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The curvature dependence of the liquid–vapor surface tension is described in the limit of small curvatures by Tolman’s length. Measurements of it, either experimentally or in a simulation, have not yet given a clear idea of its magnitude, even its sign is being debated. Previous attempts to relate Tolman’s length to a pressure tensor have led to ill-defined expressions. From an analysis of the pressure difference over the interface of a liquid drop, a pressure tensor expression is obtained for Tolman’s length that does not suffer from the previously encountered inconsistencies. This pressure difference is studied in a simulation of liquid drops, leading to an estimate of Tolman’s length. It appears to be small and bounds are given on it.

An ensemble approach for force networks in static granular packings is developed. The framework is based on the separation of packing and force scales, together with an a priori flat measure in the force phase space under the constraints that the contact forces are repulsive and balance on every particle. In this paper we will give a general formulation of this force network ensemble, and derive the general expression for the force distribution P͑f͒. For small regular packings these probability densities are obtained in closed form, while for larger packings we present a systematic numerical analysis. Since technically the problem can be written as a noninvertible matrix problem (where the matrix is determined by the contact geometry), we study what happens if we perturb the packing matrix or replace it by a random matrix. The resulting P͑f͒'s differ significantly from those of normal packings, which touches upon the deep question of how network statistics is related to the underlying network structure. Overall, the ensemble formulation opens up a different perspective on force networks that is analytically accessible, and which may find applications beyond granular matter.

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We have adjusted the Density Matrix Renormalization method to handle two dimensional systems of limited width. The key ingredient for this extension is the incorporation of symmetries in the method. The advantage of our approach is that we can force certain symmetry properties to the resulting ground state wave function. Combining the results obtained for system sizes up-to 30 × 6 and finite size scaling, we derive the phase transition point and the critical exponent for the gap in the Ising model in a Transverse Field on a two dimensional square lattice.