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Author: Franz Dieter Fischer
Title of the contribution: Convexity of energy dissipation and stability of propagating interfaces
Abstract:
In equilibrium thermodynamics, equilibrium states are given by the minimum of a convex free energy function and taking into account suitable boundary conditions. A non-convex free energy function leads to phase transformations and in certain cases to the co-existence of several phases, where the classical Gibbs phase rule allows to construct the equilibrium properties of these phases (e.g., density or pressure). Within the framework of non-equilibrium thermodynamics, the maximization of energy dissipation (under suitable boundary conditions) can be used as an extremal principle to find stationary states. Extending this argument due to Onsager, we show that stationary states generally exist for convex energy dissipation functions and that non-convexity leads to metastable and unstable states. A geometric argument, similar in spirit to Gibbs double¯tangent construction, well known from equilibrium thermodynamics, yields the stability limits of stationary
states. This argument is applied here to study a classical problem of materials science, namely the motion of a grain boundary under the influence of solute drag. As a further case of a non-convex dissipation function the hydraulic jump is discussed as a somewhat simple example to which the Onsager concept is applied.
The mathematical challenge is how to find convex envelops of non-convex functions.
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