Order parameter function

Now the meaning of the order parameter function q x) can be understood for instance in terms of dynamics if one takes before the time t tends to... 

The Parisi order parameter function q x) with 0 =s x =s 1 is shown to be related to overlap functions between different valleys. Defining an overlap... 

Fig. 17. Plot of the deAltneida-Thouless (AT) line for the SK model (Ising spins) with / = 0 (eq. 15). To the right of the AT line (P) the SK solution with a single order parameter is correct, while to the left of the AT line (SG) the Parisi s order-parameter function is believed exact. The AT line signals the onset of irreversibility.

Figure A3.3.6 Free energy as a function of the order parameter cji for the homogeneous single phase (a) and for the two-phase regions (b), 0.

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

Fig. 3. Order parameter as a function of temperature for -methoxybeiizylidene-/) - -butylariiline (MBBA), a room temperature nematic Hquid crystal. S(T) is determined from tbe polarization of tbe absorption (dicbroism) of small quantities of a dye molecule of similar stmcture (/n / .f-dimetby1aminonitrosti1bene) wbicb bas been dissolved in tbe Hquid crystal bost (1).

Positional Distribution Function and Order Parameter. In addition to orientational order, some Hquid crystals possess positional order in that a snapshot at any time reveals that there are parallel planes which possess a higher density of molecular centers than the spaces between these planes. If the normal to these planes is defined as the -axis, then a positional distribution function, can be defined, where is proportional to the... 

The central quantity is the order parameter as a function of temperature (see Fig. 13). The phase transition temperature Tq of the classical system can be located around 38 K. At high temperatures, the quantum curve of the order parameter merges with the classical curve, whereas it starts to deviate below Tq. Qualitatively, quantum fluctuations lower the ordering and thus the quantum order parameter is always smaller than its classical counterpart. The inclusion of quantum effects results in a nearly 10% lowering of Tq (see Fig. 13). 

To start the review of the PIMC results [328], we note that the detailed study of the quantum APR model (Eq. (41)) was partly motivated by the strong changes in shape of the orientational order parameter as a function of temperature as the rotational constant was increasing from its classical value 0 = 0 (see Fig. 3 in Ref. 327). For small enough 0 it was found that the order parameter decays monotonically with increasing temperature, similarly to the classical case. This is qualitatively different for larger 0, where T ) becomes a non-monotonic function of temperature. 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Here the functions g(0) and /(0) are defined in a suitable way to produce the desired phase behavior (see Chapter 14). The amphiphile concentration does not appear expHcitly in this model, but it influences the form of g(0)— in particular, its sign. Other models work with two order parameters, one for the difference between oil and water density and one for the amphiphile density. In addition, a vector order-parameter field sometimes accounts for the orientional degrees of freedom of the amphiphiles [1]. 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

Additional isothermal treatments at neighbouring temperatures small step annealing) yield plateau values of resistivity corresponding to equilibrium values at certain temperatures which reflect the order parameter in thermal equilibrium as a function of temperature ( equilibrium curve , curve 4 in Figure 1). This study can be used for an analysis of the kinetics of order-order relaxations (see Figure 3 below). 

One possible order parameter, proposed by Paris [par83], is not so much an order parameter as an order function, Define Qap to be the overlap between the states a and f3 ... 

The first order (i.c. ]> 1) approximation of the CML system defined by equation 8.44 (using either of the two methods defined above) is given by an elementary fc = 2, r = 1 CA. Since there are only 32 such rules, the particular CA rule corresponding to a CML system with parameters e and s may be found directly by calculating the outcome of each of the five possible local states. Looking at the first-order step function fi x) in equation 8.47, we can identify the absorbing state X = X with the CA state ct = 0, and x = 1/2 with a = 1. 

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