Dynamics of Macrovariables for Nonuniform Systems: Scaling Theory

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Abstract
A scale transformation of the nonequilibrium macroscopic system to larger similar systems is introduced to find kinetic equations for the evolution and fluctuation of the macrovariables. In the scale transformation, we postulate that the probability distribution for the fluctuation of the macroscopic degrees of freedom and the quantities determined by the microscopic degrees of freedom par unit volume are invariant. The characteristic length of the macroscopic state l, the macroscopic state variables yk and their fluctuation variables zk are transformed by lL = Ll, (L ≫1), ykL = L-αykandzkL = L-βzk, respectively. The probability distribution then takes the form P({ zklβ }, { ql }, Ω/ld, t/lθ), where q, Ω, d and t denote the wave vectors, volume, dimensionality and time, respectively. If α<β, then the master equation is reduced to a linearly generalized Fokker-Planck equation with time-dependent coefficients and the probability distribution is normal around the mean evolution. If αβ, then the nonlinear drift terms are important and a renormalization of kinetic coefficients must be done to determine the mean evolution. For the isotropic Heisenberg ferromagnets near the Curie point, α= β= (d - 2 + η)/2 and θ= (d + 2 - η)/2, where η is the correlation critical exponent. For the isotropic homogeneous turbulence, α= 1, β= -1/3 and θ= 2/3, where Kolmogorov's spectrum is assumed. For example, this indicates that the turbulent viscosity has the form q-4/3V(ωq-2/3), ω being the frequency.

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