Time-dependent probability density functions, information geometry and entropy production in a stochastic prey–predator model of fusion plasmas

A stochastic, prey–predator model of the L–H transition in fusion plasma is investigated. The model concerns the regulation of turbulence by zonal and mean flow shear. Independent delta-correlated Gaussian stochastic noises are used to construct Langevin equations for the amplitudes of turbulence and zonal flow shear. We then find numerical solutions of the equivalent Fokker–Planck equation for the time- dependent joint probability distribution of these quantities. We extend the earlier studies [Kim and Hollerbach, Phys. Rev. Res. 2, 023077 (2020) and Hollerbach et al., Phys. Plasmas 27, 102301 (2020)] by applying different functional forms of the time-dependent external heating (input power), which is increased and then decreased in a symmetric fashion to study hysteresis. The hysteresis is examined through the probability distribution and statistical measures, which include information geometry and entropy. We find strongly non-Gaussian probabil- ity distributions with bimodality being a persistent feature across the input powers; the information length to be a better indicator of dis- tance to equilibrium than the total entropy. Both dithering transitions and direct L-–H transitions are (also) seen when the input power is stepped in time. By increasing the number of steps, we see less hysteresis (in the statistical measures) and a reduced probability of H-mode access; intermittent zonal flow shear is seen to have a role in the initial suppression of turbulence by zonal flow shear and stronger excitation of intermittent zonal flow shear for a faster changing input power.