Scaling laws and energy dissipation in the dynamics of SLE curve as a 1D turbulence model
In this study, we demonstrate the characteristics of the stochastic Loewner evolution (SLE) as a one-dimensional (1D) turbulence model. First, we show that the diffusion process $x(t)$ obtained by the time coordinate change in the SLE becomes the time dependent diffusion process that satisfies Richardson's law of turbulence.Subsequently, we derive the condition on the diffusivity parameter $κ$ such that the diffusion process $x(t)$ satisfies Kolmogorov's 4/5th law, which determines the relation between the mean energy dissipation and velocity in the particles of turbulence. Further mathematical analysis showed that this condition is seen as the condition on Loewner entropy $S_{\mathrm{Loew}}$, which includes the white Gaussian noise term $W_{s}$ in the Loewner driving function of the present model. In addition, the numerical simulations were performed to verify the scaling relations of this 1D turbulence model. These results suggest that the standard SLE becomes a beneficial model of 1D turbulence by introducing an appropriate time coordinate change and imposing the conditions on $κ$ and $S_{\mathrm{Loew}}$.