Critical exponent mean-field value

The coexistence curve is nearly flat at its top, with an exponent p = 1/8, instead of the mean-field value of 1/2. The critical isothemi is also nearly flat at the exponent 8 (detemiined later) is 15 rather than the 3 of the analytic theories. The susceptibility diverges with an exponent y = 7/4, a much stronger divergence than that predicted by the mean-field value of 1. 

Comparing to equation 7.12, we thus see that the mean-field value for the critical exponent (3 exists and is given by... 

In the literature there have been repeated reports on an apparent mean-field-like critical behavior of such ternary systems. To our knowledge, this has first been noted by Bulavin and Oleinikova in work performed in the former Soviet Union [162], which only more recently became accessible to a greater community [163], The authors measured and analyzed refractive index data along a near-critical isotherm of the system 3-methylpyridine (3-MP) + water -I- NaCl. The shape of the refractive index isotherm is determined by the exponent <5. Bulavin and Oleinikova found the mean-field value <5 = 3 (cf. Table I). Viscosity data for the same system indicate an Ising-like exponent, but a shrinking of the asymptotic range by added NaCl [164],... 

The diffusion coefficient D is plotted in Fig. 3 as a function of the reduced temperature e. The upper x-axis shows the correlation length = o -0 63 with = 1.5nm. The short downward arrow marks the approximate locus of the transition from the asymptotic critical to mean field behavior at Nl/ 2RS [4], Below this value, at smaller values of e and larger correlation lengths, the data are compatible with the asymptotic scaling law of (9). For large values of the slope continuously increases due to the transition to the mean field exponent and the growing influence of thermal activation [81]. 

These powers a, (3, 7, p and i/ are called the critical exponents. These exponents are observed to be universal in the sense that although Pc de-pends on the details of the models or lattice considered, these exponents depend the only on the lattice dimensionality (see Table 1.2). It is also observed that these exponent values converge to the mean field values (obtained for the loopless Bethe lattice) for lattice dimensions at and above six. This suggests the upper critical dimension for percolation to be six. 

The critical isotherm is characterized by t = 0 and H = 2brj. This immediately shows that the critical exponent for the magnetic field variation with r = M is 5 = 3, in conformity with the standard mean field value encountered in the literature. Returning to Eq. (7.5.10) and differentiating with respect to M one obtains the magnetic susceptibility as... 

Measurements of the liquid gas transition of salts, e.g. of NH4CI at 1150 K [12] that yielded the mean field value of 3 can hardly achieve the mK accuracy required in work on critical phenomena. Therefore, measurements of the liquid-liquid coexistence occurring near ambient conditions are more apt for measuring critical exponents. 

In a very different context, in statistical mechanics theory of critical phenomena, corrections to classical exponents are calculated using a systematic series of mean field approximations. In this case, the deviation r from the mean-field value of a critical exponent is called coherent anomaly [173], Remember that ER(X) in Eqs. (115)—(118) corresponds to a bound state if XR < Xc and corresponds to a virtual state if XR >XC. Note that there is no other formal difference between bound and virtual states other than the sign in the logarithmic derivate of the wave function at r = R. Therefore there are no technical problems related with this fact. A relation between XR and Xc can be established for compact support potentials. In this case, using variational arguments, we obtain... 

Note that the critical exponents do not take mean-field values, even when the fractal dimension of the lattice becomes greater than 4. This is clearly because of the special structure of the n-simplex lattice, where, even though the fractal dimension becomes greater than 4 for large n, the spectral dimension remains below 2, and probability of intersection of paths of random walks remains large. Also that the non-analytical dependence of the type in the critical exponents on the lattice cannot be obtained... 

The results of the fit to equation 4 then imply a correlation-length critical exponent v = 0.51 0.02. This is in good agreement with the theoretical mean-field value [2] of Vff, = 1/2 and in clear disagreement with the theoretical three-dimensional percolation value [2] of v, = 0.88. It is not well understood why this particular composite appears to obey mean-field percolation and not three-dimensional percolation. A number of other disordered conductor-insulator composites [10] also give critical... 

This immediately shows that the critical exponent for the magnetic field variation with 77 = Mis 6 = 3. This is in conformity with the theoretical mean field value encountered in the literature. 

The critical exponent both above and below the critical point for the heat capacity contributions is given by Y = y = 1, which is the ordinary mean field value cited in earlier Sections. 

Thermal fluctuations modify critical exponents from the values predicted by mean-field theory [2], Based on the dimensionality (d = 3) and the order parameter symmetry (n = 2), the NA transition should belong to the isotropic 3DXY (n = 2, d = 3) universality class. 

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory.

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. 

With the relations given in Table 2.2-1 and the critical exponent values given in Table 2.2-2, the thermodynamic behaviour of a pure component close to the critical point can be described exactly, however further away from the critical point also the mean field contributions have to be taken into account. A theory which is in principle capable to describe... 

It is known that the classical molecular field theory discussed above is not suited for describing a close vicinity of the critical point. Experimentally obtained values of the parameter (3 (called the critical exponent) are essentially less than (3q = 1/2 predicted by the mean-field theory. On the other hand, the experimental values of (3 = 0.33-0.34 turn out to be universal for many different systems (except for quantum liquid-helium where (3... 

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. 

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