Dynamics of correlations for integrable and non-integrable systems: A two levels formulation of laws of nature

Hamiltonian systems can be classified according Poincaré into integrable and non-integrable systems. On the other hand, our previous work introduced a formulation of dynamics based on the evolution of correlations. It is shown that for integrable systems this method is equivalent to the diagonalisation problem of the Hamiltonian for integrable systems but our method can be easily extended to non-integrable systems through analytic continuation. This leads to a description of unstable dressed excited states, as well as to excitations for interacting fields. The mechanism of the formation of dressed objects will be analysed in terms of two different time scales. The analogy with these dissipative structures will be emphasized. We may distinguish two levels in the formulation of laws of nature. The first is in terms of Hamiltonian dynamics, the second, necessary for classes of non-integrable systems, The first is in terms of Hamiltonian dynamics, the second, necessary for classes of non-integrable systems, arises from limiting processes and leads to time symmetry breaking and to an intrinsically probabilistic description. Kinetic theories and thermodynamics correspond to an extension of dynamics and not to some approximation scales.