A constrained optimization problem in quantum statistical physics

In this paper, we consider the problem of minimizing quantum free energies under the constraint that the density of particles is fixed at each point of Rd, for any d $\ge$ 1. We are more particularly interested in the characterization of the minimizer, which is a self-adjoint nonnegative trace class operator, and will show that it is solution to a nonlinear self-consistent problem. This question of deriving quantum statistical equilibria is at the heart of the quantum hydrody-namical models introduced by Degond and Ringhofer. An original feature of the problem is the local nature of constraint, i.e. it depends on position, while more classical models consider the total number of particles in the system to be fixed. This raises difficulties in the derivation of the Euler-Lagrange equations and in the characterization of the minimizer, which are tackled in part by a careful parametrization of the feasible set

Small volume expansions for elliptic equations

This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form \nabla \cdot D^\eps \nabla u^\eps=0 or (-\Delta + q^\eps) u^\eps=0 with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order \eps^{2d}, where $d$ is dimension and \eps is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most \eps^{d+1} rather than the expected \eps^d. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.Comment: 42 page

Strategic Port Graph Rewriting: An Interactive Modelling and Analysis Framework

We present strategic portgraph rewriting as a basis for the implementation of visual modelling and analysis tools. The goal is to facilitate the specification, analysis and simulation of complex systems, using port graphs. A system is represented by an initial graph and a collection of graph rewriting rules, together with a user-defined strategy to control the application of rules. The strategy language includes constructs to deal with graph traversal and management of rewriting positions in the graph. We give a small-step operational semantics for the language, and describe its implementation in the graph transformation and visualisation tool PORGY.Comment: In Proceedings GRAPHITE 2014, arXiv:1407.767