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Author: Stephan, Holger × Subject: Jensen’s inequality ×

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    A generalization of Lagrange's algebraic identity and connections with Jensen's inequality
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Niculescu, Constantin P.; Stephan, Holger
    We discuss a generalization of Lagrange's algebraic identity that provides valuable insights into the nature of Jensen's inequality and of many other inequalities of convexity.
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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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    Inequalities for Markov operators, majorization and the direction of time
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Stephan, Holger
    In this paper, we connect the following partial orders: majorization of vectors in linear algebra, majorization of functions in integration theory and the order of states of a physical system due to their temporal-causal connection. Each of these partial orders is based on two general inequalities for Markov operators and their adjoints. The first inequality compares pairs composed of a continuous function (observables) and a probability measure (statistical states), the second inequality compares pairs of probability measure. We propose two new definitions of majorization, related to these two inequalities. We derive several identities and inequalities illustrating these new definitions. They can be useful for the comparison of two measures if the Radon-Nikodym Theorem is not applicable. The problem is considered in a general setting, where probability measures are defined as convex combinations of the images of the points of a topological space (the physical state space) under the canonical embedding into its bidual. This approach allows to limit the necessary assumptions to functions and measures. In two appendices, the finite dimensional non-uniform distributed case is described, in detail. Here, majorization is connected with the comparison of general piecewise affine convex functions. Moreover, the existence of a Markov matrix, connecting two given majorizing pairs, is shown.