/g(\ell )=1/g_0-(n-2)\ell /(2\pi )$. The flow reaches order-one coupling at a length whose logarithm is $2\pi /[(n-2)g_0]+O(1)$ at this order. Restoring the microscopic scale ${\Lambda }^{-1}$ gives the displayed leading exponential correlation length. Higher loops affect its prefactor, and this conclusion does not apply to the compact $n=2$ model with its Berezinskii–Kosterlitz–Thouless transition.

Finite-size rounding replaces a limiting discontinuity or divergence by a finite crossover or peak. For a spatial system near a continuous phase transition, the bulk description ceases to apply unchanged when the correlation length becomes comparable with the sample size. How the crossover width and location depend on size also depends on the model, geometry and boundary conditions; there is no universal $1/N$ rule for every phase transition.

An exact finite example is the Curie–Weiss model with $s_i=\pm 1$, $J>0$, and a conjugate field $h$ in energy units. Set $K=J/(k_{B}T)$ and $u=h/(k_{B}T)$. Its statistical Hamiltonian and finite partition function areThe binomial coefficient counts the microstates with $k$ positive Ising spins. The diagonal terms in the square add a configuration-independent constant to the energy; retaining them fixes the normalization used here. Divide each summand by $Z_N$ to obtain the probability mass function of $m_k=(2k-N)/N$. In particular,The variance expression makes the finite response nonnegative, while spin inversion symmetry gives $⟨m⟩=0$ exactly at $u=0$ for every plotted $N$.

Every curve is computed from the finite partition function, without selecting one minimum of the Bragg-Williams free energy of the Ising model. The field panel holds $K=1.5$ fixed and shows increasingly rapid switching between opposite spin magnetizations. The temperature panel uses $u=0$: its nonzero $⟨|m|⟩$ measures the typical magnitude, while the mean spin magnetization stays zero. The reference $k_{B}T/J=1$ marks the limiting critical temperature, not a singular point of any finite curve.

A short limiting argument explains the field panel. Combining the binomial coefficient with its leading entropy gives a Helmholtz free energy per spin whose stationarity equation is $m=\tanh (Km+u)$. For $K>1$ and $u=0$ the two ordered minima have $m=\pm m_{\ast }$, where $m_{\ast }=\tanh (K{m}_{\ast })$. For large $N$ well below the critical temperature, their relative Boltzmann weight is approximately $e^{2Nu{m}_{\ast }}$, giving $⟨m⟩\approx m_{\ast }\tanh (Nu{m}_{\ast })$. Thus this two-peak approximation has a field width of order $1/(N{m}_{\ast })$ in $u$. It is not valid at the critical temperature, where the peaks merge; the figure uses the exact finite sum throughout.

For $J>0$ and $S_i^z=\pm 1/2$, a parallel bond has energy $-J/2$ while an antiparallel bond has $+J/2$. A wall therefore costs J, rather than the $2J$ of a differently normalized $-J\sum s_i{s}_{i+1}$ model. At zero field its density is $[1-\tanh (\beta J/2)]/2$, positive at every nonzero temperature. Walls occur in pairs on a periodic ring, but that global parity restriction does not remove their nonzero density in the thermodynamic limit. The low-temperature correlation length is approximately $a{e}^{\beta J}/2$.

The Kibble mechanism explains how a spontaneous symmetry breaking transition can form topological defects. Regions that order without coordinating their choices can settle at different points of a vacuum manifold. Smooth relaxation removes many mismatches, but some cannot be removed while the field remains in that vacuum manifold everywhere.^[ref]

The possible defects depend on the homotopy groups of the vacuum manifold: disconnected components permit domain walls, noncontractible loops permit cosmic strings, and nontrivial surrounding two-spheres permit magnetic monopoles in three spatial dimensions. This is the homotopy classification of cosmological defects, rather than a prediction that every permitted defect must occur.^[ref]

For a direct winding example, an order parameter represented by a complex scalar field with fixed nonzero magnitude has a circular vacuum manifold. If its order-parameter phase changes by $2\pi$ around a closed spatial loop, it cannot extend continuously across a spanning disk while keeping its magnitude nonzero. A zero of the order parameter supplies the core; in three dimensions the corresponding defect can be a string.

A formation estimate needs a domain correlation length $\xi$ and a model for the chance of trapping a defect. For point defects, a dimensional estimate is $n_{\mathrm{defect}}\sim p_{\mathrm{defect}}/{\xi }^3$, with dimensionless formation efficiency $p_{\mathrm{defect}}$. Causality restricts how ordering can spread; taking a horizon-sized $\xi$ supplies a scale estimate, not by itself a guaranteed number of defects in every horizon volume. Subsequent annihilation or expansion can further change the abundance. Kibble's original discussion separately estimates the initial domain scale from fluctuations and follows later evolution.^[ref]

For a stable homogeneous scalar quartic saddle with gradient coefficient one, the Ornstein--Zernike correlation function has denominator $q^2+r_0+u_0{m}_0^2/2$. At zero source and $u_0>0$, the correlation length is $r_0^{-1/2}$ in the disordered phase and $(-2r_0{)}^{-1/2}$ on either selected ordered branch. Linear thermal tuning of $r_0$ gives correlation-length critical exponent $\nu =1/2$ with different amplitudes on the two sides.

For constant electron density, common path length $L$, and a solenoidal isotropic spectral tensor, the rotation-measure correlation contains the window $|{\int }_0^L{e}^{i{k}_{z}z}dz{|}^2$. As $L$ becomes large compared with the correlation length, this window acts as $2\pi L\delta (k_z)$ within the integral. The transverse projector then has $zz$ component one, yielding the displayed two-dimensional Fourier transform. For finite $L$, the windowed integral retains longitudinal modes and endpoint corrections. The method of using such rotation measure correlations to infer magnetic spectra is developed by Enßlin and Vogt.

For the positive quadratic kernel $\tilde{\Delta }(p)={\kappa }^{-1}{p}^2+m^2$, $\kappa ,m>0$, the connected correlation function is its inverse. Fourier inversion gives $G(r)=\kappa (\sqrt{\kappa }m/r{)}^{D/2-1}{K}_{D/2-1}(\sqrt{\kappa }mr)/(2\pi {)}^{D/2}$. The Modified Bessel function of the second kind has an exponentially decaying large-argument tail, so the correlation length is $1/(\sqrt{\kappa }m)$ at fixed positive gradient coefficient. In three dimensions this becomes $\kappa e^{-\sqrt{\kappa }mr}/(4\pi r)$. The statement concerns the continuum long-distance theory; regulator-dependent contact terms do not define the physical correlation length.