Microscopic derivation of a Schr\"odinger equation in dimension one with a nonlinear point interaction

We derive an effective equation for the dynamics of many identical bosons in dimension one, in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a three-body interaction. Assuming a simultaneous mean-field and short-range scaling, with the short-range proceeding slower than the mean-field, and choosing an initial fully condensed state, we prove propagation of chaos and obtain an effective one-particle Schr\"odinger equation with a nonlinearity concentrated at a point. More precisely, we prove convergence of one-particle density operators in the trace-class topology, and estimate the fluctuations as superexponential. This is the first derivation of the so-called nonlinear delta model, widely investigated in the last decades

Combined mean-field and semiclassical limits of large fermionic systems

We study the time dependent Schr\"odinger equation for large spinless fermions with the semiclassical scale $\hbar = N^{-1/3}$ in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schr\"odinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.Comment: Article has been peer-reviewed and publishe

Micromechanics-Based Homogenization of the Effective Physical Properties of Composites With an Anisotropic Matrix and Interfacial Imperfections

Micromechanics-based homogenization has been employed extensively to predict the effective properties of technologically important composites. In this review article, we address its application to various physical phenomena, including elasticity, thermal and electrical conduction, electric, and magnetic polarization, as well as multi-physics phenomena governed by coupled equations such as piezoelectricity and thermoelectricity. Especially, for this special issue, we introduce several research works published recently from our research group that consider the anisotropy of the matrix and interfacial imperfections in obtaining various effective physical properties. We begin with a brief review of the concept of the Eshelby tensor with regard to the elasticity and mean-field homogenization of the effective stiffness tensor of a composite with a perfect interface between the matrix and inclusions. We then discuss the extension of the theory in two aspects. First, we discuss the mathematical analogy among steady-state equations describing the aforementioned physical phenomena and explain how the Eshelby tensor can be used to obtain various effective properties. Afterwards, we describe how the anisotropy of the matrix and interfacial imperfections, which exist in actual composites, can be accounted for. In the last section, we provide a summary and outlook considering future challenges