Contribution of control theory to the problem of microscopic reversibility-macroscopic irreversibility

The problem of microscopic reversibility-macroscopic irreversibility appears in control theory via the control of a dynamic system containing a subsystem which is not in a state analogous to thermodynamic equilibrium (example: bunch of charged particles in an accelerator). In such a case the appropriate macroscopic description has to be established simultaneously with the control configuration. The macroscopic state of the non-equilibrium subsystem is defined implicitly by a Hamiltonian description valid on the microscopic level (single particles acted upon by collective forces in the example of an accelerator). The excess fine structure of the microscopic state, i.e. the complexity irrelevant for the control problem must be properly smoothed. The problem of proper smoothing coincides with a classical problem of statistical mechanics: determination of the master equation from a given system of Hamiltonian equations. After a short review of some basic assumptions and definitions of control theory, four particular dynamic systems of known microstructure are discussed. These four systems illustrate a complexity generating mechanism which, from a less precise point of view, appears to be stochastic. A real or apparent stochasticity is required for the application of statistical smoothing methods. The study of the illustrative examples suggests that the choice of correct smoothing methods is not governed by simple rules. (57 refs).