Phenomenological approach to reduced description of large-scale systems. I: Description of the approach (Q1090647)

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.