Growth-induced stability in modified SLE curve

Abstract: In this study, the non-equilibrium free energy corresponding to the curve generated by a modified stochastic Loewner evolution (SLE), which is driven by the Langevin equation, is theoretically investigated. Under certain conditions, we prove that the time derivative of the (generalized) free energy expressed by Kullback-Leibler divergence between the probability distributions of the curve and driving function has a positive value, indicating the negativity of Gibbs entropy production. In addition, it was implied that, in a certain restriction, the free energy can be expressed as a function of a Lyapunov-type exponent of the driving function. These results show a dissipative nature of conformal dynamics, and indicate the growth-induced stability of the modified SLE curve.