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Use inverse temperature ${\beta }_T=(k_{B}T{)}^{-1}$, reserving $\beta$ without a subscript for the critical exponent. Write $c=qJ$; each undirected bond occurs once, so the number of bonds is $Nq/2$. In the mean-field theory of the Ising model, write each Ising spin as ${\sigma }_i=m+\delta {\sigma }_i$ and neglect the product of fluctuations:The resulting independent-spin statistical Hamiltonian isThe constant corrects the double counting of interaction energy. The Ising spin sums now factorize and can all be evaluated:This is the approximate partition function at an assumed mean field; equilibrium fixes $m$ self-consistently. The corresponding Ising auxiliary mean-field free energy isIts equilibrium value gives the mean-field Helmholtz free energy in the imposed field. Away from a stationary point its parameter $m$ is an assumed field variable, not necessarily the actual mean Ising spin of that independent-spin distribution.

For the full small-$m$ expansion at fixed $h$, set $t_h=\tanh ({\beta }_{T}h)$ and $s_h={\mathrm{sech}}^2({\beta }_{T}h)$. Differentiating $\log \cosh z$ gives successive derivatives $t_h$, $s_h$, $-2t_h{s}_h$, and $2s_h(2-3s_h)$ at $z={\beta }_{T}h$. ThereforeAt zero field, spin inversion symmetry eliminates odd powers and this simplifies toThe quadratic coefficient changes sign and the quartic coefficient is positive atThus the zero-field mean-field prediction is a continuous, continuous phase transition: the stable zero spin magnetization develops two symmetry-related nonzero minima continuously below $T_c$.

Both requested routes give the same mean-field self-consistency equation. First, the one-spin expectation in the effective field isSecond, differentiating the Ising auxiliary mean-field free energy giveswhose stationary condition is precisely that mean-field self-consistency equation. Choose its stable, lowest-free-energy branch rather than every algebraic solution.

An equally useful mean-field approximation parametrizes the trial distribution by its actual mean Ising spin. Its probabilities are $(1\pm m)/2$, giving the Bragg-Williams free energy of the Ising modelThis energy-minus-entropy function has expansionIts stationary equation is $h=k_{B}T\mathrm{arctanh}m-cm$, again equivalent to the mean-field self-consistency equation. The two functions differ away from equilibrium, but agree on stationary branches: for $z={\beta }_T(h+cm)$ and $m=\tanh z$, the entropy bracket equals $mz-\log (2\cosh z)$. Their small-$m$ coefficients therefore need not agree at arbitrary temperature; at $T_c$ they have the same leading critical quartic coefficient $c/12$. This distinction prevents confusing the auxiliary-field expansion with the physical-magnetization variational expansion.

For the order-parameter critical exponent, expand the equation of state at $h=0$:On a nonzero stable branch,Hence $\beta =1/2$. The spontaneous magnetization is understood by selecting a branch with an infinitesimal field after the thermodynamic limit; a finite symmetric sample has zero exact zero-field mean Ising spin.

For the magnetic susceptibility per site, define $\chi =\partial m/\partial h$ with $h$ in energy units. Implicit differentiation givesAbove $T_c$, $m=0$ and $\chi =1/[k_B(T-T_c)]$. Below $T_c$, evaluated on a selected ordered branch, the expansion of $m^2$ gives $\chi \sim 1/[2k_B(T_c-T)]$. Thus the magnetic-susceptibility critical exponent is $\gamma =1$ on both sides, with different amplitudes. At $T=T_c$, the equation of state becomesso the critical-isotherm exponent is $\delta =3$. These are mean-field critical exponents, not a claim that neglecting fluctuations gives the exact Ising transition in every dimension.

Write $t=T-T_c$, take ${\alpha }_2=at$ with $a>0$, and include the magnetic term $-Bm$. At $B=0$, minimizing the Landau free energy gives $m=0$ for $t>0$ andfor $t<0$, so the order-parameter critical exponent is $\beta =1/2$. Substitution gives a singular free-energy density proportional to $-t^2$ below $T_c$, whose second temperature derivative has a finite jump, so the heat-capacity exponent is $\alpha =0$. Above $T_c$, the equation ${\alpha }_{2}m+{\alpha }_4{m}^3=B$ gives the magnetic susceptibility $\chi =(\partial m/\partial B{)}_{B=0}=1/{\alpha }_2\sim t^{-1}$, hence $\gamma =1$. At $t=0$, $B={\alpha }_4{m}^3$, so $m\sim B^{1/3}$ and $\delta =3$.

A coarse-grained scalar order parameter $\varphi (\mathbf{x})$ describes a short-range Ising-like transition with sign symmetry. Write the dimensionless Landau-Ginzburg functional asHere $r=a_{t}t$, with the relevant critical-mass shift understood when fluctuations are included. Higher local powers and gradients are negligible only within a justified long-distance expansion. The partition function is $Z=\int \mathcal{D}\varphi e^{-\mathcal{H}}$ with a microscopic cutoff. The local functional is not already the fluctuation-integrated thermodynamic free energy.

Mean-field theory replaces this integral by its stable uniform saddle, $r{\varphi }_0+u{\varphi }_0^3=h$. At $h=0$,The branch curvature is $r_L=r+3u{\varphi }_0^2$: it is $r$ above and $2|r|$ below. HenceTogether with the finite heat-capacity jump from $f_{\min }$, this yieldsThe equal-$|t|$ susceptibility amplitude ratio is two. Uniform saddles also miss domain walls and spatial texture. For an $n$-component order parameter, symmetry allows a vector generalization; ordered transverse directions additionally carry Goldstone fluctuations.

To include small fluctuations set $\varphi ={\varphi }_0+\delta \varphi$. The Gaussian action is $\frac{1}{2}\int [c(\nabla \delta \varphi {)}^2+r_L(\delta \varphi {)}^2]$, givingThe displayed $G(q)$ is a spectral covariance per Fourier delta, avoiding an extra volume factor. As $r_L\to 0$, a growing correlation length lets increasingly large correlated regions fluctuate. The singular Gaussian heat correction contains $\int d^{d}q/(c{q}^2+r_L{)}^2$, which is infrared singular for $d<4$ and logarithmic at four. This signals failure of a simple saddle expansion; the Gaussian heat exponent obtained by extrapolation is not the exact interacting critical exponent.

The Ginzburg criterion compares the fluctuation of a correlation-volume average with ${\varphi }_0^2$, rather than using all ultraviolet fluctuations at a point. Keeping $q\lesssim {\xi }^{-1}$ below the transition gives, up to finite convention-dependent constants,Dividing by ${\varphi }_0^2=|r|/u$ yieldsFor $d<4$, writing $r=a_{t}t$ gives a nonuniversal Ginzburg reduced-temperature windowInside $|t|\lesssim \mathrm{Gi}$, interacting fluctuations must be treated. For example, in three dimensions $\mathrm{Gi}\sim u^2/(c^3{a}_t)$ in this normalization. Above four dimensions the long-wavelength ratio vanishes at criticality; at four it is marginal and interaction-generated logarithms require further analysis.

The spatial gradient term supplies correlations absent from uniform Landau theory. The Ginzburg test bounds where its mean-field saddle remains self-consistent; it does not compute nonclassical exponents or a universal material-independent window.^[ref]

Take the ordered phase selected by a small positive real field, so $\varphi$ is real and positive before sending $B\to 0^{+}$. Stationarity gives $2a\varphi +2{\varphi }^3=B$. Differentiation yields $\chi =(2a+6{\varphi }^2{)}^{-1}$. ThereforeBoth have mean-field critical exponent one, with a factor-of-two amplitude difference. The below-transition result is the longitudinal response of a selected pure phase. The transverse phase direction is soft, so a transverse zero-field response is singular; an unselected phase average is not the same susceptibility. At $a=0$, $\varphi =(B/2{)}^{1/3}$ and the linear susceptibility diverges.

Mean-field theory replaces fluctuating interactions with the average field produced by the other degrees of freedom. For an interacting spin system, write ${\mathbf{S}}_i=\mathbf{S}+\delta {\mathbf{S}}_i$ and neglect connected products $\delta {\mathbf{S}}_i\cdot \delta {\mathbf{S}}_j$. This reduces the many-body partition function to independent single-site factors coupled through a self-consistency equation for $\mathbf{S}$.

The approximation is most reliable when many effectively independent neighbors contribute, for sufficiently long-range interactions, or above the upper critical dimension away from the fluctuation-dominated critical region. It usually becomes least reliable near a continuous transition in low dimensions, where correlations extend over long distances. It can estimate ordering scales and reveal candidate broken symmetries while giving incorrect critical exponents or even a spurious transition. The two-dimensional isotropic model in the later parts is an especially important example of this limitation.

Write $a=a_0(T-T_c)$, $a_0>0$, at fixed $b\ge 0$, and define the critical exponent by $|m|\sim (T_c-T{)}^{\beta }$ on the ordered side. For fixed $b>0$ and small negative $a$, the stationary equation gives $m^2=-a/b+O(a^2)$. ConsequentlyAt the tricritical point approached along the tuned path $b=0$, the stationary equation instead gives $m^4=-a/c$, soFor fixed $b<0$, the transition occurs before $a$ reaches zero and the magnetization jumps; it is not a second-order transition and has no vanishing-order-parameter exponent of this form. A path that does not keep the quartic coefficient tuned to zero generally crosses over to the ordinary result.

Use $v$ for molar volume to distinguish it from the fluctuating parcel volume below. The equation is $p=RT/(v-b)-a/v^2$. The parameter $b>0$ models excluded volume, while $a>0$ models the reduction in pressure from attractive interactions.

Below the critical temperature, a raw isotherm has a loop. The positive-slope segment is mechanically unstable and surrounding metastable portions do not give the stable equilibrium mixture. Replace the loop by a constant coexistence pressure $p_s$ satisfying the Maxwell equal-area condition,The endpoints are liquid and vapour molar volumes; intermediate mean volumes correspond to phase mixtures with changing proportions. Above the critical temperature, liquid-like and gas-like states connect continuously without a coexistence plateau.

In the pressure-temperature diagram, the liquid-vapour coexistence line ends at the critical point. Below it, the low-pressure side is vapour and the high-pressure side liquid. “Gas” conventionally denotes the region above the critical temperature, where compression alone cannot cross a liquid-vapour transition. The model has no solid phase.

The critical isotherm has an inflection: $p_v=p_{vv}=0$. Thuswhich giveFor fluctuations about a stable equilibrium value $x_0$, the system plus reservoir has probability proportional to $e^{-A(x)/(k_{B}T)}$. Expanding availability,gives a Gaussian distribution and, to this order, $⟨x⟩=x_0$ withThis requires positive curvature and sufficiently small fluctuations; it is not exact when higher-order terms matter.

The whole cylinder's volume is fixed. For the volume fluctuations of a small fluid parcel, interpret the small quantity as a material parcel containing $s$ moles, whose volume can fluctuate against the much larger remainder. Its mean volume at the critical density is $\bar{V}=s{v}_c=3sb$. At fixed temperature its availability is $A(V)=F_s(V,T)+p_{\mathrm{bath}}V$, soOn the critical isochore, approached from $T>T_c$,ConsequentlyEquivalently, $⟨\delta V^2⟩=k_{B}T{\kappa }_T\bar{V}$, with ${\kappa }_T=4b/[3R(T-T_c)]$. The mean-field variance diverges as $(T-T_c{)}^{-1}$, so its critical exponent is $1$; the root-mean-square fluctuation has exponent $1/2$. A fixed spatial observation box instead has number fluctuations, not fluctuating geometric volume. Finite size and non-Gaussian terms cut off the literal divergence; the approximation fails exactly at the critical point.

Landau theory expands the free energy in a small order parameter and keeps terms allowed by the system's symmetries. For a scalar order parameter $q$ with sign-reversal symmetry in Landau theory $q\mapsto -q$, a useful free-energy density isAt zero conjugate field $h$, the quartic Landau equilibrium branches obtained by minimizing areThe single minimum becomes two symmetry-related minima: the order parameter turns on continuously and a chosen equilibrium phase spontaneously breaks the symmetry. At the critical temperature the quadratic restoring term vanishes. The mean-field exponent is $q\propto (T_c-T{)}^{1/2}$; the susceptibility above $T_c$ is $\partial q/\partial h=1/[a(T-T_c)]$.

More general allowed terms describe other transitions. A negative quartic term, stabilized by a positive sixth-order term, can give a first-order transition with a discontinuous order parameter and metastable minima. Without the sign-reversal symmetry, a cubic term may also be allowed. The expansion is phenomenological: its coefficients depend on microscopic physics. Minimizing a uniform polynomial neglects important spatial fluctuations, particularly near a critical point, so its critical exponents need not be exact.^[ref]

These critical exponents describe the leading singular powers of percolation probability, percolation susceptibility, correlation length and critical cluster distributions near the percolation critical probability. Their existence and exact values require theorems or scaling hypotheses for the particular model; they are not consequences of the definition of the threshold. For $\theta (p)=P_p(0\leftrightarrow \infty )$ and $\chi (p)={\mathbb{E}}_p|C(0)|$, the definitions are $\theta (p)=(p-p_c{)}^{\beta +o(1)}$ above $p_c$ and $\chi (p)=(p_c-p{)}^{-\gamma +o(1)}$ below $p_c$. The correlation length has $\xi (p)=|p-p_c{|}^{-\nu +o(1)}$. At criticality, the root-cluster tail is $P(|C(0)|\ge s)=s^{-1/\delta +o(1)}$; two-point connectivity has leading power $|x{|}^{-(d-2+\eta )}$; and the expected number per site of size-$s$ clusters has leading power $s^{-\tau }$. The last distribution differs from the size-weighted distribution seen at a specified site. Relations such as $\gamma =(2-\eta )\nu$ and $2\beta +\gamma =d\nu$ have specific scaling regimes, and the latter hyperscaling relation fails in ordinary mean-field regimes above the upper critical dimension of percolation.

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral:The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2{\pi }^2$ and $8{\pi }^2/3$. Dividing by $(2\pi {)}^D$ reduces the integrals to constants times ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary upper critical dimension four visible without treating a self-consistent one-loop equation as an exact interacting critical exponent prediction.

The reduced temperature is a dimensionless distance from a critical temperature. Multiplying it by a nonzero constant does not change any critical exponent.

A renormalization group compares models at different observation lengths. A Wilsonian coarse-grained statistical Hamiltonian sums over short-distance fluctuations; a renormalization-group transformation rescales coordinates and fields for comparison. Kenneth Geddes Wilson's 1982 lecture treats fluctuations across successive length scales to explain critical phenomena.^[ref]

The experimental motivation is Andrews's carbon dioxide experiment. Dried carbon dioxide occupied a capillary tube above mercury; a screw compressed the transmitting water. A second air-filled tube gauged pressure. Flowing water controlled temperature, and a cathetometer read column positions for volume. The liquid meniscus vanished near the liquid-gas critical point. Residual air and imperfect air-gauge corrections limited accuracy.^[ref] These observations motivate critical-point theory without determining a correlation-length critical exponent. The renormalization group explains how critical measurements can become insensitive to many molecular details.

In 1972, Kenneth Geddes Wilson and Michael Ellis Fisher used an epsilon expansion about $d=4$ to compute nonclassical critical exponents. Their paper distinguishes a Gaussian fixed point from an interacting renormalization-group fixed point, tunes the thermal parameter, and extracts powers from a linearized recursion. Its Ising-like result is $\gamma =1+ϵ/6+O({ϵ}^2)$ for $d=4-ϵ$.^[ref] The worked model below isolates this mechanism with chosen coefficients; it is not a precision calculation for the three-dimensional Ising model.

For $F=F_0+b(T-T_c)M^2+d^{\prime }{M}^6$ with $b,d^{\prime }>0$, the ordered minimum has $M^4=b(T_c-T)/(3d^{\prime })$. Its singular free energy is $-2b^{3/2}(T_c-T{)}^{3/2}/\sqrt{27d^{\prime }}$. Twice differentiating in temperature gives the displayed ordered-side heat capacity singularity, with mean-field critical exponent $\alpha =1/2$. This path keeps the quartic coefficient tuned to zero; a generic path with a positive quartic term instead crosses over to the ordinary quartic result.

Systems belong to one universality class when their long-distance renormalization-group flows approach the same fixed point.

Near a renormalization-group fixed point, expand a difference between two nearby models in eigenvectors of the stability matrix. An irrelevant operator direction has $y_i<0$, so its dimensionless coefficient shrinks under increasing length factor $b$. A relevant operator direction grows and must be tuned to study the same critical behavior. This is a mechanism for losing microscopic detail, not an assertion that every parameter disappears. Critical temperature, individual amplitudes and regular contributions to free energy can remain material dependent.

For the solvable example, write $A=ϵ/g_0-1$. Its exact solution gives $g-ϵ=-ϵA{b}^{-ϵ}+O(b^{-2ϵ})$. At the thermal stopping scale $b_{\ast }=|t_0{|}^{-1/y_t}$ this is of order $|t_0{|}^{ϵ/y_t}$. If the stopped correlation length is smooth in $g$, the leading correction to its power law has that order. Thus two microscopic couplings can yield the same limiting critical exponent, yet approach it at different rates.

This mechanism makes universality useful: one tractable representative can predict critical exponents and normalized scaling forms for many systems in its universality class. A liquid-gas critical point and a scalar magnetic transition can share a long-distance description even though their microscopic constituents differ. The scope must be checked: spatial dimension, order parameter symmetry, interaction range, and extra soft variables can change the renormalization-group fixed point.

In his lecture, Kenneth Geddes Wilson connects limiting renormalization-group fixed points with common critical behavior and states the limits of his elementary truncation.^[ref]