Ultrametric Diffusion, Exponential Landscapes, and the First Passage Time Problem

Torresblanca-Badillo, Anselmo; Zúñiga–Galindo, W. A.

doi:10.1007/s10440-018-0165-2

Cited by 31 publications

(19 citation statements)

References 41 publications

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“…The mechanism of diffusion of systems of type (1.7) can be understood in the context of p-adic diffusion. Furthermore, our results on p-adic diffusion equations are new, indeed, they correspond to the so called 'degenerate energy landscapes' in the terminology of Avetisov, Kozyrev et al In the case in which the kernel J N has the form J(|x − y| p ), the corresponding diffusion equations were studied in [44]. Operators similar to L have been also studied by Bendikov [8], [9] and Kozyrev [23].…”

Section: Introductionsupporting

confidence: 53%

“…The p-adic heat equations, which include equations of types (1.8) and (1.9), and their associated Markov processes has been studied intensively in the last thirty years due to physical motivations, see e.g. [4], [5], [7], [8], [20], [22], [24], [44], [45], [48], [51] and the references therein. A central paradigm in physics of complex systems (for instance proteins) asserts that the dynamics of such systems can be modeled as a random walk in the energy landscape of the system, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis

Zúñiga–Galindo

2019

Preprint

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This work aims to show that p-adic analysis is the natural tool to study, in a rigorous mathematical way, reaction-diffusion systems on networks and the corresponding Turing patterns. By embedding the graph attached to the system into the field of p-adic numbers, we construct a family of continuous p-adic versions of the original system. In this way, we are able to study the original system and to obtain a new p-adic continuous version of it, which corresponds to a 'mean-field approximation' of the original one. The existence and uniqueness of the Cauchy problems for all these system are established. We show that Turing criteria remains essentially the same as in the classical case. However, the properties of the emergent patterns are very different. The classical Turing patterns consisting of alternating domains do not occur in the network case, instead of this, several domains (clusters) occur. Multistability, that is, coexistence of a number of different patterns for the same parameters values occur.

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Section: Introductionsupporting

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis

Zúñiga–Galindo

2019

Preprint

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“…The final purpose of these works is to provide a p-adic analytical description of experiments on the kinetics of CO binding to myoglobin, which were carried out by the group of Frauenfelder more than 30 years ago. In this article we consider the protein dynamics from a different perspective, furthermore, the reaction-diffusion equations used in the above mentioned works involve the Vladimirov operator, while the reactiondiffusion equations used here involve a transition kernels, see [36].…”

Section: Introductionmentioning

confidence: 99%

“…The study of p-adic heat equations and the attached Markov processes is a relevant mathematical matter [5], [9], [11], [25], [36], [37], [39], [42], [43], [44], and the references therein. The study of p-adic heat equations on p-adic manifolds is an open research area.…”

Section: Introductionmentioning

confidence: 99%

Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks

Zúñiga-Galindo

2022

Preprint

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In this article we introduce the ultrametric networks which are p-adic continuous analogues of the standard Markov state models constructed using master equations. A p-adic transition network (or an ultrametric network) is a model of a complex system consisting of a hierarchical energy landscape, a Markov process on the energy landscape, and a master equation. The energy landscape consists of a finite number of basins. Each basin is formed by infinitely many network configurations organized hierarchically in a infinite regular tree. The transitions between the basins are determined by a transition density matrix, whose entries are functions defined on the energy landscape. The Markov process in the energy landscape encodes the temporal evolution of the network as random transitions between configurations from the energy landscape. The master equation describe the time evolution of the density of the configurations. We focus on networks where the transition rates between two different basins are constant functions, and the jumping process inside of each basin is controlled by a p-adic radial function. We solve explicitly the Cauchy problem for the master equation attached to this type of networks. The solution of this problem is the network response to a given initial concentration. If the Markov process attached to the network is conservative, the long term response of the network is controlled by Markov chain, If the process is not conservative the network has absorbing states. We define an absorbing time, which depends on the initial concentration, if this time is finite the network reaches an absorbing state in a finite time. We identify in the response, the terms that produced this fast response, we call them, the fast transition modes. The existence of the fast transition modes is a consequence of assumption that the energy landscape is ultrametric (hierarchical), and to the best our understanding this result cannot be obtained by using standard of Markov state models. Nowadays, it is widely accepted that protein native states are kinetic hubs that can be reached quickly from any other state. The existence of fast transition modes implies that certain states on an ultrametric network work as kinetic hubs.

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“…they are p-adic counterparts of the classical heat equations. For instance, J( x p ) = x γ p e − x p , with γ > −n, corresponds to an exponential landscape in the sense of [6], in this case, the fundamental solution of (1.5) is the transition density of a bounded right-continuous Markov process without second kind discontinuities, see [34], [14] and the references therein. As a consequence of the work of many people, among them, Vlamimirov, Volovich, Zelenov, Avetisov, Kozyrev, Kochubei, Khrennikov, Albeverio, and Zúñiga-Galindo, we have now a good theory of p-adic 'linear' reaction-diffusion equations which has emerged motivated by connections between p-adic analysis and models of complex systems.…”

Section: Introductionmentioning

confidence: 99%

Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems

Zúñiga–Galindo

2018

Nonlinearity

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We initiate the study of non-Archimedean reaction-ultradiffusion equations and their connections with models of complex hierarchic systems. From a mathematical perspective, the equations studied here are the p-adic counterpart of the integro-differential models for phase separation introduced by Bates and Chmaj. Our equations are also generalizations of the ultradiffusion equations on trees studied in the 80s by Ogielski, Stein, Bachas, Huberman, among others, and also generalizations of the master equations of the Avetisov et al. models, which describe certain complex hierarchic systems. From a physical perspective, our equations are gradient flows of non-Archimedean free energy functionals and their solutions describe the macroscopic density profile of a bistable material whose space of states has an ultrametric structure. Some of our results are p-adic analogs of some well-known results in the Archimedean setting, however, the mechanism of diffusion is completely different due to the fact that it occurs in an ultrametric space.2000 Mathematics Subject Classification. Primary 80A22, 45K05; Secondary 46S10.

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Ultrametric Diffusion, Exponential Landscapes, and the First Passage Time Problem

Cited by 31 publications

References 41 publications

Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis

Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis

Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks

Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems

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