Statistical Topics and Stochastic Models for Dependent Data with Applications

9Dynamics of the Group Entropy Maximization Processes and of the Relative Entropy Group Minimization Processes Based on the Speed-gradient Principle

The maximum or minimum entropy principles answer the question, “Where does a system tend to?”, while the speed-gradient (SG) principle answers the question, “How does a system reach its steady-state mode?”. By combining these principles, the equilibrium state distribution of a non-stationary process, which follows a maximum entropy principle, and the dynamic equations of the process are obtained. SG principle is a local principle that investigates the transient modes of a physical/mechanical system, either stationary or non-stationary. In this chapter, we consider non-stationary systems, which we assume have steady-state distributions, given by the maximum entropy principle (various entropies are investigated).

Some other applications of the SG principle include the construction of statistical dynamics equations for finite systems of particles that obey the principle of maximum entropy, molecular dynamics based on the equations of classical mechanics, hock-wave transient processes in condensed matter, the Rayleigh problem on the time evolution of the shear medium flow along the plane rigid surface and microscopic description of the time evolution of a system’s internal structure.

In this chapter, we investigate the non-stationary process that obeys the group entropy maximization principle or the relative entropy group minimization ...