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Progress of Theoretical Physics

Dynamics of Critical Fluctuations. IV

Kyozi Kawasaki

Published Nov 1, 1968• 16 citations
Publisher

Abstract

Dynamics of critical fluctuations associated with the liquid-gas transition are studied employing the method developed in the previous works of this series. The CDV within which dynamics can be asymptotically closed have been found which correspond to the sound wave modes. However, these CDV are not sufficient to account for all the anomalies (divergences) of the transport coefficients. Rapid random motions also contribute to the divergences through various transport modes, thus leading to partial violation of the dynamical scaling law. The results for the anomalies essentially confirm the work of Kadanoff and Swift. The sound propagation is also studied. In some high frequency regions the sound speed is found to exceed the usual adiabatic value by a constant factor, and the attenuation does not depend upon frequency. The case of the critical mixture is also briefly discussed.

DOI
10.1143/ptp.40.930

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Physical review. B, Condensed matter•Aug 1, 1979•62 citations
Dynamics of critical fluctuations associated with the liquid-gas transition are studied employing the method developed in the previous works of this series. The CDV within which dynamics can be asymptotically closed have been found which correspond to the sound wave modes. However, these CDV are not sufficient to account for all the anomalies (divergences) of the transport coefficients. Rapid random motions also contribute to the divergences through various transport modes, thus leading to partial violation of the dynamical scaling law. The results for the anomalies essentially confirm the work of Kadanoff and Swift. The sound propagation is also studied. In some high frequency regions the sound speed is found to exceed the usual adiabatic value by a constant factor, and the attenuation does not depend upon frequency. The case of the critical mixture is also briefly discussed.
The nature of our approach to the dynamics of critical fluctuations initiated in earlier works of this series is studied by applying it to the kinetic Ising model and anisotropic Heisenberg ferromagnets, and by including the Hamiltonian density in constructing the critical dynamical variables (CDV). In the isotropic Heisenberg magnets as well as in the anisotropic Heisenberg ferromagnets where the transverse components of spins first become critical (Matsubara-Matsuda model), the dynamics of the CDV can be asymptotically closed within themselves (that is, the energy transfer among the CDV modes dominates that between the CDV modes and other microscopic modes) if the CDV is constructed from spin densities alone. The dynamical scaling law (DSL) then holds among phenomena associated with the spin densities. For other cases studied, the dynamics cannot be closed within the chosen CDV, and then the DSL may fail among the phenomena associated with the CDV. A tentative criterion is given for the CDV where the DSL holds. In particular, the characteristic frequency in the Matsubara-Matsuda model behaves as κ3/2f with κ the inverse correlation range of spin fluctuations, which agrees with that found for the liquid helium near the λ transition by Ferrell et al. using the DSL. However, the thermal conductivities of the magnetic systems studied remain finite at the critical points. Various equal-time correlations that involve certain short wave length spin fluctuations are studied in Appendices.
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 fff is the dimensionless frequency and $\ensuremath{\epsilon}$ the dimensionless temperature distance from the critical point.
The dynamics of critical fluctuations is treated theoretically by extending the usual macroscopic dynamical variables to include all other dynamical variables that could exhibit the critical slowing-down. The equations of motion for these generalized macroscopic dynamical variables are derived with the help of Mori's general theory of Brownian motion, which involve the “first moment frequency matrix” Ωq and the microscopic damping for these new dynamical variables, where q is the wave vector characterizing macroscopic inhomogeneity. The latter involves only the dynamical variables that perform rapid random motions and is disregarded, or is disposed of by means of a dimensional argument. Then the dynamics of critical fluctuations is completely governed by Ωq which is expressed in terms of equaltime correlations of long wave-length fluctuations only and allows the analysis with the help of the Widom-Kadanoff static scaling laws. The theory is applied to the problem of anomalous spin relaxations in ferro and antiferromagnets in the paramagnetic region, and the inverse characteristic times near the transition points are found to behave asymptotically as κ(5-η)/2f(q/κ) and κ3/2f(q/κ), respectively, where κ is the inverse correlation range of spin fluctuations and η the parameter measuring the deviation of spin pair correlation from the Ornstein-Zernike form. Finally a close analogy is pointed out between the present treatment of critical fluctuations are the theory of hydrodynamic turbulence.
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{T}_{c}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{T}_{c}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.
The field-theoretic description of dynamical critical effects of the influence of disorder on acoustic anomalies near the temperature of the second-order phase transition is considered for three-dimensional Ising-like systems. Calculations of the sound attenuation in pure and dilute Ising-like systems near the critical point are presented. The dynamical scaling function for the critical attenuation coefficient is calculated. The influence of quenched disorder on the asymptotic behaviour of the critical ultrasonic anomalies is discussed.
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=2z=2z=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 kkk 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{T}_{c}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 zzz. All of these apparently different transformations lead to the same modified dynamical scaling relations.