Nontrivial Scaling Research Articles

Interactions of unparticles with standard model particles

We study interactions of unparticles $\mathcal{U}$ of dimension ${d}_{\mathcal{U}}$ due to Georgi with standard model (SM) fields through effective operators. The unparticles describe the low energy physics of a nontrivial scale invariant sector. Since unparticles come from beyond the SM physics, it is plausible that they transform as a singlet under the SM gauge group. This helps tremendously in limiting possible interactions. We analyze interactions of scalar $\mathcal{U}$, vector ${\mathcal{U}}^{\ensuremath{\mu}}$, and spinor ${\mathcal{U}}^{s}$ unparticles with SM fields and derivatives up to dimension four. Using these operators, we discuss different features of producing unparticles at an ${e}^{+}{e}^{\ensuremath{-}}$ collider and other phenomenologies. It is possible to distinguish different unparticles produced at an ${e}^{+}{e}^{\ensuremath{-}}$ collider by looking at various distributions of production cross sections.

Scaling property of flux fluctuations from random walks

We study dynamical scaling of flux fluctuation sigma(t) from the one-random-walker model on regular lattices and complex networks and compare it to the surface width W(t) of a corresponding growth model. On the regular lattices, we analytically show that sigma(t) undergoes a crossover from the nontrivial scaling regime to the trivial one by increasing time t, and we verify the results by numerical simulations. In contrast to the results on the regular lattices, sigma(t) does not show any crossover behavior on complex networks and satisfies the scaling relation sigma(t) approximately t(1/2) for any t. On the other hand, we show that W(t) of the corresponding model on complex networks has two different scaling regimes, W approximately t(1/2) for t<<N and W(t) approximately t for t>>N .

Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities

By means of analytical and numerical methods we analyze the phase diagram of polaritons in one-dimensional coupled cavities. We locate the phase boundary, discuss the behavior of the polariton compressibility and visibility fringes across the critical point, and find a nontrivial scaling of the phase boundary as a function of the number of atoms inside each cavity. We also predict the emergence of a polaritonic glassy phase when the number of atoms fluctuates from cavity to cavity.

Lepton flavour violation in unparticle physics

Recently H. Georgi has introduced an unparticle U in order to describe the low energy physics of a nontrivial scale invariant sector of an effective theory. In this work we have explored the phenomenology of an unparticle using the lepton flavour violating decay μ−→e−e+e−, and found that the branching ratio of this decay is strongly dependent on the scaling dimension.

Roughness exponent measurements for the central force model

We measure the roughness exponent for fracture profiles in the two-dimensional central force lattice model using different measurement methods. We find that the profiles are self-affine for a system with narrow disorders and that broader disorders introduces overhangs in the fracture surface leading to deviation from self-affinity for small length scales and to nontrivial finite size scaling.

Localization in disordered media, anomalous roughening, and coarsening dynamics of faceted surfaces

We study a surface growth model related to the Kardar-Parisi-Zhang equation for nonequilibrium kinetic roughening, but where the thermal noise is replaced by a static columnar disorder eta(x) . This model is one of the many representations of the problem of particle diffusion in trapping or amplifying disordered media. We find that probability localization in the latter translates into facet formation in the equivalent surface growth problem. Coarsening of the pattern can therefore be identified with the diffusion of the localization center. The emergent faceted structure gives rise to nontrivial scaling properties, including anomalous surface roughening in excellent agreement with an existing conjecture for kinetic roughening of faceted surfaces. In a wider context, our study sheds light onto the scaling properties in other systems displaying this kind of patterned surface.

Time-dependent couplings and crossover length scales in nonequilibrium surface roughening

We show that time-dependent couplings may lead to nontrivial scaling properties of the surface fluctuations of the asymptotic regime in nonequilibrium kinetic roughening models. Three typical situations are studied. In the case of a crossover between two different rough regimes, the time-dependent coupling may result in anomalous scaling for scales above the crossover length. In a different setting, for a crossover from a rough to either a flat or damping regime, the time-dependent crossover length may conspire to produce a rough surface, although the most relevant term tends to flatten the surface. In addition, our analysis sheds light into an existing debate in the problem of spontaneous imbibition, where time-dependent couplings naturally arise in theoretical models and experiments.

Uncertainty Assessment of Future Hydroclimatic Predictions: A Comparison of Probabilistic and Scenario-Based Approaches

Abstract During the last decade, numerous studies have been carried out to predict future climate based on climatic models run on the global scale and fed by plausible scenarios about anthropogenic forcing to climate. Based on climatic model output, hydrologic models attempt then to predict future hydrologic regimes at regional scales. Much less systematic work has been done to estimate climatic uncertainty and to assess the climatic and hydrologic model outputs within an uncertainty perspective. In this study, a stochastic framework for future climatic uncertainty is proposed, based on the following lines: 1) climate is not constant but rather varying in time and expressed by the long-term (e.g., 30 yr) time average of a natural process, defined on a finescale; 2) the evolution of climate is represented as a stochastic process; 3) the distributional parameters of a process, marginal and dependence, are estimated from an available sample by statistical methods; 4) the climatic uncertainty is the result of at least two factors, the climatic variability and the uncertainty of parameter estimation; 5) a climatic process exhibits a scaling behavior, also known as long-range dependence or the Hurst phenomenon; and 6) because of this dependence, the uncertainty limits of the future are affected by the available observations of the past. The last two lines differ from classical statistical considerations and produce uncertainty limits that eventually are much wider than those of classical statistics. A combination of analytical and Monte Carlo methods is developed to determine uncertainty limits for the nontrivial scaling case. The framework developed is applied with temperature, rainfall, and runoff data from a catchment in Greece, for which data exist for about a century. The uncertainty limits are then superimposed onto deterministic projections up to 2050, obtained for several scenarios and climatic models combined with a hydrologic model. These projections indicate a significant increase of temperature in the future, beyond uncertainty bands, and no significant change of rainfall and runoff as they lie well within uncertainty limits.

Unparticle Physics

I discuss some simple aspects of the low-energy physics of a nontrivial scale invariant sector of an effective field theory-physics that cannot be described in terms of particles. I argue that it is important to take seriously the possibility that the unparticle stuff described by such a theory might actually exist in our world. I suggest a scenario in which some details of the production of unparticle stuff can be calculated. I find that in the appropriate low-energy limit, unparticle stuff with scale dimension dU looks like a nonintegral number dU of invisible particles. Thus dramatic evidence for a nontrivial scale invariant sector could show up experimentally in missing energy distributions.

Low-temperature electron dephasing time in AuPd revisited

Ever since the first discoveries of the quantum-interference transport in mesoscopic systems, the electron dephasing times, τ ϕ , in the concentrated AuPd alloys have been extensively measured. The samples were made from different sources with different compositions, prepared by different deposition methods, and various geometries (1D narrow wires, 2D thin films, and 3D thick films) were studied. Surprisingly, the low-temperature behavior of τ ϕ inferred by different groups over two decades reveals a systematic correlation with the level of disorder of the sample. At low temperatures, where τ ϕ is (nearly) independent of temperature, a scaling τ ϕ max ∝ D - α is found, where τ ϕ max is the maximum value of τ ϕ measured in the experiment, D is the electron diffusion constant, and the exponent α is close to or slightly larger than 1. We address this nontrivial scaling behavior and suggest that the most possible origin for this unusual dephasing is due to dynamical structure defects, while other theoretical explanations may not be totally ruled out.

Network Analysis of the State Space of Discrete Dynamical Systems

We study networks representing the dynamics of elementary 1D cellular automata (CA) on finite lattices. We analyze scaling behaviors of both local and global network properties as a function of system size. The scaling of the largest node in-degree is obtained analytically for a variety of CA including rules 22, 54, and 110. We further define the path diversity as a global network measure. The coappearance of nontrivial scaling in both the hub size and the path diversity separates simple dynamics from the more complex behaviors typically found in Wolfram's class IV and some class III CA.

Novel scaling behavior of the Ising model on curved surfaces

We demonstrate the nontrivial scaling behavior of Ising models defined on (i) a donut-shaped surface and (ii) a curved surface with a constant negative curvature. By performing Monte Carlo simulations, we find that the former model has two distinct critical temperatures at which both the specific heat C( T) and magnetic susceptibility χ( T) show sharp peaks. The critical exponents associated with the two critical temperatures are evaluated by the finite-size scaling analysis; the result reveals that the values of these exponents vary depending on the temperature range under consideration. In the case of the latter model, it is found that static and dynamic critical exponents deviate from those of the Ising model on a flat plane; this is a direct consequence of the constant negative curvature of the underlying surface.

Theory of the Collapsing Axisymmetric Cavity

We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we compute the local scaling exponent alpha=dlnh_{0}/dlnt' of the minimum radius h_{0} of the cavity, where t' is the time from collapse. The exponent alpha very slowly approaches a universal value according to alpha=1/2+1/[4 square root [-ln(t')]]. Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single nontrivial scaling exponent. Our predictions are confirmed by numerical simulations.

Gauge-gravity duality and thermalization of a boost-invariant perfect fluid

We derive the equation for the quasinormal modes corresponding to the scalar excitation of a black hole moving away in the fifth dimension. This geometry is the AdS/CFT dual of a boost-invariant expanding perfect fluid in N=4 SUSY Yang-Mills theory at large proper-time. On the gauge-theory side, the dominant solution of the equation describes the decay back to equilibrium of a scalar excitation of the perfect fluid. Its characteristic proper-time can be interpreted as a thermalization time of the perfect fluid, which is a universal (and numerically small) constant in units of the unique scale of the problem. This may provide a new insight on the short thermalization-time puzzle encountered in heavy-ion collision phenomenology. A nontrivial scaling behavior in proper-time is obtained which can be interpreted in terms of a slowly varying adiabatic approximation.

The two-well problem with surface energy

Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε &gt; 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 &lt; ζ1 &lt; 1 &lt; ζ2 &lt; ∞, .In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω &gt; 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

Crossover behavior in failure avalanches

Composite materials, with statistically distributed thresholds for breakdown of individual elements, are considered. During the failure process of such materials under external stress (load or voltage), avalanches consisting of simultaneous rupture of several elements occur, with a distribution D(Delta) of the magnitude Delta of such avalanches. The distribution is typically a power law D(Delta) proportional to Delta (-xi). For the systems we study here, a crossover behavior is seen between two power laws, with a small exponent xi in the vicinity of complete breakdown and a larger exponent xi for failures away from the breakdown point. We demonstrate this analytically for bundles of many fibers where the load is uniformly distributed among the surviving fibers. In this case xi=3/2 near the breakdown point and xi=5/2 away from it. The latter is known to be the generic behavior. This crossover is a signal of imminent catastrophic failure of the material. Near the breakdown point, avalanche statistics show nontrivial finite size scaling. We observe similar crossover behavior in a network of electric fuses, and find xi=2 near the catastrophic failure and xi=3 away from it. For this fuse model power dissipation avalanches show a similar crossover near breakdown.

Phase Separation of a Binary Fluid in the Inertia-Dominated Regime

The phase separation kinetics of a binary fluid is studied analytically through an effective one-fluid model with a random force spectrum determined self-consistently: the rate of kinetic energy injection by the random force is consistent with the droplet coalescence rate. Our detailed results for the rates of energy dissipation and kinetic energy decay are consistent with previous numerical studies. We find that simple nontrivial scaling, if any, is associated with a new universality class where the velocity length scale follows .

Supersymmetric Kerr–anti-de Sitter solutions

We prove the existence of one quarter supersymmetric type IIB configurations that arise as nontrivial scaling solutions of the standard five-dimensional Kerr--anti-de Sitter black holes by the explicit construction of its Killing spinors. This neutral, spinning solution is asymptotic to the static anti-de Sitter space-time with cosmological constant $\ensuremath{-}\frac{1}{{\ensuremath{\ell}}^{2}}$, it has two finite equal angular momenta ${J}_{1}=\ifmmode\pm\else\textpm\fi{}{J}_{2}$, mass $M=\frac{1}{\ensuremath{\ell}}(|{J}_{1}|+|{J}_{2}|)$ and a naked singularity. We also address the scaling limit associated with one-half supersymmetric solution with only one angular momentum.

Surface scaling analysis of hydrogels: From multiaffine to self-affine scaling

We show that smoothing of multiaffine surfaces that are generated by simulating a crosslinked polymer gel by a frustrated, triangular network of springs of random equilibrium lengths [G. M. Buendía, S. J. Mitchell, P. A. Rikvold, Phys. Rev. E, 66 (2002) 046119] changes the scaling behavior of the surfaces such that they become self-affine. The self-affine behavior is consistent with recent atomic force microscopy (AFM) studies of the surface structure of crosslinked polymer gels into which voids are introduced through templating by surfactant micelles [M. Chakrapani, S. J. Mitchell, D. H. Van Winkle, P. A. Rikvold, J. Colloid Interface Sci. 258 (2003) 186]. The smoothing process mimics the effect of the AFM tip that tends to flatten the soft gel surfaces. Both the experimental and the simulated surfaces have a non-trivial scaling behavior on small length scales, with a crossover to scale-independent behavior on large scales.

Finite-size scaling exponents and entanglement in the two-level BCS model

We analyze the finite-size properties of the two-level BCS model. Using the continuous unitary transformation technique, we show that nontrivial scaling exponents arise at the quantum critical point for various observables such as the magnetization or the spin-spin correlation functions. We also discuss the entanglement properties of the ground state through the concurrence which appears to be singular at the transition.