Time-Varying Energy Landscapes and Temperature paths: Dynamic Transition Rates in locally Ultrametric Complex Systems

TL;DR

This paper develops a $p$-adic ultrametric framework to model time-dependent transition rates in complex systems, capturing dynamic relaxation and folding behaviors in materials and biological systems.

Contribution

It introduces a $p$-adic parametrization of energy landscapes for time-varying transition rates, providing analytical solutions for complex system dynamics.

Findings

Observed anomalous relaxation slowing during rapid cooling.

Simulated folding-unfolding 'whiplash' effects across melting temperatures.

Demonstrated ultrametricity simplifies intra-metabasin dynamics.

Abstract

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. 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.