On the local dynamics of polynomial difference equations with fading stochastic perturbations

We examine the stability-instability behaviour of a polynomial difference equa- tion with state-independent, asymptotically fading stochastic perturbations. We find that the set of initial values can be partitioned into a stability region, an instability region, and a region of unknown dynamics that is in some sense \small". In the ¯rst two cases, the dynamic holds with probability at least 1 ¡ °, a value corresponding to the statistical notion of a confidence level. Aspects of an equation with state-dependent perturbations are also treated. When the perturbations are Gaussian, the difference equation is the Euler-Maruyama dis- cretisation of an It^o-type stochastic differential equation with solutions displaying global a.s. asymptotic stability. The behaviour of any particular solution of the difference equation can be made consistent with the corresponding solution of the differential equation, with probability 1 ¡ °, by choosing the stepsize parameter sufficiently small. We present examples illustrating the relationship between h, ° and the size of the stability region

Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number $z$. $d$-dimensional lattices with $z=2d$ are at the threshold of mechanical stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios and auxetic elasticity, depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties, and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.Comment: 12 pages, 7 figure

Evaluation of a locally homogeneous flow model of spray combustion

A model of spray combustion which employs a second-order turbulence model was developed. The assumption of locally homogeneous flow is made, implying infinitely fast transport rates between the phase. Measurements to test the model were completed for a gaseous n-propane flame and an air atomized n-pentane spray flame, burning in stagnant air at atmospheric pressure. Profiles of mean velocity and temperature, as well as velocity fluctuations and Reynolds stress, were measured in the flames. The predictions for the gas flame were in excellent agreement with the measurements. The predictions for the spray were qualitatively correct, but effects of finite rate interphase transport were evident, resulting in a overstimation of the rate development of the flow. Predictions of spray penetration length at high pressures, including supercritical combustion conditions, were also completed for comparison with earlier measurements. Test conditions involved a pressure atomized n-pentane spray, burning in stagnant air at pressures of 3, 5, and 9 MPa. The comparison between predictions and measurements was fair. This is not a very sensitive test of the model, however, and further high pressure experimental and theoretical results are needed before a satisfactory assessment of the locally homogeneous flow approximation can be made

Constraints on the average magnetic field strength of relic radio sources 0917+75 and 1401-33 from XMM-Newton observations

We observed two relic radio sources, 0917+75 and 1401-33, with the XMM-Newton X-ray observatory. We did not detect any X-ray emission, thermal or non-thermal, in excess of the local background level from either target. This imposes new upper limits on the X-ray flux due to inverse Compton scattering of photons from the cosmic microwave background by relativistic electrons in the relic sources, and new lower limits on the magnetic field strength from the relative strength of the radio and X-ray emission. The combination of radio and X-ray observations provides a measure of the magnetic field independent of equipartition or minimum energy assumptions. Due to increasing sensitivity of radio observations, the known population of cluster relics has been growing; however, studies of non-thermal X-ray emission from relics remain scarce. Our study adds to the small sample of relics studied in X-rays. In both relics, our field strength lower limits are slightly larger than estimates of the equipartition magnetic field.Comment: 11 pages, 5 figures. Accepted by MNRA

Constraining f(R) Gravity as a Scalar Tensor Theory

We search for viable f(R) theories of gravity, making use of the equivalence between such theories and scalar-tensor gravity. We find that models can be made consistent with solar system constraints either by giving the scalar a high mass or by exploiting the so-called chameleon effect. However, in both cases, it appears likely that any late-time cosmic acceleration will be observationally indistinguishable from acceleration caused by a cosmological constant. We also explore further observational constraints from, e.g., big bang nucleosynthesis and inflation.Comment: 15 pages, 5 figure

Spin-spin correlators in Majorana representation

In the Majorana representation of a spin 1/2 we find an identity which relates spin-spin correlators to one-particle fermionic correlators. This should be contrasted with the straightforward approach in which two-particle (four-fermion) correlators need to be calculated. We discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit's state

Detection of microgauss coherent magnetic fields in a galaxy five billion years ago

Magnetic fields play a pivotal role in the physics of interstellar medium in galaxies, but there are few observational constraints on how they evolve across cosmic time. Spatially resolved synchrotron polarization maps at radio wavelengths reveal well-ordered large-scale magnetic fields in nearby galaxies that are believed to grow from a seed field via a dynamo effect. To directly test and characterize this theory requires magnetic field strength and geometry measurements in cosmologically distant galaxies, which are challenging to obtain due to the limited sensitivity and angular resolution of current radio telescopes. Here, we report the cleanest measurements yet of magnetic fields in a galaxy beyond the local volume, free of the systematics traditional techniques would encounter. By exploiting the scenario where the polarized radio emission from a background source is gravitationally lensed by a foreground galaxy at z = 0.439 using broadband radio polarization data, we detected coherent $\mu$G magnetic fields in the lensing disk galaxy as seen 4.6 Gyrs ago, with similar strength and geometry to local volume galaxies. This is the highest redshift galaxy whose observed coherent magnetic field property is compatible with a mean-field dynamo origin.Comment: 29 pages, 5 figures (including Supplementary Information). Published in Nature Astronomy on August 28, 201

Quantum Reciprocity Conjecture for the Non-Equilibrium Steady State

By considering the lack of history dependence in the non-equilibrium steady state of a quantum system we are led to conjecture that in such a system, there is a set of quantum mechanical observables whose retarded response functions are insensitive to the arrow of time, and which consequently satisfy a quantum analog of the Onsager reciprocity relations. Systems which satisfy this conjecture can be described by an effective Free energy functional. We demonstrate that the conjecture holds in a resonant level model of a multi-lead quantum dot.Comment: References revised to take account of related work on Onsager reciprocity in mesoscopics by Christen, and in hydrodynamics by Mclennan, Dufty and Rub