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For candidate (i), $\mathbf{p}=p_0\hat{\mathbf{p}}\cos q_{0}x$. Since $G(q_0)=\hat{a}:=a-a_c$, $⟨{\cos }^2{q}_{0}x⟩=1/2$, and $⟨{\cos }^4{q}_{0}x⟩=3/8$, its mean-field free-energy density isFor $\hat{a}<0$, stationarity givesand substitution yieldsFor $\hat{a}\ge 0$, the minimum is $p_0=0$. The amplitude therefore vanishes continuously as $p_0\propto (a_c-a{)}^{1/2}$ on approaching $a_c$ from below, which is a continuous mean-field transition.

Write ${\mathbf{\sigma }}_i=\mathbf{m}+\delta {\mathbf{\sigma }}_i$ and neglect products of two fluctuations. ThenEach site has $q$ neighbours and each bond is counted once, so the mean-field approximation givesChoose the $x$ axis along $\mathbf{m}$. For the four-state clock model, the single-site partition function isConsequentlyand therefore

Write the spin-one-half Heisenberg antiferromagnet ason a bipartite lattice of coordination number $z$. A one-spin density operator is $\rho =(I+\mathbf{r}\cdot \mathbf{\sigma })/2$, where $\mathbf{r}$ is its Bloch vector and $|\mathbf{r}|\le 1$. In a two-sublattice product state,The minimum is $-1/4$, attained by pure antiparallel Bloch vectors, so the minimizing product state is a Néel state. In the $z\to \infty$ limit this mean-field approximation becomes exact and the ground-state energy per bond is therefore

The phrase “energy density” requires a coupling convention. For the unscaled Hamiltonian above, every site belongs to $z/2$ bonds andwhich diverges as $z\to \infty$. With the standard Kac normalizationthe finite energy density isIf the convention divides by the spatial dimension $d=z/2$ instead, the answer is $-J/4$ per site. A Hamiltonian written with ${\mathbf{\sigma }}_i\cdot {\mathbf{\sigma }}_j$ rather than ${\mathbf{S}}_i\cdot {\mathbf{S}}_j$ multiplies all these energies by four.

The mean-field approximation replacesby its first three terms. Since every site has $q$ neighbours,The sites then decouple, and their three possible weights sum toHence $c_1=1$, $c_2=2$, and

At fixed uniform $\varphi$, the nonconserved variable $p$ can relax freely, so mean-field approximation minimizeswith respect to $p$. The stationary solutions obeyThus $p=0$ when $A+c\varphi \ge 0$, while $p^2=-(A+c\varphi )/B$ when $A+c\varphi <0$. Substitution givesWriting this in the requested form yieldsbecause $A+c\varphi =c(\varphi -{\varphi }_o)$. The value ${\varphi }_o$ is the composition at which the coefficient of $p^2$ changes sign, so it is the mean-field threshold for polar ordering.

This is the one-mole, or equivalently molar volume, form of the Van der Waals equation, an approximate equation of state for a real gas:Here $p$ is the gas pressure, $T$ its absolute temperature, $R$ the molar gas constant, and $V$ the volume occupied by one mole. If $V$ denotes molar volume explicitly, its units are ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$.

The positive parameter $b$ represents excluded volume arising from the finite size and short-range repulsion of the molecules. The volume available for translational motion is reduced from $V$ to $V-b$. This is an effective excluded volume, not simply the sum of the molecules' geometric volumes. Its molar units are ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$.

The positive parameter $a$ measures intermolecular attraction in a mean-field approximation. Attractions reduce the force exerted on the walls, lowering the measured pressure by $a/V^2$ relative to the repulsion-only expression. Thus $p+a/V^2$ is the pressure corrected for this attraction. The squared inverse-volume dependence reflects a contribution proportional to the square of the molecular number density. The molar units of $a$ are $\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$.

For an arbitrary amount $n$ measured in moles, with total container volume $V_{\mathrm{tot}}$, the corresponding formula isAt low density, when $b\ll V$ and $a/V^2\ll RT/V$, it reduces to the ideal gas law. The model captures qualitative attraction, repulsion and liquid-gas behavior, but does not supply an exact equation of state for every material.

The fractional quantum Hall effect occurs when interactions produce an incompressible correlated state within a partially occupied Landau level. The orbital kinetic energy is already quantized; the important degeneracy is among many possible arrangements of electrons within the level. Repulsion can lift that degeneracy and create a many-body excitation gap. The interaction scale is of order $e^2/(4\pi ϵ{\ell }_B)$, where magnetic length ${\ell }_B=(\hslash /(e|B|){)}^{1/2}$; it is not a gap predicted by independent-electron fractional occupation alone.

A composite fermion description incorporates strong correlations by attaching $2p$ statistical flux quanta to each electron, with positive integer $p$. The attachment is a representation of correlated wavefunction phases and zeros, not a literal solenoid attached to a particle. Attaching an even number preserves fermionic exchange character. In a mean-field approximation for positive applied field,The particles consequently experience a reduced effective field. Define the signed effective filling ${\nu }^{\ast }=n_s{\varphi }_0/B^{\ast }$ and electronic filling $\nu =n_s{\varphi }_0/B$. Their relation isWhen $q$ effective levels are filled with the same field orientation, ${\nu }^{\ast }=q$, giving $\nu =q/(2pq+1)$. Reversed effective-field orientation has ${\nu }^{\ast }=-q$, giving $\nu =q/(2pq-1)$. Thus the Jain sequence isFor $p=1$, examples are $1/3$, $2/5$, $3/7$ on one branch and $2/3$, $3/5$ on the reversed branch. Fractional electronic plateaus can be understood as integer effective Hall states of these composite objects. The original proposal explicitly makes this connection.^[ref]

At a gapped fractional state, longitudinal dissipation becomes very small and the Hall magnitude approachesDisorder can localize quasiparticles and allow the plateau to persist over a filling interval, provided it does not overwhelm the interaction gap. The Laughlin wavefunction at $\nu =1/(2p+1)$ is a particularly simple correlated example; its charged excitations carry fractions of the electron charge. More generally the basic quasiparticle charge in the simple Jain sequence has magnitude $e/(2pq\pm 1)$. Such excitations are distinct from noninteracting bare electrons occupying a fraction of a level.

At $\nu =1/(2p)$, the mean effective field $B^{\ast }$ vanishes. This permits a compressible composite-fermion Fermi-sea description, rather than an ordinary integer effective-level gap. The external field remains large: cancellation is in the effective description. Not every fractional state is explained by the simplest integer composite fermion sequence, and the effective masses and gaps reflect interactions rather than simply the GaAs band mass. The utility of the construction is that it organizes many observed fractions and their nearby excitations into one understandable pattern.

For the uniform mean-field approximation field $\mathbf{M}=M{\hat{\mathbf{e}}}_z$, the inverse fermion propagator is diagonal in momentum, frequency and spin: $-i{\omega }_l+{\xi }_k-\sigma M$. The Grassmann Gaussian integral therefore givesHere $d\mu (M)$ denotes the residual uniform-field measure; integration over its orientation only affects a nonextensive prefactor. HenceThe exchange field separates the two spin bands by $2M$. Integrating out the fermions is exact for a specified auxiliary field; restricting that field to be constant is the mean-field approximation.

The mean-field approximation replaces the fluctuating environment of each microscopic variable by a uniform average field. For spins, write $S_i=S+\delta S_i$ and drop products $\delta S_i\delta S_j$ between different sites. The resulting single-site problem can be solved independently, and its average spin must reproduce the assumed order parameter $S$: this is the self-consistency equation.

It is reliable when correlated thermal fluctuations about the mean are small, for example with many interacting neighbours, long-range interactions or high spatial dimension. Near a thermodynamic critical point the quantitative test is the Ginzburg criterion. For short-range scalar ordering the upper critical dimension is four, with logarithmic qualifications at four; a two-dimensional nearest-neighbour lattice is below this and its asymptotic critical region of a phase transition is not described accurately by simple mean-field theory.^[ref]

The square lattice has coordination number of a lattice $z=4$. The mean-field decoupling isAccounting for the factor $1/2$ in the directed-neighbour sum gives the Hamiltonian in the mean-field approximation:The constant term is necessary to correct the double counting of the interaction. Let $h=4JS+B$. The one-site partition function isso $Z_{\mathrm{MF}}=e^{-2\beta JN{S}^2}{Z}_1^N$. The Helmholtz free energy is thereforeThe equilibrium mean-field approximation is obtained by minimizing this expression over the allowed $S$. The $s=0$ state supplies the extra $1$ in $Z_1$; omitting it would instead describe the two-state Ising model. This is the Blume-Capel mean-field transition at zero single-ion anisotropy.

The self-consistency equation requires that the mean spin computed using the single-site Boltzmann distribution equals the spin $S$ used to form the effective field. With $h=4JS+B$,ThusEquivalently, differentiating part (b) gives $N^{-1}\partial F_{\mathrm{MF}}/\partial S=4J[S-⟨s{⟩}_h]$, so stationarity of the Helmholtz free energy in the mean-field approximation is exactly this condition. Among multiple stationary points one must select the stable minimum.

At zero field, write the right-hand side of the self-consistency equation as $G_T(S)=g(4\beta JS)$, with $g(x)=2\sinh x/[1+2\cosh x]$. The origin is always a solution, and the slope there isThe critical crossing occurs when this equals one, givingThis is a continuous phase transition in the mean-field approximation. For $x>0$,so the positive branch is a strictly concave function, while $g$ is odd and bounded by one. For $T\ge T_c$, the slope at zero is at most one and strict concavity puts $G_T(S)$ below $S$ for every $S>0$, leaving only the unmagnetized solution. For $T<T_c$, it initially lies above $S$ but has $G_T(1)<1$; hence there is one positive intersection and, by spin inversion symmetry, one negative intersection.

The curvature of the Helmholtz free energy in the mean-field approximation is $4J[1-G_T^{\prime }(S)]$ per site. It is negative at $S=0$ below $T_c$ and positive at the nonzero intersections, identifying the ordered minima and the unstable zero solution.

The critical temperature and branches here belong to the mean-field approximation. In the exact finite system, spin inversion symmetry forces zero average magnetization at zero field; selecting a broken-symmetry phase requires the thermodynamic limit with a vanishing field of specified sign.

Choose a coarse-grained magnetization $m(\mathbf{x})$ as the order parameter: it vanishes in the paramagnetism phase and develops a spontaneous nonzero value in the ferromagnetism phase. In zero applied magnetic field, spin reversal sends $m\mapsto -m$, so the local Landau free energy contains only even powers. A minimal spatially inhomogeneous free-energy functional isThe positive gradient coefficient penalizes rapid spatial variations. Near an ordinary transition take $a(T)=a_0(T-T_0)$, $a_0>0$, and positive $b$; retain the sextic term when $b$ can approach or cross zero. The source $h$ is conjugate to the magnetization and explicitly breaks spin reversal.

Determine equilibrium by minimizing this free-energy functional, imposing the appropriate boundary conditions. Its Euler-Lagrange equation isFor a uniform system at $h=0$, compare the values of all stable stationary solutions, rather than accepting every solution of this equation. The selected global minimum gives the equilibrium magnetization; derivatives of its free energy give entropy, heat capacity and magnetic susceptibility. At an ordinary transition the two ordered minima emerge continuously from zero. Spatial dependence also permits domain walls and an estimate of the correlation length.

This is a mean-field approximation: it selects a minimizing profile and neglects statistical fluctuations around it. The symmetry constrains the expansion, but its numerical coefficients are phenomenological. Close enough to a transition, thermal fluctuations near a critical point can invalidate the resulting mean-field critical exponents.^[ref]

Take $a=a_0(T-T_c)$ and hold $b,c$ fixed. The heat capacity comes from the minimized free energy, $C=-T{d}^2{F}_{\min }/d{T}^2$, with any smooth background added separately. For $b>0$ and $T<T_c$,Above $T_c$, $m=0$ and the polynomial contributes zero. There is a finite jump, rather than a divergence: the conventional ordinary mean-field critical exponents areThe upper branch has zero order-parameter contribution; a finite regular heat capacity has exponent zero.

Along the tuned tricritical path $b=0$, the ordered minimum givesUpper-side qualification: the vanishing disordered-side heat-capacity amplitude in Landau theory matters here: for $b=0$ and $a>0$, the minimizing solution is still exactly $m=0$, so this saddle-point polynomial gives $F_{\min }-F_{\mathrm{reg}}=0$ and $C_{\mathrm{sing},+}=0$. A zero function cannot determine a nonzero-amplitude power-law exponent. For example, adding the allowed smooth background $F_{\mathrm{reg}}=-C_{0}T\log (T/T_{\ast })$, with $C_0>0$, leaves the entire phase diagram unchanged but makes the upper-side heat capacity exactly $C_0$, not a $(T-T_c{)}^{-1/2}$ divergence.

One convention assigns the tricritical scaling exponent $\alpha =1/2$ to both sides, writing $C_{\mathrm{sing},\pm }=A_{\pm }|T-T_c{|}^{-1/2}$ with $A_{+}=0$ at this mean-field approximation saddle. If the source's literal $C\sim |T-T_c{|}^{-{\alpha }_{+}}$ is meant to have a nonzero leading coefficient, the given polynomial establishes no upper-side tricritical singular exponent; with a nonzero smooth background its leading upper-side exponent is zero. A nonzero upper-side singularity requires fluctuation information beyond this minimization. This distinction prevents claiming an upper-side divergence that the displayed Landau free energy does not produce.

The reduced BCS pairing Hamiltonian retains scattering of opposite-momentum, opposite-spin Cooper pairs:A mean-field approximation introduces the BCS anomalous average and pairing field ${\Delta }_k=-{\sum }_{k^{\prime }}{V}_{k,k^{\prime }}⟨b_{k^{\prime }}⟩$. The subtraction $-\sum V_{k,k^{\prime }}⟨b_k^{†}⟩⟨b_{k^{\prime }}⟩$ corrects the double counting of the mean-field interaction energy.

A self-consistency equation requires the average field assumed in a mean-field approximation to equal the expectation value calculated in that field.

For $\varphi =A\cos (q_{0}z)$, the gradient-quartic Brazovskii model has volume-averaged free-energy densityThe averages are $⟨{\cos }^2⟩=⟨{\sin }^2⟩=1/2$ and $⟨{\cos }^2{\sin }^2⟩=1/8$. For $B>0$, the stable amplitude is zero for $a\ge a_c$ and satisfies $A^2=(a_c-a)/(B{q}_0^2)$ below it. The mean-field approximation therefore predicts a continuous phase transition.

With positive $g$, the quadratic coefficients select the component with the smaller negative coefficient. Equal negative coefficients give a circle of minima in the mean-field approximation; the negative diagonal is a spin-flop transition, ending at a bicritical point at the origin.

For the auxiliary-field convention with cost $M^2/(2U)$ and coupling $-M\bar{\psi }{\sigma }_z\psi$, the homogeneous saddle point equation equates M to U times spin polarization per site. M has energy units. It is not itself a magnetic moment, and its coefficient changes if the contact interaction or density-of-states spin convention is changed. The exact three-component decoupling must be distinguished from selecting one constant spin direction as a mean-field approximation.