Critical Divergence and Exponents

The collapse of dimensionality associated with the critical singularity (11.114) has many dramatic consequences in Ms- In this limit, all conjugate vectors and response functions become mathematically ill-defined (divergent), corresponding to infinities in physical properties associated with the conjugate metric M (cf. Table 11.1)  

Although the critical extensive vectors S)C, V)C and associated responses CPc, fiTc, aPc are strictly undefined at the critical state, we can consider this state as a target for approach along a chosen thermodynamic path. This will enable us to characterize physical and mathematical details of the asymptotic divergences (11.131) that are the hallmark of critical phenomena. 

The natural path variable or order parameter to characterize proximity to the critical limit is the minor eigenvalue e2 of the thermodynamic metric. This suggests that we examine the functional dependence of conjugate responses on e2, 

Of course, we should identify the particular path chosen to approach the critical limit, and for this purpose it is convenient to introduce a dimensionless path parameter such as  

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)] 

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A very extensive numerical study has been recently carried out by Singh and co-workers. ° Except for the critical divergence Ising exponent, excellent agreement is found between PRISM theory based on the R-... 

If the finite size of the system is ignored (after all, A is probably 10 or greater), the compressibility is essentially infinite at the critical point, and then so are the fluctuations. In reality, however, the compressibility diverges more sharply than classical theory allows (the exponent y is significantly greater dian 1), and thus so do the fluctuations. 

Immediately when the dynamic interpretation of Monte Carlo sampling in terms of the master equation, Eq. (31), was realized an application to study the critical divergence of the relaxation time in the two-dimensional Ising nearest-neighbor ferromagnet was attempted . For kinetic Ising and Potts models without any conservation laws, the consideration of dynamic universality classespredicts where z is the dynamic exponent , but the... 

In the AOT-water-oil system, when the temperature is increased sufficiently a cloud point (critical point) is reached. At temperatures well below this transition temperature the viscosity data are in agreement with a simple hard-sphere model. Upon approaching the critical point one first notices a relatively moderate increase of viscosity by about a factor of 4 followed by a critical divergence of viscosity very close to the cloud point. The critical divergence of such a system (with decane as oil) at the critical temperature was studied and was shown to scale almost Ising-like according to rj [ Tc - T)/Tc] with a critical exponent of 0.03 [69]. 

The asymptotic relations (Equations 7.9 and 7.10) are exact for any one-component liquid with spherical interactions, except when the Taylor expansion (Equation 7.7) fails to hold, which is near a critical point (Fisher 1964). At the critical point, the DCF becomes a nonanalytic function, and the correlations develop a nonexponential algebraic decay in l/r +i where ii = 0.041 is a critical exponent (Fisher 1964). In all that follows, we will stay away from criticality, hence the OZ forms. Equations 7.9 and 7.10, are essentially exact. When the critical point is approached, the isothermal compressibility diverges, and the analysis above shows that the correlation length diverges as the square root of the compressibility. The divergence of leads to a Coulomb decay of the pair correlation in Equation 7.10, but the correct exponent is slightly faster than pure Coulomb decay. 

These equations hold for small introduce absolute value signs [p — pd, ie, in the vicinity of GP. Materials near GP are often called nearly critical gels. The exponents depend not only on the dynamic critical exponent (relaxation exponent Wc) but also on the dynamic exponents s and z for the viscosity (Pc — p) and the equilibrium modulus Ge (p - PcY- If one, in addition, assumes symmetry of the diverging Xmax near the gel point... 

Many important properties, such as critical dimensionality and critical exponents describing the divergence of correlation length and other quantities can thus be obtained from renormalization group analysis. It is worth noting that for X well below X only the quadratic terms of the potential contribute to the asymptotic properties of P, which reduces therefore to a multigaussian distribution in accordance with the central limit theorem [13]. There exists,however,a (frequently very narrow) vicinity of X ... 

The nematic-smectic-A (NA) transition is one that has been studied theoretically with fluctuations being accounted for within different levels of approximation. It has also been studied experimentally by a diverse array of high-resolution techniques in laboratories around the world. While much has been understood about the transition, almost every probe of the NA transition, whether mean-field behaviour of solutes, nature of divergences and values of critical exponents or phase transition order, has met with conflicting experimental results. It appears that another generation of resolution and precision enhancement is required before the complete story is told. As remarked several years ago by deGennes and Frost [3] It seems that we almost understand, but not quite . 

In order to find the critical probability and critical exponents we have to consider equation (6) more thoroughly. The critical exponents, like k and t defined earlier, play an essential role in modern theory of phase transitions in general, " and will be defined later. In gelation they define how quantities like the molecular weight diverge at the critical point. Therefore we rewrite equation (5) as = w and find... 

The main difference between this model and the one described in Sect. C. V. 5 is that in the previous case the length describing the correlations between different sites diverges (with exponent v) at the Potts critical point whereas in the present case the correlations extend over at most one lattice spacing. Accordingly, one would expect that this type of polychromatic percolation belongs always to the same universality class as random percolation, and Monte Carlo calculations as well as renormalization group methods have confirmed this expectation. 

This equation is analogous to the compressibility equation for fluids and diverges with the same exponent y as the critical temperaUire is approached from above ... 

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what m pure-fluid systems is the isothennal compressibility k, and what in mixtures is the osmotic compressibility, and detennines how fast these quantities diverge as the critical point is approached (i.e. as > 1). 

However, the discovery in 1962 by Voronel and coworkers [H] that the constant-volume heat capacity of argon showed a weak divergence at the critical point, had a major impact on uniting fluid criticality widi that of other systems. They thought the divergence was logaritlnnic, but it is not quite that weak, satisfying equation (A2.5.21) with an exponent a now known to be about 0.11. The equation applies both above and... 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

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