Dynamical scaling functions above Tc for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio R of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent upsilon of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of upsilon for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region.
Dynamical scaling functions above Tc for the characteristic frequencies and the dynamical correlation functions of the order parameter and the conserved density of model C are calculated in one loop order. By a proper exponentiation procedure these results can be extended in order to consider the changes in these functions using the fixed point values and exponents in two loop order. The dynamical amplitude ratio R of the characteristic frequencies is generalized to the critical region. Surprisingly the decay of the shape functions at large scaled frequency does not behave as expected from applying scaling arguments. The exponent upsilon of the decay does not change when going from the critical to the hydrodynamic region although the shape functions change. The value of upsilon for the order parameter is in agreement with its value in the critical region, whereas for the conserved density it is equal to 2, the value in the hydrodynamic region.
We analyze the field theoretic functions of the dynamical model C in two-loop order. Our results correct long-standing errors in these functions published by several authors. We discuss, in particular, the fixed points for the ratio w* of the two time scales involved, as well as their stability. The regions of the "phase diagram," whose axes are the spatial dimension d and number of order parameter components n, correspond to these fixed points; previous authors have found, in addition, an anomalous region in which the scaling properties were unclear. We show that such an anomalous region does not exist. There are only two regions: one with a finite fixed-point w* where the dynamical exponent z=2+alpha/nu, and another where w*=0 and z is equal to the model A value. We show how the one-loop result is recovered from the two-loop result in the limit epsilon=4-d going to zero.
The critical dynamics at elastic phase transitions of second order is studied by the renormalization-group theory. The dynamical theory is based on a stochastic equation of motion for damped phonons. For systems with one-dimensional soft sectors usual dynamical scaling holds with the critical dynamical exponent z=2. For two-dimensional sectors logarithmic corrections appear. Novel features are found for an isotropic-phonon model. The dynamical susceptibility and the characteristic frequency depend singularly on an irrelevant parameter, which, however, diverges at the fixed point. Consequently dynamical scaling breaks down; e.g., the characteristic frequency is no longer a homogeneous function of the wave number k and the inverse correlation length ${\ensuremath{\xi}}^{\ensuremath{-}1}$. Instead it is a homogeneous function of these variables and an irrelevant parameter. Consequently, the critical dynamics can be characterized by the relaxational exponent $z=2+c\ensuremath{\eta}$ at Tc, while in the hydrodynamic region the sound frequency is characterized by $z=2\ensuremath{-}\frac{1}{2}\ensuremath{\eta}$ and the damping by yet another exponent. This breakdown of scaling is also reflected by the fact that different fixing conditions, i.e., different choices of the frequency, lead to different values of z. All of these apparently different transformations lead to the same modified dynamical scaling relations.
A general theory is presented for sound attenuation and dispersion near the critical point in the region of frequencies which are much greater than the characteristic frequency of order-parameter dynamics, but are much smaller than the frequency of sound with the wavelength equal to the range of correlation of local order-parameter fluctuations. The method used is the most general version of the mode-mode coupling theory plus the static and dynamic scaling hypotheses and some general properties of time-correlation functions. If we ignore the small critical exponent $\ensuremath{\alpha}$ associated with the heat-capacity singularity, we find that the attenuation behaves as $f{\ensuremath{\epsilon}}^{0}$ and the dispersion (relative sound-velocity change with frequency) as ${f}^{0}{\ensuremath{\epsilon}}^{0}$, where f is the dimensionless frequency and $\ensuremath{\epsilon}$ the dimensionless temperature distance from the critical point.
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.
Equipped with the field theory representation of non-linear Langevin equations, the tools of dynamic perturbation theory, and the dynamic renormalization group introduced in Chapters 4 and 5, we are now in the position to revisit models for dynamic critical behavior that entail reversible mode couplings and other conserved hydrodynamic modes. We have already encountered some of these in Section 3.3. In models C and D, respectively, a non-conserved or conserved n -component order parameter is coupled to a conserved scalar field, the energy density. Through a systematic renormalization group analysis, we may critically assess the earlier predictions from scaling theory, and discuss the stability of fixed points characterized by strong dynamic scaling, wherein the order parameter and conserved non-critical mode fluctuate with equal rates, and weak dynamic scaling regimes, where these characteristic time scales differ. Next we investigate isotropic ferromagnets (model J), with the conserved spin density subject to reversible precession in addition to diffusive relaxation. Exploiting rotational invariance, we can now firmly establish the scaling relation z = ( d + 2 − η )/2. Similar symmetry arguments yield a scaling relation for the dynamic exponents associated with the order parameter and the non-critical fields in the O ( n )-symmetric SSS model that encompasses model E for planar ferromagnets and superfluid helium 4 (for n = 2), and model G for isotropic antiferromagnets ( n = 3). There exist competing strong- and weak-scaling fixed points, with the former stable to one-loop order, and characterized by z = d /2 for all slow modes.
The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from Tc, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near Tc, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.