Replica symmetry breaking in the `small world' spin glass

We apply the cavity method to a spin glass model on a `small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by M\'ezard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and $\pm J$ long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.Comment: 21 pages, 3 figure

Spin models on random graphs with controlled topologies beyond degree constraints

We study Ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution $p(k)$ can be chosen arbitrarily, but which allows for further fine-tuning of the topology via preferential attachment of edges on the basis of an arbitrary function Q(k,k') of the degrees of the vertices involved. We solve these models using finite connectivity equilibrium replica theory, within the replica symmetric ansatz. In our ensemble of graphs, phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution, but also on the choice made for Q(k,k'). The increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real-world interacting particle systems by spin systems on suitably defined random graphs.Comment: 21 pages, 4 figures, submitted to J Phys

Slowly evolving random graphs II: Adaptive geometry in finite-connectivity Hopfield models

We present an analytically solvable random graph model in which the connections between the nodes can evolve in time, adiabatically slowly compared to the dynamics of the nodes. We apply the formalism to finite connectivity attractor neural network (Hopfield) models and we show that due to the minimisation of the frustration effects the retrieval region of the phase diagram can be significantly enlarged. Moreover, the fraction of misaligned spins is reduced by this effect, and is smaller than in the infinite connectivity regime. The main cause of this difference is found to be the non-zero fraction of sites with vanishing local field when the connectivity is finite.Comment: 17 pages, 8 figure

Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices

We calculate equilibrium solutions for Ising spin models on `small world' lattices, which are constructed by super-imposing random and sparse Poissonian graphs with finite average connectivity c onto a one-dimensional ring. The nearest neighbour bonds along the ring are ferromagnetic, whereas those corresponding to the Poisonnian graph are allowed to be random. Our models thus generally contain quenched connectivity and bond disorder. Within the replica formalism, calculating the disorder-averaged free energy requires the diagonalization of replicated transfer matrices. In addition to developing the general replica symmetric theory, we derive phase diagrams and calculate effective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random sparse long-range bonds (i.e. `small world' spin-glasses).Comment: 22 pages, LaTeX, IOP macros, eps figure

Dynamical replica analysis of processes on finitely connected random graphs II: Dynamics in the Griffiths phase of the diluted Ising ferromagnet

We study the Glauber dynamics of Ising spin models with random bonds, on finitely connected random graphs. We generalize a recent dynamical replica theory with which to predict the evolution of the joint spin-field distribution, to include random graphs with arbitrary degree distributions. The theory is applied to Ising ferromagnets on randomly diluted Bethe lattices, where we study the evolution of the magnetization and the internal energy. It predicts a prominent slowing down of the flow in the Griffiths phase, it suggests a further dynamical transition at lower temperatures within the Griffiths phase, and it is verified quantitatively by the results of Monte Carlo simulations.Comment: 30 pages, 4 figures, submitted to J.Phys.

Warp propagation in astrophysical discs

Astrophysical discs are often warped, that is, their orbital planes change with radius. This occurs whenever there is a non-axisymmetric force acting on the disc, for example the Lense-Thirring precession induced by a misaligned spinning black hole, or the gravitational pull of a misaligned companion. Such misalignments appear to be generic in astrophysics. The wide range of systems that can harbour warped discs - protostars, X-ray binaries, tidal disruption events, quasars and others - allows for a rich variety in the disc's response. Here we review the basic physics of warped discs and its implications.Comment: To be published in Astrophysical Black Holes by Haardt et al., Lecture Notes in Physics, Springer 2015. 19 pages, 2 figure

The Economic Resource Receipt of New Mothers

U.S. federal policies do not provide a universal social safety net of economic support for women during pregnancy or the immediate postpartum period but assume that employment and/or marriage will protect families from poverty. Yet even mothers with considerable human and marital capital may experience disruptions in employment, earnings, and family socioeconomic status postbirth. We use the National Survey of Families and Households to examine the economic resources that mothers with children ages 2 and younger receive postbirth, including employment, spouses, extended family and social network support, and public assistance. Results show that many new mothers receive resources postbirth. Marriage or postbirth employment does not protect new mothers and their families from poverty, but education, race, and the receipt of economic supports from social networks do

Measurements of MeV photon flashes in petawatt laser experiments

Planar targets illuminated by the Petawatt laser system emit directed beams of photons with energies of MeVs. The laser pulses have durations of 0.5 or 5 psec, on target energies in excess of 100 joules, and focal-spot sizes that vary from 10 to 100 µm, producing peak intensities greater than 1019 watts/cm;2. Arrays of PIN diodes, dosimeters and nuclear-activation detectors measure the angular distributions of photons with energies greater than 0.5 MeV. The PIN diodes, with 1 cm2 by 500-µm sensitive volume, are housed in lead pigs with 2.5-cm thick walls. Measured emission intensities have been as high as 5x1013 (gamma) MeV/steradian. The angular distributions are highly directed in forward directions, with significant variations on a shot-to-shot basis. Backward radiated intensities tend to be more than a decade lower than in forward direc

Laboratory observation of secondary shock formation ahead of a strongly radiative blast wave

High Mach number blast waves were created by focusing a laser pulse on a solid pin, surrounded by nitrogen or xenon gas. In xenon, the initial shock is strongly radiative, sending out a supersonic radiative heat wave far ahead of itself. The shock propagates into the heated gas, diminishing in strength as it goes. The radiative heat wave also slows, and when its Mach number drops to 2 with respect to the downstream plasma, the heat wave drives a second shock ahead of itself to satisfy mass and momentum conservation in the heat wave reference frame; the heat wave becomes subsonic behind the second shock. For some time both shocks are observed simultaneously. Eventually the initial shock dimimishes in strength so much that it can longer be observed, but the second shock continues to propagate long after this time. This sequence of events is a new phenomenon that has not previously been discussed in literature. Numerical simulation clarifies the origin of the second shock, and its position is consistent with an analytical estimate