Phenomenological approach to reduced description of large-scale systems. I: Description of the approach
type: Item
instance of: scholarly article
title: Phenomenological approach to reduced description of large-scale systems. I: Description of the approach
zbMATH Open document ID: 0621.93004
author: V. V. Marbukh
published in: Automation and Remote Control
publication date: 1986-01-01T00:00:00Z
review text: The probability evolution for Markov processes with a finite number of states is described by the Kolmogorov equation \(\dot p=Lp\) where p is a vector of a large dimension. An approximate solution is found in the form \(\tilde p(t)=\Omega (\Lambda (t))\) where \(\Omega\) is a given function of a small dimension vector \(\Lambda\). If \(<Y>\) is an expected value, \(<Y>=\Omega p\), then \(<\dot Y>=\Phi Lp\) and \(<\tilde Y>=\Phi \Omega (\Lambda)\), \(<\dot Y\tilde {\;}>=\Phi L\Omega (\Lambda)\) for the approximate distribution. The condition \(<\dot Y\tilde {\;}>=<\tilde Y\dot {\;}>\hat=\Phi \nabla \Omega {\dot \Lambda}\) yields the equation for \(\Lambda(t):\) \[ {\dot \Lambda}=R^{-1}(\Lambda)B(\Lambda),\quad R\hat=\Phi \nabla \Omega,\quad B=\Phi L\Omega. \] Some heuristic ways are shown for a rational choice of \(\Omega(\Lambda)\) which are connected with the traditional apparatus of statistical physics. Partially, it is suggested to find \(\Omega(\Lambda)\) from the principle of maximum entropy of p with respect to the limit distribution \(p^*\) or some approximation of that distribution under a condition that an expected value \(<Y>\) is fixed. The parameters \(\Lambda\) are associated with Lagrange multipliers of the optimal problem. The author promises to give some examples of applications in the second part of the paper.
Mathematics Subject Classification ID: 93A15
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 93E03
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 82B35
zbMATH DE Number: 4008186
zbMATH Keywords: canonical distribution
zbMATH Keywords: Markov processes with a finite number of states
zbMATH Keywords: approximate solution
zbMATH Keywords: maximum entropy
zbMATH Keywords: Lagrange multipliers
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4008186