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A product trial distribution with actual mean Ising spin $m$ givesIts entropy is the sum of independent-spin entropies, with $0\log 0=0$. Differentiation gives $k_{B}T\mathrm{arctanh}m=qJm+h$, the self-consistency equation. The variational mean parameter is distinct from the assumed field parameter of the Ising auxiliary mean-field free energy until the equilibrium condition is imposed.

For $H=\int [a{\varphi }^2/2+\kappa |\nabla \varphi {|}^2/2+\gamma ({\nabla }^2\varphi {)}^2/2+B{\varphi }^2|\nabla \varphi {|}^2]$ and an even positive trial kernel $J(q)$, define $S_0={\sum }_q^{+}1/J(q)$ and $S_2={\sum }_q^{+}{q}^2/J(q)$ using a positive-wavevector sum for a real field. Isserlis theorem gives $⟨H_4{⟩}_0=4B{S}_0{S}_2/V$. The Feynman-Bogoliubov inequality therefore has stationary kernelsThus both the mass and gradient coefficient shift. In the thermodynamic limit, their self-consistency equations areA coarse-graining ultraviolet cutoff $\Lambda$ is necessary for the first integral in three dimensions: its large-$k$ radial integrand tends to a constant. Physical solutions require $J(q)>0$.

The approximate partition function obtained by decoupling each Ising bond has this free energy. The quadratic constant compensates double counting. Its stationary condition is the self-consistency equation; away from stationarity $m$ parametrizes the assumed effective field and need not equal that distribution's mean Ising spin. It agrees at stationary points with the Bragg-Williams free energy of the Ising model, although their off-equilibrium expansions differ.

Mean-field theory replaces fluctuating interactions with a self-consistent average field. For example, an Ising spin feels a field determined by the average neighboring spin magnetization; a self-consistency equation approximates the interacting Ising model. Neglecting correlated fluctuations can fail near a continuous phase transition. The mean-field approximation in Landau theory is a particular use of this general framework.

The mean-field approximation neglects products of Ising spin deviations, replacing each Ising spin's neighbours by the common spin magnetization. With coordination number of a lattice $q$, the effective field is $h+qJm$. The self-consistency equation equates the assumed mean Ising spin with its independent-spin expectation in that effective field. It predicts $k_B{T}_c=qJ$ and the usual quartic mean-field critical exponents, while neglecting spatially correlated fluctuations.

The mean-field approximation replaces each site's fluctuating surroundings by a self-consistent average. On a cubic lattice each spin has coordination number of a lattice $z=2D$, even though summing only over positive basis directions counts each bond once. Write ${\sigma }_n=M+\delta {\sigma }_n$ and drop products of deviations. Each Ising bond becomes ${\sigma }_n{\sigma }_m\simeq M{\sigma }_n+M{\sigma }_m-M^2$, so the approximate statistical Hamiltonian isThe constant corrects counting each bond twice. The resulting independent-spin partition function is $Z_{\mathrm{MF}}=e^{-{\beta }_{T}NzJ{M}^2/2}[2\cosh ({\beta }_{T}zJM){]}^N$, where ${\beta }_T=1/(k_{B}T)$. The one-spin probabilities giveEquating this average with the assumed $M$ givesHere $J>0$ is assumed; a ferromagnetic order parameter is not the appropriate choice for an antiferromagnetic coupling. Put $A=T_c/T$. For $A\le 1$, $\tanh (AM)<AM\le M$ at any $M>0$, so only $M=0$ is possible. For $A>1$, the function $\tanh (AM)-M$ has positive slope at zero, is strictly concave on $M>0$ and is negative at $M=1$. It therefore has exactly one positive zero and a spin-reversed negative zero. Expanding at the bifurcation givesThe transition is continuous, with order-parameter critical exponent $\beta =1/2$. Stability can also be checked from the Bragg-Williams free energy of the Ising model per Ising spin:Its derivative is $-zJM+k_{B}T\mathrm{arctanh}M$, giving the same self-consistency equation. Near zero,The quartic coefficient is positive at $T_c$, the stable minima approach zero continuously, and the minimized free energy has no first-derivative jump. Thus mean-field approximation predicts a second-order transition at $T_c=2DJ/k_B$. This is the mean-field Ising critical bifurcation.

For short-range interactions, large dimension gives many neighbours and suppresses the relative fluctuations of their average. More quantitatively, Gaussian field theory fluctuations averaged over a correlation volume have variance of order ${\xi }^{2-D}$, while the squared ordinary mean-field order parameter is of order $|t|\sim {\xi }^{-2}$. Their ratio is proportional to ${\xi }^{4-D}$. The Ginzburg criterion therefore makes neglect of critical fluctuations asymptotically consistent above four dimensions, inconsistent below four, and marginal at four, where logarithms occur. Large dimension is a route to validity; sufficiently long-range or infinite-range interactions can also produce mean-field behaviour. In low-dimensional short-range systems even the existence and location of a transition can differ from this approximation.

A phase diagram partitions a space of control parameters into regions with distinct equilibrium phases. Its boundaries mark nonanalyticities of the thermodynamic-limit free energy; a first-order phase transition has a jump in a first derivative, whereas a continuous phase transition has a continuously vanishing order parameter and singular fluctuations.

An explicit three-parameter example is the scalar Landau free energywith parameter axes $(r,u,h)$. These can be realized locally by the temperature, single-ion anisotropy and magnetic field of the Blume–Capel model. The three dimensions here refer to control-parameter space, not necessarily to three spatial dimensions. At $h=0$, minimizing this potential gives a continuous phase transition at $r=0$ for $u>0$. For $u<0$, coexistence occurs at $r=3u^2/(16v)$, whereThere are three global minima, $M=0$ and $M=\pm \sqrt{-3u/(4v)}$. Their order parameter jump decreases to zero as $u\uparrow 0$. Thus $(r,u,h)=(0,0,0)$ is a tricritical point. The two transition curves in the $h=0$ plane meet there.

In the three-dimensional phase diagram, crossing $h=0$ inside the ordered region switches between opposite signs of $M$, so that region is itself a first-order phase transition sheet. For $u<0$, two tricritical wings at opposite nonzero fields separate weakly and strongly magnetized phases. Their wing critical edges can be obtained rather than merely sketched: require $V^{\prime }=V^{\prime \prime }=V^{\prime \prime \prime }=0$. Since $V^{\prime \prime \prime }=6uM+20v{M}^3$, the nonzero solutions satisfyThe fourth derivative there is $-12u>0$, giving ordinary continuous critical endpoints. The wings meet the zero-field coexistence sheet along the tricritical three-phase line and terminate at these edges. Their exact tricritical wing coexistence factorization supplies the surfaces in the figure: for two nonnegative minima $a\le b$, set $s=a+b$, $p=ab$ andDirect expansion gives $V(M)-V(a)=(v/6)(M-a{)}^2(M-b{)}^2[(M+s{)}^2+p]\ge 0$, proving that these surfaces describe coexistence of global, not merely local, minima.

For the unheaded Ising model calculation, assume ferromagnetic exchange $J>0$ and let $M$ denote the mean Ising spin. On the hypercubic lattice the coordination number of a lattice is $z=2D$, even though the original positive-direction bond sum counts each bond only once. The mean-field approximation neglects products of deviations from $M$:There are $ND$ bonds, so the approximate energy is $H_{\mathrm{MF}}=-2DJM{\sum }_i{\sigma }_i+NDJ{M}^2$. Each independent Ising spin therefore sees the effective field $2DJM$. With ${\beta }_{\mathrm{th}}=1/(k_{B}T)$, averaging its two Boltzmann weights gives the self-consistency equationThe slope of its right side at zero is $T_c/T$, whereAbove $T_c$, $\tanh x<x$ for $x>0$ excludes a nonzero solution. Below $T_c$ the initial slope exceeds one, and concavity of the hyperbolic tangent on the positive axis gives one positive solution and its negative partner. The Ising auxiliary mean-field free energy $DJ{M}^2-k_{B}T\log [2\cosh (2{\beta }_{\mathrm{th}}DJM)]$ has positive quartic coefficient near $T_c$, while its quadratic coefficient changes sign there. Its stable minima therefore move continuously away from zero; this is a continuous phase transition in the mean-field theory of the Ising model.

Put $\tau =(T_c-T)/T_c>0$ and $c=T_c/T=(1-\tau {)}^{-1}$. The Taylor expansion $\tanh x=x-x^3/3+O(x^5)$ gives, on the nonzero branch,Consequently $M\sim \sqrt{3}{\tau }^{1/2}$ and the mean-field order-parameter exponent is $\beta =1/2$. The inverse-temperature symbol ${\beta }_{\mathrm{th}}$ used above is distinct from this order-parameter critical exponent.

Differentiate the bound with respect to one $J(q)$, using $\partial S_0/\partial J(q)=-J(q{)}^{-2}$ and $\partial S_2/\partial J(q)=-q^{2}J(q{)}^{-2}$:Thus an interior minimizing Gaussian variational kernel for a gradient quartic interaction hasUse the half-space mode density given in the question. In the thermodynamic limit, the coupled self-consistency equations areThe admissible solution must have $J(q)>0$. The coefficient $\gamma$ is unchanged because the interaction's derivative contributes only a constant and a $q^2$ term.

The formal continuum integrals require the coarse-graining ultraviolet cutoff $\Lambda$ in three dimensions: the first radial integrand tends to a constant at large $k$, so taking $\Lambda =\infty$ would leave $\bar{a}$ divergent. The low-wavevector fluctuation argument is independent of this short-distance regularization.

In the usual three-dimensional isotropic setting, the transition is expected to be discontinuous, by the Brazovskii fluctuation-induced first-order transition mechanism.

If $\bar{\kappa }<0$, write $q_{\ast }^2=-\bar{\kappa }/(2\gamma )$ and $r=\bar{a}-{\bar{\kappa }}^2/(4\gamma )$. Then $J(q)=r+\gamma (q^2-q_{\ast }^2{)}^2$. A continuous finite-wavevector transition would require $r\to 0$ with $q_{\ast }>0$. But the soft-mode shell integral behaves asThe positive fluctuation corrections in the self-consistency equations diverge in that limit. In particular $\bar{\kappa }=\kappa +2B{I}_0$ cannot stay at a finite negative value while $I_0$ diverges. Thus the isotropic disordered state cannot lose stability through a zero-mass mode on a fixed nonzero shell in this approximation.

Ordering is instead expected through a free-energy crossing with a finite-amplitude smectic phase, giving a first-order phase transition. This is a reasoned fluctuation prediction for the finite-wavevector transition; the disordered Gaussian variational approximation alone does not locate the coexistence point or calculate the amplitude jump. A zero-wavevector instability or a lower-dimensional setting would require a separate analysis.

Differentiating the mean-field free energy and imposing stationarity givesThe hyperbolic-function identity $\sinh x/(1+\cosh x)=\tanh (x/2)$ turns this into the self-consistency equation

SetThe factor two in the last expression restores the omitted negative wavevectors. For one independent mode $J_k=J(k)$,Differentiating the bound from parts (d) and (e) givesIts stationary point therefore obeys the self-consistency equationThe interaction has generated a positive, wavevector-independent mass shift.

Write the optimized kernel asNear the soft-mode sphere, the supplied asymptotic result saysThus the self-consistency equationcannot reach $\bar{a}=a_c$ for any $g>0$: the fluctuation correction diverges first. The isotropic static structure factor therefore remains finite, and the isotropic state never undergoes the continuous Gaussian instability predicted in part (a).

The modulated minimum from part (b), however, has negative free energy for sufficiently low $a$ and amplitude $A_{\ast }=\pi (a_c-a)/(8g)$. For small positive $g$, its free energy must cross that of the still locally stable isotropic variational state. At the crossing the isotropic inverse susceptibility is positive and the smectic amplitude is nonzero. The order parameter consequently jumps, which is the Brazovskii fluctuation-induced first-order transition.

Using the field derivative while imposing the self-consistency equation givesAt $B=0$, $m=0$ is always a solution. Linearizing the right-hand side for small $m$ gives slopeA continuous phase transition occurs when this slope crosses one, so, with $k_B=1$ and ${\kappa }_c=e^{-g/T_c}$,On the side where the slope exceeds one, the nonzero solutions lower the mean-field free energy and the trivial solution is unstable. The equation has a physical continuous-transition solution only in the corresponding parameter range of this spin-one model.