Dynamics of Critical Fluctuations. I

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Abstract
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.

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