Informationsverlust, abstrakte Entropie und die mathematische Beschreibung des zweiten Hauptsatzes der Thermodynamik

We intend to discuss the relations between the more abstract notion of entropy and the ideas of information using nonlinear partial differential equations as an example. Hyperbolic equations of the first order, where vanishing of the characteristics in a discontinuity may be interpreted as loss of initial information, will serve as illustration. The problem of non‐physical rarefaction shocks in discussed, and with Lax′ shock condition a first entropy condition is introduced. The approximation of a physical problem with friction by an inviscid problem leads to a demand for an entropy inequality characterizing the solutions of the model with friction in the limit, when friction vanishes. This demand for an additional condition, namely for an entropy inequality as mathematical pendant to the second fundamental law also dominates the numerics of the presented equations. It is only the fulfilment of a discrete entropy condition — i.e. of a discretized form of the second fundamental law — that provides for the convergence of finite difference procedures.