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Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations

In this work we propose a model for open Markov chains that can be interpreted as a system of non-interacting particles evolving according to the rules of a Markov chain. The number of particles in the system is not constant, because we allow the particles to arrive or leave the state space accordin...

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Autor principal: Salgado-García, Raúl
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2021
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7927112/
https://www.ncbi.nlm.nih.gov/pubmed/33672246
http://dx.doi.org/10.3390/e23020256
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author Salgado-García, Raúl
author_facet Salgado-García, Raúl
author_sort Salgado-García, Raúl
collection PubMed
description In this work we propose a model for open Markov chains that can be interpreted as a system of non-interacting particles evolving according to the rules of a Markov chain. The number of particles in the system is not constant, because we allow the particles to arrive or leave the state space according to prescribed protocols. We describe this system by looking at the population of particles on every state by establishing the rules of time-evolution of the distribution of particles. We show that it is possible to describe the distribution of particles over the state space through the corresponding moment generating function. This description is given through the dynamics ruling the behavior of such a moment generating function and we prove that the system is able to attain the stationarity under some conditions. We also show that it is possible to describe the dynamics of the two first cumulants of the distribution of particles, which in some way is a simpler technique to obtain useful information of the open Markov chain for practical purposes. Finally we also study the behavior of the time-dependent correlation functions of the number of particles present in the system. We give some simple examples of open chains that either, can be fully described through the moment generating function or partially described through the exact solution of the cumulant dynamics.
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spelling pubmed-79271122021-03-04 Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations Salgado-García, Raúl Entropy (Basel) Article In this work we propose a model for open Markov chains that can be interpreted as a system of non-interacting particles evolving according to the rules of a Markov chain. The number of particles in the system is not constant, because we allow the particles to arrive or leave the state space according to prescribed protocols. We describe this system by looking at the population of particles on every state by establishing the rules of time-evolution of the distribution of particles. We show that it is possible to describe the distribution of particles over the state space through the corresponding moment generating function. This description is given through the dynamics ruling the behavior of such a moment generating function and we prove that the system is able to attain the stationarity under some conditions. We also show that it is possible to describe the dynamics of the two first cumulants of the distribution of particles, which in some way is a simpler technique to obtain useful information of the open Markov chain for practical purposes. Finally we also study the behavior of the time-dependent correlation functions of the number of particles present in the system. We give some simple examples of open chains that either, can be fully described through the moment generating function or partially described through the exact solution of the cumulant dynamics. MDPI 2021-02-23 /pmc/articles/PMC7927112/ /pubmed/33672246 http://dx.doi.org/10.3390/e23020256 Text en © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Salgado-García, Raúl
Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
title Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
title_full Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
title_fullStr Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
title_full_unstemmed Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
title_short Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
title_sort open markov chains: cumulant dynamics, fluctuations and correlations
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7927112/
https://www.ncbi.nlm.nih.gov/pubmed/33672246
http://dx.doi.org/10.3390/e23020256
work_keys_str_mv AT salgadogarciaraul openmarkovchainscumulantdynamicsfluctuationsandcorrelations