Macrosystems theory and its applications. Equilibrium models
type: Item
instance of: scholarly article
title: Macrosystems theory and its applications. Equilibrium models
zbMATH Open document ID: 0841.90087
published in: Lecture Notes in Control and Information Sciences
publication date: 1995-06-22T00:00:00Z
review text: The book is mainly concerned with systems in which stochastic behavior of the components is transformed into a determined behavior of the system as a whole. The equilibria of the macrosystems was investigated using mathematical modeling. The macrosystem structure is studied with the help of a stochastic model, thus the mapping \(\mathcal H\) which characterizes the ``input-output'' transformation is decomposed into \({\mathcal H}_1\) (``input-state'') -- which is decomposed further into \({\mathcal H}_{11}\) (``input-stochastic state'') and \({\mathcal H}_{12}\) (``stochastic state-state'') -- and \({\mathcal H}_2\) (``state-output''). The stationary states for homogeneous macrosystems are determined by means of the principle of entropy maximization. The models of the stationary states are based upon constrained extremum problems having entropy objective functions. For unconstrained macrosystems (i.e. with no influence from the outside) the study approaches three cases: (1) macrosystems having Fermi-states (i.e. each state in \(\sigma_n\) can be occupied by only one element with probability one); (2) macrosystems having Einstein-states, for which each state in \(\sigma_n\) can be occupied by an arbitrary number of elements; (3) macrosystems having Boltzmann-states, where the occupation numbers \(N_n\) are significantly less than the capacity \(G_n\) of the subset \(\sigma_n\). The morphological properties -- namely the existence of the maximum and its ``sharpness'' -- are interesting for determining the fluctuations from the most probable states (there is a study for each type of macrosystem). If the resources are unconstrained, the set \(\mathcal N\) contains all possible macrostates generated by the stochastic mechanism for distributing the elements among the states. If the resources are constrained, not all the macrostates from \(\mathcal N\) are feasible, but only a subset \(\mathcal D\) in \(\mathcal N\). The model of stationary states (MSS) is represented on t…
Mathematics Subject Classification ID: 90B99
Mathematics Subject Classification ID: 93A13
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 91B74
Mathematics Subject Classification ID: 90B30
Mathematics Subject Classification ID: 68T10
Mathematics Subject Classification ID: 90C90
zbMATH DE Number: 766959
zbMATH Keywords: stationary states for homogeneous macrosystems
zbMATH Keywords: entropy maximization
zbMATH Keywords: Fermi-states
zbMATH Keywords: Einstein-states
zbMATH Keywords: Boltzmann-states
zbMATH Keywords: probabilistic hierarchical structures
zbMATH Keywords: image reconstruction
author: Yuri S. Popkov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 766959