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 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 dynamical theory of critical fluctuations put forward in a previous work is rephrased and extended so as to be applicable below the transition points as well. The theory is illustrated for the isotropic Heisenberg spin systems where the critical dynamical variables can be chosen in such a way as to reproduce the correct characteristic times governing the dynamics of critical fluctuations which were obtained by the dynamical scaling laws. In this connection the general forms of various equal time correlations of long wave length fluctuations have been obtained with the help of the scaling law equation of state. The theory is useful as a starting point for devising approximate schemes of evaluating time correlation functions of physical quantities of our interest, where at least the correct scaling properties of the characteristic times of the problem are retained. This is illustrated by deriving a new self-consistent equation for the time-displaced spin auto-correlation function, and the truncated moments for describing the part of the spectral line shape that is associated with critical fluctuations are introduced and 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.
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.
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.
The dynamical scaling theory of phase transitions begins below the phase transition and extrapolates the frequencies, by means of the temperature continuity at a finite wave number, through the critical point into the temperature region above the transition. Applied to the ferromagnet, this approach leads to the observed five-halves critical dispersion at the Curie point. In the paramagnetic region it yields the critical slowing down of spin diffusion with a critical exponent of one-third. An alternative approach beginning in the paramagnetic region has been developed from the basic statistical mechanics of spin interaction. In the present paper, we present a simplified version of this approach, based on the fluctuation-dissipation theorem. The relevant form of this theorem is easily derived along familiar lines and involves obtaining an expression for the spin current. The spin diffusi on coefficient is then expressed in terms of the ratio of the fluctuations in spin current aud spin density. The correlation function for the current fluctuations can be evaluated by factoring (decoupled-mode approximation). The results are shown to be equivalent of those obtained by the other methods, and a simple explanation is given for the rise of the scaling function outside of the hydrodynamic region.
Abstract Time evolution, near a phase transition in the critical domain of critical systems not far from equilibrium, using a Langevin-type evolution is studied. Typical quantities of interest are relaxation rates towards equilibrium, time-dependent correlation functions and transport coefficients. The main motivation for such a study is that, in systems in which the dynamics is local (on short time-scales, a modification of a dynamic variable has an influence only locally in space) when the correlation length becomes large, a large time-scale emerges, which characterizes the rate of time evolution. This phenomenon called critical slowing down leads to universal behaviour and scaling laws for time-dependent quantities. In contrast with the situation in static critical phenomena, there is no clean and systematic derivation of the dynamical equations governing the time evolution in the critical domain, because often the time evolution is influenced by conservation laws involving the order parameter, or other variables like energy, momentum, angular momentum, currents and so on. Indeed, the equilibrium distribution does not determine the driving force in the Langevin equation, but only the dissipative couplings are generated by the derivative of the equilibrium Hamiltonian, and directly related to the static properties. The purely dissipative Langevin equation specifically discussed, corresponding to static models like the f4 field theory and two-dimensional models. Renormalization group (RG) equations are derived, and dynamical scaling relations established.