In this work, we study the dynamics of complex systems with time-dependent transition rates, focusing on p-adic analysis in modeling such systems. Starting from the master equation that governs the stochastic dynamics of a system with a large number of interacting components, we generalize it by p-adically parametrizing the metabasins to account for states that are organized in a fractal and hierarchical manner within the energy landscape. This leads to a not necessarily time homogeneous Markov process described by a time-dependent operator acting on an ultrametric space.
We prove well-posedness of the initial value problem and analyze the stochastic nature of the master equation with time-dependent transition-operator. We demonstrate how ultrametricity simplifies the description of intra-metabasin dynamics without increasing computational complexity.
We apply our theoretical framework to two scenarios: glass relaxation under rapid cooling and protein folding dynamics influenced by temperature variations.
In the glass relaxation model, we observe anomalous relaxation behavior where the dynamics slow down during cooling, with lasting effects depending on how drastic the temperature drop is. In the protein folding model, we incorporate temperature-dependent transition rates to simulate folding and unfolding processes across the melting temperature.