Order-parameter critical exponent

E. V. Aibano. Determination of the order-parameter critical exponent of an irreversible dimer-monomer surface reaction model. Phys Rev E 49 1738-1739, 1994. 

Cv Pc 8 a an P Y c f specific heat reduced density critical exponent for the critical isotherm critical exponent for the specific heat critical exponent for the specific heat along isot r critical exponent for the order parameter critical exponent for the susceptibility reduced temperature friction coefficient... 

The critical behaviours of higher order satellites near a continuous phase transition are of particular interest, as they provide the opportunity to study the crossover exponents of the respective symmetry breaking perturbations in the spin Hamiltonian. Each order of satellite has an associated order parameter critical exponent given by / = 2 - a -where a is the specific heat exponent, and crossover exponent As an example, if the transition is described by the 3DXY model, then the exponent 2 measures the crossover caused by a perturbation of uniaxial symmetry. For this model, theory predicts that = o P, with... 

P is the critical exponent and t denotes the reduced distance from the critical temperature. In the vicinity of the critical point, the free energy can be expanded in tenns of powers and gradients of the local order parameter m (r) = AW - I bW ... 

A is a constant and p is the critical exponent which adopts values from 0.3 to 0.5. Values around p = 0.5 are observed for long-range interactions between the particles for short-range interactions (e.g. magnetic interactions) the critical exponent is closer to p 0.33. As shown in the typical curve diagram in Fig. 4.2, the order parameter experiences its most relevant changes close to the critical temperature the curve runs vertical at Tc. 

Above the transition temperature cf> = 0 and below the critical temperature 4> > 0. At T = 0, = 1. Any order parameter defined to have a magnitude of unity at zero temperature is said to be normalized. Criteria for testing the theory include a number of critical exponents, which are accurately known from experimental measurement. 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

For a second-order transition, this problem is conveniently studied in terms of the order parameter distribution function, Pi,( ). Finite size scaling theory implies that near the critical point P/.( ) longer depends on the three variables L, 1 — T/T separately but rather is a scaled function of two variables (1 — only where v is the critical exponent of... 

It should be mentioned that the calculated entanglement here has a corresponding critical exponent ys = 0. This means that the entanglement is constant at the critical point over all sizes of the system. But it is not a constant over all values of U/t. There is an abrupt jump across the critical point as L —> oo. If we divide the regime of the order parameter into noncritical regime and critical... 

A common mathematical assumption is that the dependence of the order parameter e2 on the path parameter r can be described by a critical exponent relationship of the form [see, e.g., H. E. Stanley. Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971)]... 

Nearly the same limits of r exist in real solid state experiments. However, the relevant maximal time tm which could be achieved in such computer simulations (see equation (5.1,60)) for a given Tq, could turn out to be not long enough for determining the asymptotic laws under question. For instance, existence of so-called small critical exponents in physics of phase transitions [14] was not experimentally confirmed since to obtain these critical exponents, the process covering several orders of the parameter t — fo should be... 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

Early attempts to develop theories that accounted for the power-law behavior and the actual magnitudes of the various critical exponents include those by van der Waals for the (liquid + gas) transition, and Weiss for the (ferromagnetic + paramagnetic) transition. These and a later effort called the Landau theory have come to be known as mean field theories because they were developed using the average or mean value of the order parameter. These theories invariably led to values of the exponents that differed significantly from the experimentally obtained values. For example, both van der Waals and Weiss obtained a value of 0.50 for (3, while the observed value was closer to 0.35. 

Several paradoxes have become apparent from modern descriptions of phase transitions, and these have driven much of the research activity in this field. The intermolecular interactions that are responsible for the phase transition are relatively short-ranged, yet they serve to create very long-range order at the transition temperature. The quantum mechanical details of the interactions governing various transitions are very different, and the length scales over which they operate vary considerably, yet the observation of scaling laws and the equivalences of a given critical exponent value within a fixed dimensionality of the order parameter show that some additional principle not described by quantum mechanics must also be at work. Also, the partition... 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

PI3.2 The phase transition of 4He into its superfluid phase from the normal liquid is continuous. The order parameter, rj, of the transition has been found to vary as the square root of the ratio of the superfluid density, ps, to the total density, px. Some data for rj as a function of (Tc — T) are given below. Obtain the critical exponent, /3, for the order parameter from this data. 

Maybe the major achievement provided by the discovery of the fe rfim is the possibility to study the complete set of critical exponents on a ferroic system for the first time after their prediction [9,10]. Table 15.1 shows the results as compared with predictions from theory and simulations. Most remarkably, the order parameter exponent ft (Figure 15.10) clearly deviates from the prediction ft 0 and achieves a value which comes close to that observed recently on the standard rfim system, the dilute uniaxial antiferromagnet Fci ,Zn.,F2, x = 0.15, in an external magnetic field [50], Further, the most disputed value, namely the specific heat exponent a [48] (Figure 15.12) clearly describes the same logarithmic divergence as that found on Fci. Zn. I 2, a 0 [10], which still lacks theoretical confirmation. 

As a final example in this subsection, Fig. 9 examines the distortion of the phase diagram due to these wall effects, for three choices of the film thickness [58]. One notes that the shape of the coexistence curves is always parabolic near the critical point, as expected, since this shape reflects the mean field order parameter exponent, (3=1/2 [186]. In real systems, one expects - except for extremely long... 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

What does your best value for the exponent j8 indicate about the nature of the critical behavior underlying the N-I transition (mean-field second-order or tricritical) Note that /3 and P describe the temperature variation of the nematic order parameter S, which is a basic characteristic of the liquid crystal smdied. Thus, the same j8 and P values could have been obtained from measurements of several other physical properties, such as those mentioned in the methods section. 

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