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.

For $u>0$ and $t>0$, minimizing a uniform field gives zero magnetization when $|r|\le t$. For $r<-t$ only component 1 orders, with $m_1^2=-(t+r)/(4u)$; for $r>t$ only component 2 orders, with $m_2^2=(r-t)/(4u)$. Both transition lines are continuous. The positive-$t$ disordered wedge separates the two easy-axis phases.

For $f=a|\varphi {|}^2+|\varphi {|}^4/2+c|{\partial }_x\varphi {|}^2+|{\partial }_x^2\varphi {|}^2$, a plane-wave order parameter has quadratic coefficient $a+c{k}^2+k^4$. If $c<0$, this is minimized at $k^2=-c/2$, and a nonzero amplitude appears when $a<c^2/4$. If $c\ge 0$, the minimizing wave number is zero and the ordered mean-field phase requires $a<0$. The arbitrary global phase is a degeneracy, not a missing minimization condition. These are functional-minimization results; finite-temperature fluctuations can invalidate long-range order in low dimensions.

A stationary point on the domain $\varphi \ge 0$ must be tested using allowed one-sided variations. For $F=A{\varphi }^2/2-B{\varphi }^3+\lambda {\varphi }^4/4$ with $B>0$, the boundary is a local minimum when $A>0$ and a local maximum when $A<0$. At $A=0$, the leading negative cubic term makes it a one-sided local maximum even though $F^{\prime \prime }(0)=0$; on an unrestricted real line it is instead a stationary inflection point.

For $\varphi \ge 0$ and positive $c,B,\lambda ,T_0$, define $\gamma =2B^2/(\lambda c)$. Completing the square givesFor $0<\gamma <1$, phase coexistence occurs at $T^{\ast }=T_0/\sqrt{1-\gamma }$ with a discontinuous minimum at ${\varphi }^{\ast }=2B{T}^{\ast }/\lambda$. For $\gamma \ge 1$, a nonzero global minimum has lower free energy at every finite temperature. Following the global equilibrium branch then does not force a restoration transition.

When $\gamma \ge 1$ in the stated model, the broken, nonzero minimum remains globally preferred at every finite temperature. A metastable zero state can still exist for $T>T_0$, but its lifetime and a nucleation temperature are kinetic questions. Its disappearance at the spinodal point is not an equilibrium coexistence temperature.

At an equilibrium minimum, $d{F}_{\min }/dT={\partial }_{T}F$ because ${\partial }_{\varphi }F=0$. For $0<\gamma <1$, the entropy decrease at coexistence is $S_0-S_{+}=c(1-\gamma )T^{\ast }({\varphi }^{\ast }{)}^2$. Multiplying by $T^{\ast }$ gives the positive latent heat released on cooling. The units are total heat or heat density according to the normalization of $F$.

For $f=f_0+a(T-T_c)|\mathbf{m}{|}^2+b|\mathbf{m}{|}^4+c|\nabla \mathbf{m}{|}^2$ with positive coefficients and uniform free boundaries, the minimum is $\mathbf{m}=0$ at $T\ge T_c$. Below $T_c$ it is any constant vector with $|\mathbf{m}{|}^2=a(T_c-T)/(2b)$. A chosen direction leaves an $O(2)$ subgroup of $O(3)$ invariant; for proper rotations the pattern is $SO(3)\to SO(2)$. The mean-field transition is continuous, with square-root onset. Long-wavelength fluctuations can invalidate the ordered state in low dimensions.

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.

At coexistence, the landau free energy can factor as $F(\varphi )=\lambda {\varphi }^2(\varphi -v{)}^2/4$. The minima at $0,v$ have equal free energy. Differentiating shows the intermediate maximum at $v/2$, with barrier height $\lambda v^4/64$. The barrier is a local free-energy barrier; predicting spatial nucleation also requires interfacial or gradient information.

For $a,c>0$, $b\le 0$ and the molecular bound $Q\in [-1/2,1]$, coexistence with $Q=0$ occurs at $b_t=-{\min }_{0<Q\le 1}(a/Q+3cQ)$. Negative $Q$ has positive free energy and cannot be the stable phase in this coefficient range. Thus $a\le 3c$ gives $b_t=-\sqrt{12ac}$ and jump $0\to \sqrt{a/(3c)}$, whereas $a>3c$ gives $b_t=-(a+3c)$ and jump $0\to 1$. The interior case is the usual small-order-parameter Landau theory result. A formal unconstrained quartic minimization can violate the molecular bounds when extrapolated too far.

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.

Spinodal decomposition is spontaneous growth of composition fluctuations in a homogeneous mixture whose local mixing free energy has negative composition curvature. For a mean $\varphi$, equal regions at $\varphi \pm \delta \varphi$ change the mean free energy by $f^{\prime \prime }(\varphi )(\delta \varphi {)}^2/2+O((\delta \varphi {)}^4)$, which is negative inside the spinodal. Interfacial costs suppress sufficiently short wavelengths, while conservation of composition constrains the dynamics. Outside the spinodal but inside the binodal, the uniform state is instead metastable and requires nucleation to begin separation.

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 state is locally stable against small disturbances but has a higher thermodynamic potential than another accessible phase. In a free-energy landscape it occupies a local minimum separated from the stable state by a barrier.

A liquid undergoing supercooling illustrates the distinction: small fluctuations can relax without freezing, but a sufficiently large crystal nucleus can grow. The nucleation barrier depends on the competition between interfacial cost and bulk free-energy gain. Waiting time is therefore affected by sample volume, impurities, surfaces and observation duration. A spinodal point is the mean-field limit where the metastable minimum loses local stability; it is distinct from the coexistence condition where the stable phases have equal appropriate intensive potentials. Metastability explains hysteresis without making the equilibrium phase boundary path dependent.

A vapor whose partial pressure exceeds its vapor pressure at the local temperature; nucleation can permit condensation.

The fractional excess of water-vapor pressure above equilibrium saturation over a flat pure-water surface at the same temperature: $s=p_v/p_{\mathrm{sat}}(T)-1$. Positive $s$ can drive droplet growth, but activation of a particular aerosol particle requires exceeding its critical supersaturation. Cooling, condensation and mixing jointly determine parcel supersaturation.

The ambient water vapor partial pressure divided by the equilibrium saturation pressure over a stated reference surface at the same temperature. Relative to flat pure water, atmospheric supersaturation is $s=S-1$. A curved solute-containing cloud droplet generally has a different equilibrium saturation ratio from unity.

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.

Formation of most nuclei within an interval short compared with subsequent particle growth. A common birth period can help narrow a particle size distribution if mixing and later growth are sufficiently uniform and secondary nucleation is suppressed.

A schematic time history of growth-species concentration showing buildup, a short nucleation interval above an effective nucleation threshold, then growth above the equilibrium solubility but below substantial new nucleation. The concentration history suggests how to separate particle birth from growth; it does not establish that every particle synthesis follows that mechanism.

Formation of a critical nucleus at a pre-existing surface or interface. Favorable wetting can replace part of the cost of a free nucleus and lower its barrier; later deposition of a free particle on a support is a different process.

Ice forms from ice-supersaturated vapour at a suitable foreign surface. The relevant temperature and surface properties differ from liquid-droplet immersion freezing.

A suitable particle colliding with a supercooled droplet can trigger heterogeneous freezing. This is distinguished from nucleation around an inclusion already inside it.

Ice nucleates on an inclusion already immersed in supercooled water. The interfacial nucleation barrier, inclusion properties, droplet volume and observation time affect freezing probability.

Formation of a critical nucleus within the bulk parent phase without assistance from a pre-existing interface. Its interfacial cost competes with the bulk free-energy gain.

Ice embryos form without a foreign nucleating surface once thermal fluctuations can overcome the free-energy barrier on the relevant timescale. An apparent freezing threshold depends on droplet volume and measurement duration.

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.

Magnetic hysteresis means that a material's magnetization depends on its field history as well as the present applied field. A closed quasistatic magnetic cycle dissipates energy per unit volume equal to its $H$–$B$ loop area. It is relevant to remanence in permanent magnets and core losses in electrical transformers.

Magnetic coercivity is the magnitude of a reverse applied magnetic field intensity required to bring a previously magnetized specimen's magnetization to zero on a specified hysteresis branch. It depends on material, preparation and measurement history. It differs from remanent magnetization, which is the remaining magnetization at zero applied field.

The magnitude of reverse magnetic field strength at which a previously magnetized specimen's magnetic flux density $B$ vanishes on a specified magnetic hysteresis branch. Intrinsic magnetic coercivity instead refers to zero magnetization. Because $B={\mu }_0(H+M)$, their zero conditions generally occur at different applied fields. The preparation, branch and specimen geometry must be specified when comparing measurements.

Superheating is persistence of a phase above the equilibrium temperature at which a higher-temperature phase becomes favorable. The familiar case is a liquid above its boiling point at the imposed pressure, although delayed melting can also be discussed as superheating.

The liquid can remain in a metastable state because producing a vapor bubble costs interfacial energy. A sufficiently large bubble overcomes the nucleation barrier and grows, converting the liquid and consuming latent heat. Smooth clean containers can delay nucleation, while cavities and boiling chips provide heterogeneous nucleation sites. The observed temperature of sudden bubble formation therefore need not equal the equilibrium boiling point. The amount of superheating is a kinetic and preparation-dependent observation, rather than a new equilibrium phase boundary.

A liquid maintained above its equilibrium boiling temperature at the imposed pressure. A barrier to nucleation can delay its conversion into vapor, making this a metastable state.

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 persistence of a phase below the equilibrium temperature where a lower-temperature phase becomes favorable. A liquid remaining unfrozen below its equilibrium melting temperature is the standard example.

Creating a crystal nucleus gains bulk free energy but costs interfacial energy. Small embryos can disappear, while one exceeding the critical size can grow; this nucleation barrier permits a metastable liquid. A surface or suspended inclusion can lower the barrier through heterogeneous nucleation, whereas homogeneous nucleation occurs within the bulk. Measured freezing onset consequently depends on droplet size, purity and waiting time. Growth releases latent heat and can temporarily warm the remaining material toward the coexistence temperature, separating the onset of nucleation from the later heat-removal problem.

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.

For a dilute-to-dense liquid coexistence problem, the dilute-phase coexistence concentration at specified conditions. It is a branch of the coexistence boundary, not generally the concentration at the critical endpoint.