Open Markov Chains: Cumulant Dynamics, Fluctuations and Correlations
1Centro de Investigación en Ciencias-IICBA, Universidad Autónoma del Estado de Morelos, Avenida Universidad 1001, Colonia Chamilpa, Cuernavaca 62209, Mexico.
Entropy (Basel, Switzerland)
|March 6, 2021
Summary
We introduce a model for open Markov chains, describing particle systems with variable populations. This model uses moment generating functions and cumulants to analyze particle distribution dynamics and achieve stationarity.
Area of Science:
- Mathematical Physics
- Statistical Mechanics
- Probability Theory
Background:
- Markov chains are fundamental stochastic processes.
- Modeling systems with non-constant particle numbers presents challenges.
- Open systems require specialized frameworks to account for particle exchange.
Purpose of the Study:
- To develop a mathematical model for open Markov chains.
- To analyze the dynamics of particle distributions in such systems.
- To investigate methods for describing system stationarity and particle behavior.
Main Methods:
- Formulation of an open Markov chain model.
- Utilizing moment generating functions to describe particle distribution.
- Analysis of the time-evolution of particle populations.
- Derivation and analysis of cumulant dynamics.
- Study of time-dependent correlation functions.
Main Results:
- Demonstrated that particle distribution can be described by moment generating functions.
- Proved that the system can attain stationarity under specific conditions.
- Showcased the utility of analyzing the first two cumulants for practical insights.
- Investigated correlation functions for particle number dynamics.
- Provided examples of open chains solvable via moment generating functions or cumulant dynamics.
Conclusions:
- The proposed model effectively describes open Markov chains with variable particle numbers.
- Moment generating functions and cumulant dynamics offer powerful tools for analyzing these systems.
- The model provides a framework for understanding particle behavior and system stationarity in open probabilistic systems.
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