Landau free energy

A Landau free energy is a symmetry-constrained analytic expansion in an order parameter. Its minima determine the mean-field equilibrium phases.

Fix the spatial mean of a scalar order parameter to $M$ and sum the Boltzmann weights of all configurations with that mean. The resulting constrained partition function $Z_M$ defines this free-energy density, with an additive normalization fixed consistently. For an applied conjugate field $h$, $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the chosen measure normalization. The thermodynamic limit gives $f(h)={\inf }_M[F_c(M)-hM]$. A homogeneous Landau free energy is an approximation to this constrained potential; phase-separated configurations can convexify it in the full thermodynamic limit.

In a finite box, retain the constant mode $M$ while integrating all nonzero field modes using the Wilsonian coarse-grained statistical Hamiltonian. Below the smallest nonzero momentum, the remaining effective energy is $V{U}_{\Lambda }(M)$ and its Boltzmann weight is precisely the constrained order-parameter free energy weight. Thus the displayed equality holds with consistent field-independent normalization. It is not an identity between a bare statistical Hamiltonian and an unconstrained equilibrium free energy. An analytic polynomial Landau free energy further assumes a homogeneous branch and regular effective coefficients; long-distance fluctuations near and below an upper critical dimension can invalidate that analytic approximation.

For a real unrestricted scalar order parameter and $A=A_2{m}^2/2+A_3{m}^3/3+A_4{m}^4/4$ with $A_3\ne 0$ and $A_4>0$, the disordered global minimum is replaced discontinuously by a nonzero minimum at the displayed phase coexistence condition. At coexistence $A=A_4{m}^2[m+2A_3/(3A_4){]}^2/4$. Ordered stationary points first appear at the spinodal point $A_2=A_3^2/(4A_4)$, while the disordered point loses local stability at $A_2=0$; neither local limit is the equilibrium transition condition.

For the stable scalar Landau free energy $f=b{\lambda }^4+c{\lambda }^3+a{\lambda }^2+d\lambda$, $b>0$, a stable zero-curvature equilibrium must have $f^{\prime }=f^{\prime \prime }=f^{\prime \prime \prime }=0$. Solving these equations givesAt this point $f=f({\lambda }_c)+b(\lambda -{\lambda }_c{)}^4$. Varying two control parameters allows a line of first-order phase transitions to terminate here. A spinodal point satisfies zero curvature but need not satisfy the third-derivative condition or remain a local minimum.

For a scalar Landau free energy $f=a{\varphi }^2/2+b{\varphi }^4/4+c{\varphi }^3/3+d\varphi$ with $b>0$, the shift $\varphi =\psi -c/(3b)$ removes the cubic term. The quadratic coefficient becomes $a^{\prime }=a-c^2/(3b)$ and the linear coefficient becomes $d^{\prime }=d-ac/(3b)+2c^3/(27b^2)$. In a closed binary fluid mixture, the linear term is constant at fixed total composition, leaving the symmetric quartic phase coexistence construction in the shifted variable.

For $n\ge 2$ and ${\lambda }_n>0$, this Landau free energy has mean-field critical exponents $\beta =1/[2(n-1)]$ and $\alpha =(n-2)/(n-1)$ when $r$ crosses zero linearly with temperature. For $n>2$, obtaining this behavior requires tuning the lower even interactions to zero.

For the Landau free energy $G(\varphi )={\varphi }^6/6+g{\varphi }^4/4+\epsilon {\varphi }^2/2$, positive $g$ gives a continuous phase transition at $\epsilon =0$. For negative $g$, phase coexistence between the disordered global minimum $\varphi =0$ and the ordered global minima occurs on $\epsilon =3g^2/16$, with ${\varphi }^2=-3g/4$. Indeed, on that locus the polynomial is $G(\varphi )={\varphi }^2({\varphi }^2+3g/4{)}^2/6\ge 0$, so these three zeros are global minima. For $g<0$, the disordered and ordered spinodal points are respectively $\epsilon =0$ and $\epsilon =g^2/4$; equality of stationary free energies determines phase coexistence, while loss of a local minimum determines a spinodal point.

The equilibrium magnetization is a value of the magnetization that minimizes the Landau free energy. A stable equilibrium has nonnegative second derivative with respect to the magnetization; degenerate nonzero minima describe symmetry-related magnetized phases.

The spinodal is the boundary where a metastable phase loses local stability. In a mean-field free-energy description the relevant curvature vanishes there, separating local minima from unstable configurations.

A spinodal point is the limit of local metastability of a phase. At a mean-field spinodal, a local minimum merges with a neighboring maximum and the corresponding second derivative of the free energy vanishes.

For the Landau free energy $V(M)=r{M}^2/2+u{M}^4/4-hM$ with $r<0$ and $u>0$, stationary branches satisfy $h=rM+u{M}^3$. Their local stability requires $r+3u{M}^2>0$. At a spinodal point, $M_{\mathrm{sp}}=\pm \sqrt{|r|/(3u)}$ and $h_{\mathrm{sp}}=(2r/3)M_{\mathrm{sp}}$, giving the displayed field magnitude. Following local minima without nucleation produces a hysteresis loop switching at these limits. Equilibrium instead switches between the two global minima at $h=0$.

A metastable phase is a local but nonglobal minimum of the thermodynamic potential. A finite nucleation barrier can delay conversion to the globally stable phase until fluctuations create a sufficiently large nucleus or the metastable minimum reaches a spinodal point.

Nucleation forms a sufficiently large region of a new phase within an old metastable phase. In classical nucleation theory, positive surface energy competes with bulk free-energy gain, producing a critical nucleus and a barrier to conversion. The Arrhenius nucleation time depends exponentially on this barrier; boundaries and impurities can alter it. This explains how metastability and hysteresis can persist despite a different globally stable phase.

Dependence of a system's observed state on the path by which its control parameters were reached. In a first-order phase transition, a metastable local minimum may persist after another phase becomes globally stable, so increasing and decreasing the control parameter give different switching points. Nucleation and sweep rate determine actual switching; a mean-field spinodal point is a local-stability limit, not the equilibrium phase coexistence condition.

Superheating is the metastable persistence of a low-temperature phase above its equilibrium transition temperature.

A solid can locally exceed its solidus because a thin solute-diffusion layer changes composition faster than thermal conduction changes temperature. Near a melting phase boundary, a depleted solid can join the equilibrium solidus at the interface while the adjacent, more concentrated solid remains above its own lower solidus. The assumed single sharp interface is then thermodynamically unstable to additional melting and may be replaced by a mushy layer.

Supercooling is the metastable persistence of a high-temperature phase below its equilibrium transition temperature.

Two compositions coexist when one straight line is tangent to the free-energy density at both compositions. Equality of its slope gives equal chemical potentials, and equality of its intercept gives equal pressures.

The binodal is the coexistence boundary: its two branches give the compositions of phases joined by the common-tangent construction.