WIAS Preprint No. 1629, (2011)

A mathematical framework for general classical systems and time irreversibility as its consequence


  • Stephan, Holger
    ORCID: 0000-0002-6024-5355

2010 Mathematics Subject Classification

  • 82C03 60J25 60B05


  • statistical state, second law of thermodynamics, topological bidual space, Markov operator, Jensen's inequality




It is well known that important models in statistical physics like the Fokker-Planck equation satisfy an H-theorem, i.e., have a decreasing Lyapunov function (or increasing entropy). This illustrates a symmetry break in time and reflects the second law of thermodynamics. In this paper, we show that any physically reasonable classical system has to have this property. For this purpose, we develop an abstract mathematical framework based on the theory of compact topological spaces and convex analysis. Precisely, we show:
1) Any statistical state space can be described as the convex hull of the image of the canonical embedding of the bidual space of its deterministic state space (a compact topological Hausdorff space).
2) The change of any statistical state is effected by the adjoint of a Markov operator acting in the space of observables.
3) Any Markov operator satisfies a wide class of inequalities, generated by arbitrary convex functions. As a corollary, these inequalities imply a time monotone behavior of the solution of the corresponding evolution equations.
Moreover, due to the general abstract setting, the proof of the underlying inequalities is very simple and therefore illustrates, where time symmetry breaks: A model is time reversible for any states if and only if the corresponding Markov operator is a deterministic one with dense range.
In addition, the proposed framework provides information about the structure of microscopic evolution equations, the choice of the best function spaces for their analysis and the derivation of macroscopic evolution equations.

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