  The fundamental properties of 2-dimensional (2D) Ising system were formulated
using the Loewner theory. We focus on the role of the complexity measure of the
2D geometry, referred to as the Loewner entropy, to derive the
statistical-mechanical relations of the 2D Ising system by analyzing the
structure of the interface (i.e., the phase separation line). For the mixing
property of the discrete Loewner evolution, we assume that the Loewner driving
force ${\it\eta_s(n)}$ obtained from the interface has a stationary property,
where the autocorrelation function $\langle{\it\eta_s(0)\eta_s(n)}\rangle $
converges in the long-time limit. Using this fact, we reconstruct the
continuous Loewner evolution driven by the diffusion process whose increments
correspond to the sequence of ${\it\eta_s(n)}$, and the fractal dimension of
the generated curve was derived. We show that these formulations lead to a
novel expression of the Hamiltonian, grand canonical ensemble of the system,
which also are applicable for the non-equilibrium state of the system. In
addition, the relations on the central limit theorem (CLT) governing the local
fluctuation of the interface, the non-equilibrium free energy, and fluctuation
dissipation relation (FDR) were derived using the Loewner theory. The present
results suggest a possible form of the complexity-based theory of the 2D
statistical mechanical systems that is applicable for the non-equilibrium
states.
