In this article, we study the small sphere limit of the WangYau quasi-local energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009, Commun Math Phys 288(3):919942, 2009). Given a point <i>p</i> in a spacetime <i>N</i>, we consider a canonical family of surfaces approaching <i>p</i> along its future null cone and evaluate the limit of the WangYau quasi-local energy. The evaluation relies on solving an optimal embedding equation whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in Chen etal. (Commun Math Phys 308(3):845863, 2011). Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at <i>p</i>. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding

We introduce a unified particle framework which integrates the phase-field method with multi-material simulation to allow modeling of both liquids and solids, as well as phase transitions between them. A simple elastoplastic model is used to capture the behavior of various kinds of solids, including deformable bodies, granular materials, and cohesive soils. States of matter or phases, particularly liquids and solids, are modeled using the non-conservative Allen-Cahn equation. In contrast, materials—made of different substances—are advected by the conservative Cahn-Hilliard equation. The distributions of phases and materials are represented by a phase variable and a concentration variable, respectively, allowing us to represent commonly observed fluid-solid interactions. Our multi-phase, multi-material system is governed by a unified Helmholtz free energy density. This framework provides the first method in computer graphics capable of modeling a continuous interface between phases. It is versatile and can be readily used in many scenarios that are challenging to simulate. Examples are provided to demonstrate the capabilities and effectiveness of this approach.

We prove the following stronger version of the positivity of quasi-local energy (mass) stated by Liu and Yau: the quasi-local energy of each connected component of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface and the boundary of the spacelike hypersurface is connected.

The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is $ 2\pi \hbar^2 a \varrho/m$ when the density of the gas is $ \varrho$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a > 0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.

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