We carry out a finite size scaling analysis of the jamming transition in frictionless bi-disperse soft core disks in two dimensions. We consider two different jamming protocols: (i) quench from random initial positions, and (ii) quasistatic shearing. By considering the fraction of jammed states as a function of packing fraction for systems with different numbers of particles, we determine the spatial correlation length critical exponent ν ≈ 1, and show that corrections to scaling are crucial for analyzing the data. We show that earlier numerical results yielding ν < 1 are due to the improper neglect of these corrections.
Recent experiments have shown that the superconducting energy gap in some cuprates is spatially inhomogeneous. Motivated by these experiments, and using exact diagonalization of a model d-wave Hamiltonian, combined with Monte Carlo simulations of a Ginzburg-Landau free energy functional, we have calculated the single-particle local density of states LDOS ͑ , r͒ of a model high-T c superconductor as a function of temperature. Our calculations include both quenched disorder in the pairing potential and thermal fluctuations in both phase and amplitude of the superconducting gap. Most of our calculations assume two types of superconducting regions: ␣ with a small gap and large superfluid density, and  with the opposite. If the  regions are randomly embedded in an ␣ host, the LDOS on the ␣ sites still has a sharp coherence peak at T = 0, but the  component does not, in agreement with experiment. An ordered arrangement of  regions leads to oscillations in the LDOS as a function of energy. The model leads to a superconducting transition temperature T c well below the pseudogap temperature T c0 and has a spatially varying gap at very low T, both consistent with experiments in underdoped Bi2212. Our calculated LDOS ͑ , r͒ shows coherence peaks for T Ͻ T c , which disappear for T Ͼ T c , in agreement with previous work considering phase but not amplitude fluctuations in a homogeneous superconductor. Well above T c , the gap in the LDOS disappears.
Starting from the single graphics processing unit (GPU) version of the Smoothed Particle Hydrodynamics (SPH) code DualSPHysics, a multi-GPU SPH program is developed for free-surface flows. The approach is based on a spatial decomposition technique, whereby different portions (sub-domains) of the physical system under study are assigned to different GPUs. Communication between devices is achieved with the use of Message Passing Interface (MPI) application programming interface (API) routines. The use of the sorting algorithm radix sort for inter-GPU particle migration and sub-domain "halo" building (which enables interaction between SPH particles of different subdomains) is described in detail. With the resulting scheme it is possible, on the one hand, to carry out simulations that could also be performed on a single GPU, but they can now be performed even faster than on one of these devices alone. On the other hand, accelerated simulations can be performed with up to 32 million particles on the current architecture, which is beyond the limitations of a single GPU due to memory constraints. A study of weak and strong scaling behaviour, speedups and efficiency of the resulting program is presented including an investigation to elucidate the computational bottlenecks. Last, possibilities for reduction of the effects of overhead on computational efficiency in future versions of our scheme are discussed.
Using Monte Carlo techniques, we study a simple model which exhibits a competition between superconductivity and other types of order in two dimensions. The model is a site-diluted XY model, in which the XY spins are mobile, and also experience a repulsive interaction extending to one, two, or many shells of neighbors. Depending on the strength and range of the repulsion and spin concentration, the spins arrange themselves into a remarkable variety of patterns at low temperatures T, including phase separation, checkerboard order, and straight or labyrinthine patterns of stripes, which sometimes show hints of nematic or smectic order. This pattern formation profoundly affects the superfluid density ␥. Phase separation tends to enhance ␥, checkerboard order suppresses it, and stripe formation increases the component of ␥ parallel to the stripes and reduces the perpendicular one. We verify that ␥͑T =0͒ is proportional to the effective conductance of a random conductance network whose conductances equal the couplings of the XY system. Possible connections between the model and real materials, such as single high-T c cuprate layers, are briefly discussed.
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