Phenomenological approach to reduced description of large-scale systems. I: Description of the approach
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.
Markov processes with a finite number of states
scientific article; zbMATH DE number 4008186