Abstract
A new conceptual framework for the foundations of statistical mechanics starting from dynamics is presented. It is based on the classification and the study of invariants in terms of the concepts of our formulation of non-equilibrium statistical mechanics. A central role is played by thecollision operator. The asymptotic behaviour of a class of states is determined by the collisional invariants independently of the ergodicity of the system. For this class of states we have an approach to thermodynamical equilibrium.
We discuss the existence of classes of states which approach equilibrium. The complex microstructure of the phase space, as expressed by the weak stability concept which was introduced by Moser and others, plays here an essential role. The formalism that we develop is meaningful whenever the “dissipativity condition” for the collision operator is satisfied. Assuming the possibility of a weak coupling approximation, this is in fact true whenever Poincaré's theorem on the nonexistence of uniform invariants holds. In this respect, our formalism applies to few body problems and no transition to the thermodynamic limit is required.
Our approach leads naturally to a ‘classical theory of measurement’. In particular a precise meaning can now be given to ‘thermodynamic variables’ or to ‘macrovariables’ corresponding to a measurement in classical dynamics.
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Prigogine, I., Grecos, A. & George, C. On the relation of dynamics to statistical mechanics. Celestial Mechanics 16, 489–507 (1977). https://doi.org/10.1007/BF01229290
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DOI: https://doi.org/10.1007/BF01229290