A Loewner-Theoretic Approach to the Nonlinear Generalized Langevin Equation: The Role of Entropy in Colored Noise Environment

Abstract

In this study, the formal derivation of a one-dimensional nonlinear generalized Langevin equation is demonstrated using a decomposition method based on the conformal transformation governed by the chordal Loewner equation. Here, we used a modified Mori-Zwanzig method whose operator is substituted by that is derived from the discrete Loewner evolution. By this approach, the different types of fluctuation-dissipation relation (FDR) were reformulated using mathematical terms affected by the conformal maps. Dealing with a memory kernel that models the cell migration experiment, the numerical simulation was performed to obtain the specific scaling law of energy dissipation that is common among the two obtained types of FDRs. In addition, the concept of Loewner entropy is used for the estimation of the canonical ensemble in the colored noise environment throughout the theoretical analyses.