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A stochastic continuum dynamical system based on a dimensional reduction procedure

Kaczmarek J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

1998

Vol. 46, nr 2

149-162

In this paper a concept of a three-scale modelling of behaviour of materials is extended to stochastic phenomena. The main element of the three-scale description consists in a dimensional reduction procedure. Consequently, evolution of probability densities obtained as a result of applying such a procedure is considered. Connections between the probability densities on different levels of description is determined. The starting point for obtaining the continuum model is a Hamiltonian dynamical system with stochastic initial conditions. The Hamiltonian system is predestined to describe behaviour of group of atoms which constitute a body. The dimensional reduction of such a system leads to fields on the continuum with considerable smaller dimension than the Hamiltonian system. An important role is assigned to internal state variables. With the help of these variables, a possibility of creation of stochastic operators on more averaged levels is shown. This is an important aspect of modelling phenomena related to complicated microstructure. Some similarities and differences of the presented model and statistical mechanics are discussed.