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Author: Stephan, Holger × Subject: Markov operator ×

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    A mathematical framework for general classical systems and time irreversibility as its consequence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Stephan, Holger
    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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    Positivity and polynomial decay of energies for square-field operators
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Stephan, Artur; Stephan, Holger
    We show that for a general Markov generator the associated square-field (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.