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

A note on stochastic Schr\"odinger equations with fractional multiplicative noise

This work is devoted to non-linear stochastic Schr\"odinger equations with multiplicative fractional noise, where the stochastic integral is defined following the Riemann-Stieljes approach of Z\"ahle. Under the assumptions that the initial condition is in the Sobolev space H^q(\Rm^n) for a dimension $n$ less than three and $q$ an integer greater or equal to zero, that the noise is a $Q-$fractional Brownian motion with Hurst index $H\in(1/2,1)$ and spatial regularity H^{q+4}(\Rm^n), as well as appropriate hypotheses on the non-linearity, we obtain the local existence of a unique pathwise solution in \calC^0(0,T,H^q(\Rm^n)) \cap \calC^{0,\gamma}(0,T,H^{q-2}(\Rm^n)), for any $\gamma \in [0,H)$. Contrary to the parabolic case, standard fixed point techniques based on the mild formulation of the SPDE cannot be directly used because of the weak smoothing in time properties of the Schr\"odinger semigroup. We follow here a different route and our proof relies on a change of phase that removes the noise and leads to a Schr\"odinger equation with a magnetic potential that is not differentiable in time

An inverse problem in quantum statistical physics

International audienceWe address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

International audienceThis work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the white-noise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô-Schrödinger equation, that is a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence