Two Types of Discontinuous Percolation Transitions in Cluster Merging Processes

Percolation is a paradigmatic model in disordered systems and has been applied to various natural phenomena. The percolation transition is known as one of the most robust continuous transitions. However, recent extensive studies have revealed that a few models exhibit a discontinuous percolation transition (DPT) in cluster merging processes. Unlike the case of continuous transitions, understanding the nature of discontinuous phase transitions requires a detailed study of the system at hand, which has not been undertaken yet for DPTs. Here we examine the cluster size distribution immediately before an abrupt increase in the order parameter of DPT models and find that DPTs induced by cluster merging kinetics can be classified into two types. Moreover, the type of DPT can be determined by the key characteristic of whether the cluster kinetic rule is homogeneous with respect to the cluster sizes. We also establish the necessary conditions for each type of DPT, which can be used effectively when the discontinuity of the order parameter is ambiguous, as in the explosive percolation model.Comment: 9 pages, 6 figure

Cluster aggregation model for discontinuous percolation transition

The evolution of the Erd\H{o}s-R\'enyi (ER) network by adding edges can be viewed as a cluster aggregation process. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel $K_{ij}\sim ij$, where $ij$ is the product of the sizes of two merging clusters. Here, we study more general cases in which $K_{ij}$ is sub-linear as $K_{ij}\sim (ij)^{\omega}$ with $0 \le \omega < 1/2$; we find that the percolation transition (PT) is discontinuous. Moreover, PT is also discontinuous when the ER dynamics evolves from proper initial conditions. The rate equation approach for such discontinuous PTs enables us to uncover the mechanism underlying the explosive PT under the Achlioptas process.Comment: 5 pages, 5 figure

Quotient-Comprehension Chains

Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic, but also in probabilistic and classical logic. This relation is presented by a long series of examples, some of them easy, and some also highly non-trivial (esp. for von Neumann algebras). We have not yet identified a unifying theory. Nevertheless, the paper contributes towards such a theory by introducing the new quotient-and-comprehension perspective on measurement instruments, and by describing the examples on which such a theory should be built.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Classical Strongly Coupled QGP: VII. Energy Loss

We use linear response analysis and the fluctuation-dissipation theorem to derive the energy loss of a heavy quark in the SU(2) classical Coulomb plasma in terms of the $l=1$ monopole and non-static structure factor. The result is valid for all Coulomb couplings $\Gamma=V/K$, the ratio of the mean potential to kinetic energy. We use the Liouville equation in the collisionless limit to assess the SU(2) non-static structure factor. We find the energy loss to be strongly dependent on $\Gamma$. In the liquid phase with $\Gamma\approx 4$, the energy loss is mostly metallic and soundless with neither a Cerenkov nor a Mach cone. Our analytical results compare favorably with the SU(2) molecular dynamics simulations at large momentum and for heavy quark masses.Comment: 18 pages, 15 figures. v2: added references, changed title, replaced figures for Fig. 7, corrected typo

Super Jackstraws and Super Waterwheels

We construct various new BPS states of D-branes preserving 8 supersymmetries. These include super Jackstraws (a bunch of scattered D- or (p,q)-strings preserving supersymmetries), and super waterwheels (a number of D2-branes intersecting at generic angles on parallel lines while preserving supersymmetries). Super D-Jackstraws are scattered in various dimensions but are dynamical with all their intersections following a common null direction. Meanwhile, super (p,q)-Jackstraws form a planar static configuration. We show that the SO(2) subgroup of SL(2,R), the group of classical S-duality transformations in IIB theory, can be used to generate this latter configuration of variously charged (p,q)-strings intersecting at various angles. The waterwheel configuration of D2-branes preserves 8 supersymmetries as long as the `critical' Born-Infeld electric fields are along the common direction.Comment: 23 pages, 10 figure

New Regime of MHD Turbulence: Cascade Below Viscous Cutoff

In astrophysical situations, e.g. in the interstellar medium (ISM), neutrals can provide viscous damping on scales much larger than the magnetic diffusion scale. Through numerical simulations, we have found that the magnetic field can have a rich structure below the dissipation cutoff scale. This implies that magnetic fields in the ISM can have structures on scales much smaller than parsec scales. Our results show that the magnetic energy contained in a wavenumber band is independent of the wavenumber and magnetic structures are intermittent and extremely anisotropic. We discuss the relation between our results and the formation of the tiny-scale atomic structure (TSAS).Comment: ApJ Letters, accepted (Feb. 10, 2002; ApJ, 566, L...); 10 pages, 3 figure

Stress-energy Tensor Correlators in N-dim Hot Flat Spaces via the Generalized Zeta-Function Method

We calculate the expectation values of the stress-energy bitensor defined at two different spacetime points $x, x'$ of a massless, minimally coupled scalar field with respect to a quantum state at finite temperature $T$ in a flat $N$-dimensional spacetime by means of the generalized zeta-function method. These correlators, also known as the noise kernels, give the fluctuations of energy and momentum density of a quantum field which are essential for the investigation of the physical effects of negative energy density in certain spacetimes or quantum states. They also act as the sources of the Einstein-Langevin equations in stochastic gravity which one can solve for the dynamics of metric fluctuations as in spacetime foams. In terms of constitutions these correlators are one rung above (in the sense of the correlation -- BBGKY or Schwinger-Dyson -- hierarchies) the mean (vacuum and thermal expectation) values of the stress-energy tensor which drive the semiclassical Einstein equation in semiclassical gravity. The low and the high temperature expansions of these correlators are also given here: At low temperatures, the leading order temperature dependence goes like $T^{N}$ while at high temperatures they have a $T^{2}$ dependence with the subleading terms exponentially suppressed by $e^{-T}$. We also discuss the singular behaviors of the correlators in the $x'\rightarrow x$ coincident limit as was done before for massless conformal quantum fields.Comment: 23 pages, no figures. Invited contribution to a Special Issue of Journal of Physics A in honor of Prof. J. S. Dowke