In understanding the improved confinement state, the exploration of statistical theory is urgent. This is because an improved confinement state is associated with various kinds of transitions, and the empirical database must be compared to the results from statistical theory. Transitions include those to establish improved states (e.g., L-H transition) and those to deteriorate barriers like neoclassical tearing mode (NTM) which is an obstacle to sustain the improved confinement state for a long period. We have developed a statistical theory to analyze the transition phenomena in far-nonequilibrium systems[1,2], and apply it to problems of the onsets of L-H transition and NTM.
First, a statistical model for the bifurcation of the radial electric field is analyzed in view of describing L-H transitions of tokamak plasmas. Noise in micro fluctuations is shown to lead to random changes of radial electric field. The probability density function for and the ensemble average of the radial electric field are obtained. The L-to-H and the H-to-L transition probabilities are calculated, and the effective phase limit is derived.[3] Ensemble average of heat flux is also obtained. Next, we apply this theoretical method to the problem of the stochastic trigger by microscopic turbulence for neoclassical tearing mode [4]. We formulate a Langevin equation for NTM as a stochastic equation in the presence of noise source induced by background fluctuations. Statistical properties of NTM perturbation, such as the probability density function (PDF), the rate of excitation, the average of amplitude, the boundary of phase, being the statistical (long time) average of transition conditions, and its formula are derived.

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