Tricritical universality class

Transition matrix, Monte Carlo 17 Tricritical point/behavior 10, 12, 21 Tricritical universality class 93 Triple lines 5, 36, 38, 42, 43, 49, 50, 53, 56, 81, 89, 91, 94 Triple points 37, 39, 40, 54 Triple pressure 39, 40, 51 Triple temperature 53, 55 Two-chaiii equation 244... 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

The SmA liquid crystalline phase results from the development of a one-dimensional density wave in the orientationally ordered nematic phase. The smectic wave vector q is parallel to the nematic director (along the z-axis) and the SmA order parameter i/r= i/r e is introduced by P( ) = Po[1+R6V ]- Thus the order parameter has a magnitude and a phase. This led de Gennes to point out the analogy with superfluid helium and the normal-superconductor transition in metals [7, 59]. This would than place the N-SmA transition in the three-dimensional XY universality class. However, there are two important sources of deviations from isotropic 3D-XY behavior. The first one is crossover from second-order to first-order behavior via a tricritical point due to coupling between the smectic order parameter y/ and the nematic order parameter Q. The second source of deviation from isotropic 3D-XY behavior arises from the coupling between director fluctuations and the smectic order parameter, which is intrinsically anisotropic [60-62]. 

The SmA-N transition is often encountered in liquid crystals and has been extensively studied both theoretically and experimentally. Based on the analogy with the superconductor to normal metal transition, de Gennes [19,21] classified the SmA-N transition in the isotropic three-dimensional XY universality class. Actually, however, deviation from the isotropic 3D-XY behaviour occurs to give the cross-over from second order to first order behaviour via a tricritical point due to coupling between smectic and nematic order parameters. This type of deviation has been predicted for the... 

From (5.5.20) and (5.5.21) it is seen that Beet, where = 2vj — v, and D oc fK If V = 2, B should be finite at the transition temperature. However, experimentally, it appears that B at the transition is almost vanishingly small within experimental limits. Few measurements are available on D to draw any definite conclusions. In any case, as pointed out earlier, the exponents are neither universal nor do they agree with the predictions of any of the theoretical models. Vithana et a/. have suggested that the widely differing values of the exponents for the different compounds may be a consequence of the fact that one is measuring effective values associated with crossover effects between the XY class and a tricritical point. A further complication is that the experiments of Evans-Lutterodt et appear to indicate that the occurrence of different... 

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