Time-varying energy landscapes and temperature paths: dynamic transition rates in locally ultrametric complex systems

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 sys-
tem 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 neces-
sarily time homogeneous Markov process described by a time-dependent oper-
ator 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 com-
plexity. We apply our theoretical framework to two scenarios: glass relaxation under rapid cooling and protein folding dynamics influenced by temperature vari-
ations. In the glass relaxation model, we observe anomalous relaxation behavior
where the dynamics slow down during cooling, with lasting effects depending on
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how drastic the temperature drop is. In the protein folding model, we incorporate
temperature-dependent transition rates to simulate folding and unfolding pro-
cesses across the melting temperature. Our results capture a ‘whiplash’ effect:
from an unfolded state, the system folds and then returns to an unfolded state
(which may differ from the initial one) in response to temperature changes. This
study demonstrates the effectiveness of p-adic parametrization and ultrametric
analysis in modeling complex systems with dynamic transition rate, providing
analytical solutions that improve our understanding of relaxation processes in
material and biological systems.