The Landau free energy of the three-dimensional Phi⁴model in wide temperature intervals

Abstract

The range of validity of Landau freeenergy potentials with the usual approximation of constam coefficients for terms higher than quadratic has recently been questioned (L Phys.: Condenr. Marter 1 (1989) 8327). The frequent observation in real systems, within large temperature intervals, clearly outside any possible critical region, of power laws of the type IC -TI" for the order parameter has also been pointed out as an indication that certain simple general features in phase transitions, not related at all ta critical phenomena, are beyond the usual approximations included in L Landau free-energy expansion. In particular, the value of the exponent h has been proposed to be related to the displacive and order-disorder degree of the system. In order to elucidate these questions, the temperature dependence of the Landau free energy corresponding to the three-dimensional O4 model has been investigated using a straightforward Monte Carlo method. Different model parameteis have been considered, ranging from typical displacive panmeters to those approaching a pure order-disorder system. Following its formal definition, the Landau free energy at each temperature has been directly derived from the order panmeter distribution in a Metropolis statistical sample. En ContriLCt with other numerical methods used in previous literature, no approximation is introduced in the calculation, except inherent to the numetical method employed. It is shown that the temperature dependence of the L m d u potential coefficients follows smooth simple laws that are outside the usual assumptions in Landau theory and can be related to the order-disorder degree of the system. The quadratic coefficient in the Landau potential exhibits a linea tempelatUE dependence in large temperature intervals but shows a marked change in slope about the transition temperature.The quatic coefficient is shown to depend on tempentlrre as strongly as the quadratic coefficient having a minimum around the transition point. The strong temperature dependence of this quartic coefficieht is responsible for the 'non-classical' behaviour of the order panmeter, which wn be described by a power law.

Coarse graining from variationally enhanced sampling applied to the Ginzburg–Landau model

“…Here we note only that the slope of g(T ) and b(T ) is roughly 10 times smaller than the one of a(T ). This nonconstant behavior has been reported also in previous literature, for other Landau free-energy models (10). It is possible to define an effective critical temperature T * c as the temperature at which the parameter a(T ) becomes zero.…”

“…1, which gives an artist's impression of a Landau free energy as a function of a onedimensional order parameter as the system crosses the critical temperature Tc. For T > Tc the order parameter fluctuations are predominantly Gaussian and the effect of the non-Gaussian quartic term is hidden in the tail of the distribution, as pointed out also by other authors (10,13). By using a p(s) that is broader than the natural Gaussian fluctuations, we can enhance the probability with which the tails are sampled, thus improving the computation of the parameter b proportional to the quartic term I4(s).…”

“…An advantage of using this model is that its strengths and limits are well known and that other researchers have already attempted to perform such a calculation (8)(9)(10)(11)(12)(13)(14). A system that undergoes a second-order phase transition is described in the GL model by the following free energy, valid in a rotationally invariant one-component real-order parameter scenario,…”

Ferroelastic phase transitions: structure and microstructure

“…Although the original applications of Landau-type theories to describe the thermodynamics of phase transitions (Landau, 1937;Devonshire, 1949) were essentially phenomenological attempts to describe the excess free energy as a function of an order parameter Q with mathematically simple functions, the success of this approach to describe phase transitions in real systems has inspired theoretical projects relating macroscopic free-energy functions to fundamental microscopic theory (Chihara et al, 1973;Bruce, 1980;Rae, 1982;Salje et al, 1991;Dove et al, 1992;Radescu et al, 1995;Pe  rez-Mato & Salje, 2000). Here, we outline the rather general approach adopted by Salje et al (1991) and adapted by Pe  rez-Mato & Salje (2000).…”

Crossover between a displacive and an order-disorder phase transition

“…This model can be defined as a 3D cubic lattice of classical anharmonic 1D oscillators with nearest-neighbour harmonic coupling [2,3,4,5,6,7]:…”