Anomalous fluctuation and relaxation in unstable systems

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The essential ideas of the scaling theory of transient phenomena proposed by the author for a single macrovariable near the instability point are extended to multi-macrovariables in nonequilibrium systems. The time region is divided into three regimes according to the scaling behavior of the fluctuating parts of the macrovariables. In the first regime, the fluctuation is Gaussian and it is described by the linearized stochastic equation (or linear Fokker-Planck equation). In the second regime, the fluctuation is non-Gaussian, but it is probabilistic or stochastic (not dynamical) in the sense that the stochastic nature comes from the probability distribution in the initial regime and that each representative motion is deterministic, namely a random force can be neglected asymptotically in the second regime. In the final regime, the fluctuation is again Gaussian. A fluctuation-enhancement theorem for multi-macrovariables is given, which states that the fluctuation becomes enhanced by the order of the system size Ω in the second regime, which is of order log Ω, if the initial system is located just at the unstable point. An anomalous fluctuation theorem for multi-macrovariables is also proven, which states that the fluctuation is anomalously enhanced in proportion to δ−2 at times of order log δ if the initial system deviates by δ from the unstable point.

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This work is partially financed by the Scientific Research Fund of the Ministry of Education.

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Suzuki, M. Anomalous fluctuation and relaxation in unstable systems. J Stat Phys 16, 477–504 (1977) doi:10.1007/BF01152285

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Key words

  • Macrovariable
  • multi-macrovariable
  • multimode
  • most probable path
  • variance
  • instability point
  • unstable system
  • fluctuation enhancement
  • anomalous fluctuation
  • relaxation
  • mode coupling
  • scaling property
  • scaling theory
  • Gaussian
  • non-Gaussian
  • linear, initial regime
  • nonlinear, second regime
  • nonequilibrium system
  • asymptotic evaluation
  • Ω-expansion
  • Fokker-Planck equation
  • Kramers-Moyal equation