Monte Carlo sampling for stochastic weight functions

Conventional Monte Carlo simulations are stochastic in the sense that the acceptance of a trial move is decided by comparing a computed acceptance probability with a random number, uniformly distributed between 0 and 1. Here, we consider the case that the weight determining the acceptance probability itself is fluctuating. This situation is common in many numerical studies. We show that it is possible to construct a rigorous Monte Carlo algorithm that visits points in state space with a probability proportional to their average weight. The same approach may have applications for certain classes of high-throughput experiments and the analysis of noisy datasets.D. F. acknowledges support by EPSRC Programme Grant EP/I001352/1 and EPSRC grant EP/I000844/1. K. J. S. acknowledges support by the Swiss National Science Foundation (Grant No. P2EZP2-152188 and No. P300P2-161078). S. M. acknowledges financial support from the Gates Cambridge Scholarship

Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming

In the late 1980s, Sam Edwards proposed a possible statistical-mechanical framework to describe the properties of disordered granular materials1. A key assumption underlying the theory was that all jammed packings are equally likely. In the intervening years it has never been possible to test this bold hypothesis directly. Here we present simulations that provide direct evidence that at the unjamming point, all packings of soft repulsive particles are equally likely, even though generically, jammed packings are not. Typically, jammed granular systems are observed precisely at the unjamming point since grains are not very compressible. Our results therefore support Edwards’ original conjecture. We also present evidence that at unjamming the configurational entropy of the system is maximal.S.M. acknowledges financial support by the Gates Cambridge Scholarship. K.J.S. acknowledges support by the Swiss National Science Foundation under Grant No. P2EZP2-152188 and No. P300P2-161078. D.F. acknowledges support by EPSRC Programme Grant EP/I001352/1 and EPSRC grant EP/I000844/1. K.R. and B.C. acknowledge the support of NSF-DMR 1409093 and the W. M. Keck Foundation

How to share underground reservoirs

Many resources, such as oil, gas, or water, are extracted from porous soils and their exploration is often shared among different companies or nations. We show that the effective shares can be obtained by invading the porous medium simultaneously with various fluids. Partitioning a volume in two parts requires one division surface while the simultaneous boundary between three parts consists of lines. We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops. While the spanning thread has fractal dimension ${1.55\pm0.03}$, the set of all lines has dimension ${1.69\pm0.02}$. The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality

Fracturing ranked surfaces

Discretized landscapes can be mapped onto ranked surfaces, where every element (site or bond) has a unique rank associated with its corresponding relative height. By sequentially allocating these elements according to their ranks and systematically preventing the occupation of bridges, namely elements that, if occupied, would provide global connectivity, we disclose that bridges hide a new tricritical point at an occupation fraction $p=p_{c}$, where $p_{c}$ is the percolation threshold of random percolation. For any value of $p$ in the interval $p_{c}< p \leq 1$, our results show that the set of bridges has a fractal dimension $d_{BB} \approx 1.22$ in two dimensions. In the limit $p \rightarrow 1$, a self-similar fracture is revealed as a singly connected line that divides the system in two domains. We then unveil how several seemingly unrelated physical models tumble into the same universality class and also present results for higher dimensions

Molecular Subtype Classification Is a Determinant of Non-Sentinel Lymph Node Metastasis in Breast Cancer Patients with Positive Sentinel Lymph Nodes

Background: Previous studies suggested that the molecular subtypes were strongly associated with sentinel lymph node (SLN) status. The purpose of this study was to determine whether molecular subtype classification was associated with nonsentinel lymph nodes (NSLN) metastasis in patients with a positive SLN. Methodology and Principal Findings: Between January 2001 and March 2011, a total of 130 patients with a positive SLN were recruited. All these patients underwent a complete axillary lymph node dissection. The univariate and multivariate analyses of NSLN metastasis were performed. In univariate and multivariate analyses, large tumor size, macrometastasis and high tumor grade were all significant risk factors of NSLN metastasis in patients with a positive SLN. In univariate analysis, luminal B subgroup showed higher rate of NSLN metastasis than other subgroup (P = 0.027). When other variables were adjusted in multivariate analysis, the molecular subtype classification was a determinant of NSLN metastasis. Relative to triple negative subgroup, both luminal A (P = 0.047) and luminal B (P = 0.010) subgroups showed a higher risk of NSLN metastasis. Otherwise, HER2 over-expression subgroup did not have a higher risk than triple negative subgroup (P = 0.183). The area under the curve (AUC) value was 0.8095 for the Cambridge model. When molecular subtype classification was added to the Cambridge model, the AUC value was 0.8475. Conclusions: Except for other factors, molecular subtype classification was a determinant of NSLN metastasis in patient

Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube?

A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here. Combining numerical simulations and rigorous results, we find off-critical scale-free behavior and a continuous transition at a critical density of holes that significantly differs from classical percolation.This is the author accepted manuscript. The final version is available from the American Physical Society via http://dx.doi.org/10.1103/PhysRevLett.116.05570