A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction of a few typos, to appear In Probability Theory and Related Field

Logarithmic Corrections to Extremal Black Hole Entropy from Quantum Entropy Function

We evaluate the one loop determinant of matter multiplet fields of N=4 supergravity in the near horizon geometry of quarter BPS black holes, and use it to calculate logarithmic corrections to the entropy of these black holes using the quantum entropy function formalism. We show that even though individual fields give non-vanishing logarithmic contribution to the entropy, the net contribution from all the fields in the matter multiplet vanishes. Thus logarithmic corrections to the entropy of quarter BPS black holes, if present, must be independent of the number of matter multiplet fields in the theory. This is consistent with the microscopic results. During our analysis we also determine the complete spectrum of small fluctuations of matter multiplet fields in the near horizon geometry.Comment: LaTeX file, 52 pages; v2: minor corrections, references adde

Functional tissue engineering of ligament healing

Ligaments and tendons are dense connective tissues that are important in transmitting forces and facilitate joint articulation in the musculoskeletal system. Their injury frequency is high especially for those that are functional important, like the anterior cruciate ligament (ACL) and medial collateral ligament (MCL) of the knee as well as the glenohumeral ligaments and the rotator cuff tendons of the shoulder. Because the healing responses are different in these ligaments and tendons after injury, the consequences and treatments are tissue- and site-specific. In this review, we will elaborate on the injuries of the knee ligaments as well as using functional tissue engineering (FTE) approaches to improve their healing. Specifically, the ACL of knee has limited capability to heal, and results of non-surgical management of its midsubstance rupture have been poor. Consequently, surgical reconstruction of the ACL is regularly performed to gain knee stability. However, the long-term results are not satisfactory besides the numerous complications accompanied with the surgeries. With the rapid development of FTE, there is a renewed interest in revisiting ACL healing. Approaches such as using growth factors, stem cells and scaffolds have been widely investigated. In this article, the biology of normal and healing ligaments is first reviewed, followed by a discussion on the issues related to the treatment of ACL injuries. Afterwards, current promising FTE methods are presented for the treatment of ligament injuries, including the use of growth factors, gene delivery, and cell therapy with a particular emphasis on the use of ECM bioscaffolds. The challenging areas are listed in the future direction that suggests where collection of energy could be placed in order to restore the injured ligaments and tendons structurally and functionally

Entanglement entropy of black holes

The entanglement entropy is a fundamental quantity which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff which regulates the short-distance correlations. The geometrical nature of the entanglement entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in 4 and 6 dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as 't Hooft's brick wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields which non-minimally couple to gravity is emphasized. The holographic description of the entanglement entropy of the black hole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.Comment: 89 pages; an invited review to be published in Living Reviews in Relativit