Stochastic Processes of Macroscopic Variables

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Abstract
Possible types of the Markov processes of macroscopic variables, including nonlinear fluctuations in open systems far from equilibrium, are explored on the basis of the scale invariance of macroscopic features. This is done by using a scaling method for asymptotic evaluation proposed previously. It is shown on an assumption that even in a many-variable system with multiple sets of scaling exponents, its asymptotic stochastic behavior is characterized by two parameters; a time scaling exponent of the fluctuation drift coefficients θh and that of the generalized diffusion coefficients σ. Then the asymptotic form of the stochastic equation for the probability distribution turns out to be of a generalized diffusion type if θh>σ, of a generalized Fokker-Planck type if θh=σ, and of a continuity type if θh≪σ. It is shown that θh≦σ for internal fluctuating forces for which a fluctuation-dissipation theorem holds. New relations between fluctuations in steady states and irreversible macroscopic motions are derived. Some features of the normal (αµ≪βµ), critical (αµ=βµ) and turbulent (αµ>βµ) fluctuations are also clarified from this new point of view.

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