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Landau theory
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===Second-order transitions=== [[File:LandauFreeEnergy.svg|thumb|Sketch of free energy as a function of order parameter <math>\eta</math>]] Consider a system that breaks some symmetry below a phase transition, which is characterized by an order parameter <math>\eta</math>. This order parameter is a measure of the order before and after a phase transition; the order parameter is often zero above some critical temperature and non-zero below the critical temperature. In a simple ferromagnetic system like the [[Ising model]], the order parameter is characterized by the net magnetization <math>m</math>, which becomes spontaneously non-zero below a critical temperature <math>T_c</math>. In Landau theory, one considers a free energy functional that is an analytic function of the order parameter. In many systems with certain symmetries, the free energy will only be a function of even powers of the order parameter, for which it can be expressed as the series expansion<ref>{{cite book|last1=Landau|first1=L.D.|last2=Lifshitz|first2=E.M.|title=Statistical Physics|volume=5|publisher=Elsevier|year=2013|isbn=978-0080570464}}</ref> :<math>F(T,\eta) - F_0 = a(T) \eta^2 + \frac{b(T)}{2} \eta^4 + \cdots</math> In general, there are higher order terms present in the free energy, but it is a reasonable approximation to consider the series to fourth order in the order parameter, as long as the order parameter is small. For the system to be thermodynamically stable (that is, the system does not seek an infinite order parameter to minimize the energy), the coefficient of the highest even power of the order parameter must be positive, so <math>b(T)>0</math>. For simplicity, one can assume that <math>b(T) = b_0</math>, a constant, near the critical temperature. Furthermore, since <math>a(T)</math> changes sign above and below the critical temperature, one can likewise expand <math>a(T) \approx a_0 (T-T_c)</math>, where it is assumed that <math>a>0</math> for the high-temperature phase while <math>a<0</math> for the low-temperature phase, for a transition to occur. With these assumptions, minimizing the free energy with respect to the order parameter requires :<math>\frac{\partial F}{\partial \eta} = 2a(T) \eta + 2b(T) \eta^3 = 0</math> The solution to the order parameter that satisfies this condition is either <math>\eta= 0</math>, or :<math>\eta_0^2 = -\frac{a}{b} = - \frac{a_0}{b_0}(T-T_c)</math> [[File:LandauTheoryTransitions.svg|thumb|Order parameter and specific heat as a function of temperature]] It is clear that this solution only exists for <math>T<T_c</math>, otherwise <math>\eta= 0</math> is the only solution. Indeed, <math>\eta= 0</math> is the minimum solution for <math>T>T_c</math>, but the solution <math>\eta_0</math> minimizes the free energy for <math>T<T_c</math>, and thus is a stable phase. Furthermore, the order parameter follows the relation :<math>\eta(T) \propto \left| T - T_c \right|^{1/2}</math> below the critical temperature, indicating a [[critical exponent]] <math>\beta = 1/2</math> for this Landau mean-theory model. The free-energy will vary as a function of temperature given by :<math>F - F_0 = \begin{cases} - \dfrac{a_0^2}{2b_0} (T-T_c)^2, & T<T_c \\ 0, & T>T_c \end{cases} </math> From the free energy, one can compute the specific heat, :<math>c_p = -T\frac{\partial^2 F}{\partial T^2} = \begin{cases} \dfrac{a_0^2}{b_0} T, & T<T_c \\ 0, & T>T_c \end{cases} </math> which has a finite jump at the critical temperature of size <math>\Delta c = a_0^2 T_c/b_0</math>. This finite jump is therefore not associated with a discontinuity that would occur if the system absorbed [[latent heat]], since <math>T_c \Delta S = 0</math>. It is also noteworthy that the discontinuity in the specific heat is related to the discontinuity in the ''second'' derivative of the free energy, which is characteristic of a ''second''-order phase transition. Furthermore, the fact that the specific heat has no divergence or cusp at the critical point indicates its critical exponent for <math>c\sim |T-T_c|^{-\alpha}</math> is <math>\alpha=0</math>.
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