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

Ángel Morán Ledezma

TL;DR

The paper addresses the dynamics of complex systems with time-varying transition rates on hierarchically organized energy landscapes by deploying a $p$-adic, ultrametric parametrization of metabasins. It introduces a non-autonomous Markov framework with a time-dependent operator $\boldsymbol{W}(t)$ acting on a locally ultrametric space, establishes well-posedness and Markov properties, and leverages the Trotter–Kato decomposition to obtain analytic descriptions of the ultrametric component via Kozyrev wavelets. The authors demonstrate the approach through a two-basin model and show how temperature paths modulate intra- and inter-basin dynamics, yielding phenomena such as anomalous relaxation under rapid cooling and a whiplash effect in protein folding. The framework provides analytical insights and computational simplifications for relaxation processes in materials and biomolecules under dynamic environmental conditions, with potential applicability to broader hierarchical systems driven by time-dependent barriers.

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.

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

TL;DR

The paper addresses the dynamics of complex systems with time-varying transition rates on hierarchically organized energy landscapes by deploying a -adic, ultrametric parametrization of metabasins. It introduces a non-autonomous Markov framework with a time-dependent operator acting on a locally ultrametric space, establishes well-posedness and Markov properties, and leverages the Trotter–Kato decomposition to obtain analytic descriptions of the ultrametric component via Kozyrev wavelets. The authors demonstrate the approach through a two-basin model and show how temperature paths modulate intra- and inter-basin dynamics, yielding phenomena such as anomalous relaxation under rapid cooling and a whiplash effect in protein folding. The framework provides analytical insights and computational simplifications for relaxation processes in materials and biomolecules under dynamic environmental conditions, with potential applicability to broader hierarchical systems driven by time-dependent barriers.

Abstract

In this work, we study the dynamics of complex systems with time-dependent transition rates, focusing on -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 -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 -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.

Paper Structure

This paper contains 10 sections, 7 theorems, 81 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Time dependent transition rates and $p$-adic transition networks.
  3. The $p$-adic numbers and trees.
  4. Ultrametric dynamics and fractal energy landscapes
  5. Time-dependent $p$-adic operators and Markov processes.
  6. The effect of ultrametricity on the evolution process.
  7. A two basin model and characteristic relaxation.
  8. Time-dependent energy barriers and temperature
  9. Anomalous relaxation caused by fast cooling
  10. A protein protein folding example with dynamic transition rates.

Key Result

Proposition 1

Let $X$ a Banach space and for every $t>0$ let $A(t)$ be a bounded linear operator on $X$. If the function $t\mapsto A(t)$ is strongly continuous for $0\le t<T$ then for every initial condition $x\in X$, the attached initial value problem is well-posed.

Figures (11)

  • Figure 1: Tree representation of $G_3$ for $p=2$
  • Figure 2: Hierarchical (rugged) energy landscape where each minima is in one-to-one correspondence with an element of $G_3$ for $p=2$
  • Figure 3: Ultrametric energy landscape, where the infinite number of energy barriers follow a self-similar and hierarchical organization in sub-basins, all minima have the same energy level (degenerate).
  • Figure 4: Locally-ultramemtric landscape: The blue and green transitions are controlled by a constant $w_{I,J}$, meanwhile inside each meta-basin (red-arrows) the transitions follow a fractal and hierarchical organization controlled by a radial function $w_{I,I}(|\cdot|_p).$
  • Figure 5: Temperature path for temperature drop from $300K$ to $200K$
  • ...and 6 more figures

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Theorem 1
  • Remark 1
  • Definition 3
  • Proposition 2
  • Theorem 2
  • proof
  • ...and 6 more