Statistics for transition of a plasma turbulence with multiple characteristic scales

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Subcritical transition of an inhomogeneous plasma where turbulences with different characteristic space-time scales coexist is analyzed with methods of statistical physics of turbulences. We derived the development equations of the probability density function (PDF) of the spectrum amplitudes of the fluctuating electro-static potential. By numerically solving the equations, the steady state PDFs were obtained. Although the subcritical transition is observed when the turbulent fluctuations are ignored, the PDF shows that the transition is smeared out by the turbulent fluctuations. It means that the approximation ignoring the turbulent fluctuations employed by traditional transition theories could overestimate the range where hysteresis is observed and statistical analyses are inevitably needed.