Exponent Delta Research Articles

Tsallis cosmology and its applications in dark matter physics with focus on IceCube high-energy neutrino data

In this paper we employ a recent proposal of C. Tsallis and formulate the first law of thermodynamics for gravitating systems in terms of the extensive but non-additive entropy. We pay a particular attention to an integrating factor for the heat one-form and show that in contrast to conventional thermodynamics it factorizes into thermal and entropic part. Ensuing two laws of thermodynamics imply Tsallis cosmology, which is then subsequently used to address the observed discrepancy between current bound on the Dark Matter relic abundance and present IceCube data on high-energy neutrinos. To resolve this contradiction we keep the conventional minimal Yukawa-type interaction between standard model and Dark Matter particles but replace the usual Friedmann field equations with Tsallis-cosmology-based modified Friedmann equations. We show that when the Tsallis scaling exponent delta sim 1.57 (or equivalently, the holographic scaling exponent alpha sim 3.13) the aforementioned discrepancy disappears.

On the infinite Borwein product raised to a positive real power

In this paper, we study properties of the coefficients appearing in the q-series expansion of prod _{nge 1}[(1-q^n)/(1-q^{pn})]^delta , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power delta . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For p=3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent delta is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the p=3 case.

The generalized second law of thermodynamics with Barrow entropy

We investigate the validity of the generalized second law of thermodynamics, applying Barrow entropy for the horizon entropy. The former arises from the fact that the black-hole surface may be deformed due to quantum-gravitational effects, quantified by a new exponent Delta . We calculate the entropy time-variation in a universe filled with the matter and dark energy fluids, as well as the corresponding quantity for the apparent horizon. We show that although in the case Delta =0, which corresponds to usual entropy, the sum of the entropy enclosed by the apparent horizon plus the entropy of the horizon itself is always a non-decreasing function of time and thus the generalized second law of thermodynamics is valid, in the case of Barrow entropy this is not true anymore, and the generalized second law of thermodynamics may be violated, depending on the universe evolution. Hence, in order not to have violation, the deformation from standard Bekenstein–Hawking expression should be small as expected.

Power-law bounds for critical long-range percolation below the upper-critical dimension

We study long-range Bernoulli percolation on {mathbb {Z}}^d in which each two vertices x and y are connected by an edge with probability 1-exp (-beta Vert x-yVert ^{-d-alpha }). It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if 0<alpha <d then there is no infinite cluster at the critical parameter beta _c. We give a new, quantitative proof of this theorem establishing the power-law upper bound Pβc(|K|≥n)≤Cn-(d-α)/(2d+α)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathbf {P}}_{\\beta _c}\\bigl (|K|\\ge n\\bigr ) \\le C n^{-(d-\\alpha )/(2d+\\alpha )} \\end{aligned}$$\\end{document}for every nge 1, where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality (2-eta )(delta +1)le d(delta -1) relating the cluster-volume exponent delta and two-point function exponent eta .

Observational constraints on Barrow holographic dark energy

We use observational data from Supernovae (SNIa) Pantheon sample, as well as from direct measurements of the Hubble parameter from the cosmic chronometers (CC) sample, in order to extract constraints on the scenario of Barrow holographic dark energy. The latter is a holographic dark energy model based on the recently proposed Barrow entropy, which arises from the modification of the black-hole surface due to quantum-gravitational effects. We first consider the case where the new deformation exponent Delta is the sole model parameter, and we show that although the standard value Delta =0, which corresponds to zero deformation, lies within the 1sigma region, a deviation is favored. In the case where we let both Delta and the second model parameter to be free we find that a deviation from standard holographic dark energy is preferred. Additionally, applying the Akaike, Bayesian and Deviance Information Criteria, we conclude that the one-parameter model is statistically compatible with Lambda hbox {CDM} paradigm, and preferred comparing to the two-parameter one. Finally, concerning the present value of the Hubble parameter we find that it is close to the Planck value.

Thermal features of Barrow corrected-entropy black hole formulation

Through the last years, it was demonstrated that quantum corrections of entropy, represented by logarithmic and power law corrections terms, constituted an association between semi-classical entropic areas and the curvature correction in Einstein–Hilbert’s Lagrangian and vice-versa. Loop quantum gravity approach provided the logarithmic corrections, which arises from quantum and thermal equilibrium fluctuations. On the other hand, Barrow’s entropy was introduced from the fact that the black hole surface can be modified due to quantum gravitational effects. The new exponent Delta that appears in Barrow’s entropy is a measure of this perturbation. In this letter we have analyzed the thermodynamical effects of the quantum fluctuations upon the geometry of a Barrow’s black hole. We demonstrated that new formulations of the equipartition law, which corresponds to the horizon energy, can be constructed from both entropic formalisms. Besides, we have calculated the heat capacity for both formulations and we discussed their thermal viability. We have also establish a condition on one of the constant pre-factors of the logarithmic correction.

Fractality in momentum space: A signal of criticality in nuclear collisions

We show that critical systems of finite size develop a fractal structure in momentum space with anomalous dimension given in terms of the isotherm critical exponent delta of the corresponding infinite system. The associated power laws of transverse momentum correlations, in high-energy nuclear collisions, provide us with a signature of a critical point in strongly interacting matter according to the laws of QCD.

Nonlocal self-improving properties

Solutions to nonlocal equations with measurable coefficients are higher differentiable. Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by integral(Rn)integral(Rn)[u(x) - u(y)][eta(x) - eta(y)]K(x, y) dx dy = integral(Rn) f eta dx for all eta is an element of C-c(infinity) (R-n), where the kernel K( . ) is a measurable function and satisfies the bounds 1/Lambda vertical bar x - y vertical bar(n+2 alpha) <= K(x, y) <= Lambda/vertical bar x - y vertical bar(n+2 alpha) with 0 1, while f is an element of L-loc(q)(R-n) for some q > 2n/(n + 2 alpha). The main result states that there exists a positive, universal exponent delta equivalent to delta(n, alpha, Lambda, q) such that for every weak solution u the self-improving property u is an element of W-alpha,W-2 (R-n) double right arrow u is an element of W-loc(alpha+delta,2+delta) (R-n) holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable. The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Holder-type inequalities in R-2n and which is implied by delicate covering and exit-time arguments. In turn, such reverse Holder inequalities are based on the concept of dual pairs, that is, pairs (mu, U) of measures and functions in R-2n which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order alpha.

Theta-point polymers in the plane and Schramm-Loewner evolution

We study the connection between polymers at the θ temperature on the lattice and Schramm-Loewner chains with constant step length in the continuum. The second of these realize a useful algorithm for the exact sampling of tricritical polymers, where finite-chain effects are excluded. The driving function computed from the lattice model via a radial implementation of the zipper method is shown to converge to Brownian motion of diffusivity κ=6 for large times. The distribution function of an internal portion of walk is well approximated by that obtained from Schramm-Loewner chains. The exponent of the correlation length ν and the leading correction-to-scaling exponent Δ_{1} measured in the continuum are compatible with ν=4/7 (predicted for the θ point) and Δ_{1}=72/91 (predicted for percolation). Finally, we compute the shape factor and the asphericity of the chains, finding surprising accord with the θ-point end-to-end values.

Multifractal finite-size scaling and universality at the Anderson transition

We describe a new multifractal finite size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L^3=120^3 and involving nearly 10^6 independent wavefunctions have yielded unprecedented precision for the critical disorder W_c=16.530 (16.524,16.536) and the critical exponent nu=1.590 (1.579,1.602). We find that the multifractal exponents Delta_q exhibit a previously predicted symmetry relation and we confirm the non-parabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.

Anderson orthogonality and the numerical renormalization group

Anderson Orthogonality (AO) refers to the fact that the ground states of two Fermi seas that experience different local scattering potentials, say |G_I> and |G_F>, become orthogonal in the thermodynamic limit of large particle number N, in that |<G_I|G_F>| ~ N^(- Delta_AO^2 /2) for N->infinity. We show that the numerical renormalization group (NRG) offers a simple and precise way to calculate the exponent Delta_AO: the overlap, calculated as function of Wilson chain length k, decays exponentially, ~ exp(-k alpha), and Delta_AO can be extracted directly from the exponent alpha. The results for Delta_AO so obtained are consistent (with relative errors typically smaller than 1%) with two other related quantities that compare how ground state properties change upon switching from |G_I> to |G_F>: the difference in scattering phase shifts at the Fermi energy, and the displaced charge flowing in from infinity. We illustrate this for several nontrivial interacting models, including systems that exhibit population switching.

Monte Carlo simulation of monolayer graphene at nonzero temperature

We present results from lattice simulations of a monolayer graphene model at non-zero temperature. At low temperatures for sufficiently strong coupling the model develops an excitonic condensate of particle-hole pairs corresponding to an insulating phase. The Berezinskii-Kosterlitz-Thouless phase transition temperature is associated with the value of the coupling where the critical exponent delta governing the response of the order parameter at criticality to an external source has a value close to 15. The critical coupling on a lattice with temporal extent N_t=32 (T=1/(N_t a_t) where a_t is the temporal lattice spacing) and spatial extent N_s=64 is very close to infinite coupling. The value of the transition temperature normalized with the zero temperature fermion mass gap Delta_0 is given by T_BKT/Delta_0=0.055(2). This value provides an upper bound on the transition temperature, because simulations closer to the continuum limit where the full U(4) symmetry is restored may result in an even lower value. In addition, we measured the helicity modulus Upsilon and the fermion thermal mass Delta_T(T), the later providing evidence for a pseudogap phase with Delta_T>0 extending to arbitrarily high T.

Low-frequency isotropic and anisotropic Raman spectra of aromatic liquids

The Raman spectra below 300 wavenumbers of six different aromatic molecular liquids have been measured with a time and polarization resolved optical Kerr effect technique. The isotropic and anisotropic contributions were determined to yield the complete third order response, and thus a more detailed description of the microscopic liquid dynamics. The anisotropic contributions accurately reproduced previously published results. Both the isotropic and anisotropic Raman spectral densities shift toward lower frequencies with decreasing molecular weights. The first moment of the isotropic spectral densities scales linearly with the inverse square root of the molecular weight, which is consistent with interaction-induced dynamics in these liquids being driven mainly by motions with a translational character. Also, the isotropic spectral densities could be fit to a single Bucaro-Litovitz function. The exponent delta of this function increases monotonically with the inverse square root of the molecular weight. A possible physical origin of this behavior is discussed.

Swelling of polymers in porous media

The swelling of polymers in random matrices is studied using computer simulations and percolation theory. The model system consists of freely jointed hard sphere chains in a matrix of hard spheres fixed in space. The average size of the polymer is a nonmonotonic function of the matrix volume fraction, phi(m). For low values of phi(m) the polymer size decreases as phi(m) is increased but beyond a certain value of phi(m) the polymer size increases as phi(m) is increased. The qualitative behavior is similar for three different types of matrices. In order to study the relationship between the polymer swelling and pore percolation, we use the Voronoi tessellation and a percolation theory to map the matrix onto an irregular lattice, with bonds being considered connected if a particle can pass directly between the two vertices they connect. The simulations confirm the scaling relation R(G) approximately (p-p(c))(delta(0))N(nu), where R(G) is the radius of gyration, N is the polymer degree of polymerization, p is the number of connected bonds, and p(c) is the value of p at the percolation threshold, with universal exponents delta(0)(approximately = -0.126+/-0.005) and nu(approximately = 0.6+/-0.01). The values of the exponents are consistent with predictions of scaling theory.

Multifactor analysis of multiscaling in volatility return intervals

We study the volatility time series of 1137 most traded stocks in the U.S. stock markets for the two-year period 2001-2002 and analyze their return intervals tau , which are time intervals between volatilities above a given threshold q . We explore the probability density function of tau , P_(q)(tau) , assuming a stretched exponential function, P_(q)(tau) approximately e;(-tau;(gamma)) . We find that the exponent gamma depends on the threshold in the range between q=1 and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how gamma depends on four essential factors, capitalization, risk, number of trades, and return. We show that gamma depends on the capitalization, risk, and return but almost does not depend on the number of trades. This suggests that gamma relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of tau , mu_(m) identical with(tautau);(m);(1m) , in the range of 10<tau< or =100 by a power law, micro_(m) approximately tau;(delta). The exponent delta is found also to depend on the capitalization, risk, and return but not on the number of trades, and its tendency is opposite to that of gamma . Moreover, we show that delta decreases with increasing gamma approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.

Aggregation of magnetic holes in a rotating magnetic field

We experimentally investigated field-induced aggregation of nonmagnetic particles confined in a magnetic fluid layer when rotating magnetic fields were applied. After application of a magnetic field rotating in the plane of the fluid layer, the single particles start to form two-dimensional clusters, like dimers, trimers, and more complex structures. These clusters aggregated again and again to form bigger clusters. During this nonequilibrium process, a broad range of cluster sizes was formed, and the scaling exponents z and z;{'} of the number of clusters N(t) approximately t;{-z;{'}} and average cluster size S(t) approximately t;{z} were calculated. The process could be characterized as diffusion-limited cluster-cluster aggregation. We found that all sizes of clusters that occurred during an experiment fall on a single curve, as the dynamic scaling theory predicts. However, the characteristic scaling exponents z;{'},z and crossover exponents Delta were not universal. A particle tracking method was used to find the dependence of the diffusion coefficients D_{s} on cluster size s . The cluster motions show features of Brownian motion. The average diffusion coefficients D_{s} depend on the cluster size s as a power law D_{s} proportional, variants;{gamma} where values of gamma as different as gamma=-0.62+/-0.19 and gamma=-2.08+/-0.51 were found in two of the experiments.

Dynamic behavior of the interface of striplike structures in driven lattice gases

In this work, the dynamic behavior of the interfaces in both the standard and random driven lattice gas models (DLG and RDLG, respectively) is investigated via numerical Monte Carlo simulations in two dimensions. These models consider a lattice gas of density rho=12 with nearest-neighbor attractive interactions between particles under the influence of an external driven field applied along one fixed direction in the case of the DLG model, and a randomly varying direction in the case of the RDLG model. The systems are also in contact with a reservoir at temperature T . Those systems undergo a second-order nonequilibrium phase transition between an ordered state characterized by high-density strips crossing the sample along the driving field, and a quasilattice gas disordered state. For T less, similarT_{c} , the average interface width of the strips (W) was measured as a function of the lattice size and the anisotropic shape factor. It was found that the saturation value W_{sat};{2} only depends on the lattice size parallel to the external field axis L_{y} and exhibits two distinct regimes: W_{sat};{2} proportional, variantlnL_{y} for low temperatures, that crosses over to W_{sat};{2} proportional, variantL_{y};{2alpha_{I}} near the critical zone, alpha_{I}=12 being the roughness exponent of the interface. By using the relationship alpha_{I}=1(1+Delta_{I}) , the anisotropic exponent for the interface of the DLG model was estimated, giving Delta_{I} approximately 1 , in agreement with the computed value for anisotropic bulk exponent Delta_{B} in a recently proposed theoretical approach. At the crossover region between both regimes, we observed indications of bulk criticality. The time evolution of W at T_{c} was also monitored and shows two growing stages: first one observes that W proportional, variantlnt for several decades, and in the following times one has W proportional, variantt;{beta_{I}} , where beta_{I} is the dynamic exponent of the interface width. By using this value we estimated the dynamic critical exponent of the correlation length in the perpendicular direction to the external field, giving z_{ perpendicular};{I} approximately 4 , which is consistent with the dynamic exponent of the bulk critical transition z_{ perpendicular};{B} in both theoretical approaches developed for the standard model. A similar scenario was also observed in the RDLG model, suggesting that both models may belong to the same universality class.

Observation of a Universal Aggregation Mechanism and a Possible Phase Transition in Au Sputtered by Swift Heavy Ions

Two exponents delta for the size distribution of n-atom clusters, Y(n) approximately n{-delta}, have been found in Au clusters sputtered from embedded Au nanoparticles under swift heavy ion irradiation. For small clusters, below 12.5 nm in size, delta has been found to be 3/2, which can be rationalized as occurring from a steady state aggregation process with size independent aggregation. For larger clusters, a delta value of 7/2 is suggested, which might come from a dynamical transition to another steady state where aggregation and evaporation rates are size dependent. In the present case, the observed decay exponents do not support any possibility of a thermodynamic liquid-gas-type phase transition taking place, resulting in cluster formation.

Anomalous mobility of dislocation kink solitons in disordered solid solutions

Dislocation kink solitons in disordered solid solutions provide an example of quasiparticles showing anomalous kinetics-i.e., the nonlinear dependence of the displacement x on the time t , x approximately t(delta) (delta<1) . To describe the dynamic phase transition from the ordinary linear to anomalous regime, the dynamics of a quasiparticle in an energy landscape that performs a correlated random walk on the energy scale was theoretically studied. The phase diagram was characterized by the calculated temperature dependence of the threshold driving force F(th) below which the average velocity of quasiparticles vanishes. The exponent delta of the kinetic equation for the anomalous phase, x approximately t(delta), was determined by simple statistical arguments using the concepts of the "optimal fluctuation method." The dependence of the threshold driving force F(th) on the concentration of solute atoms and statistical properties of a random energy landscape relevant to disordered solid solutions was calculated. The correlations between steps of the random potential were shown to modify the concentration dependence of F(th), thereby providing a qualitative explanation of experimental data on the dislocation pinning in solid solutions.

Scaling in the aggregation dynamics of a magnetorheological fluid

We present experimental results on the aggregation dynamics of a magnetorheological fluid, namely, an aqueous suspension of micrometer-sized superparamagnetic particles, under the action of a constant uniaxial magnetic field using video microscopy and image analysis. We find a scaling behavior in several variables describing the aggregation kinetics. The data agree well with the Family-Vicsek scaling ansatz for diffusion-limited cluster-cluster aggregation. The kinetic exponents z and z' are obtained from the temporal evolution of the mean cluster size S(t) and the number of clusters N(t), respectively. The crossover exponent Delta is calculated in two ways: first, from the initial slope of the scaling function; second, from the evolution of the nonaggregated particles, n1(t). We report on results of Brownian two-dimensional dynamics simulations and compare the results with the experiments. Finally, we discuss the differences obtained between the kinetic exponents in terms of the variation in the crossover exponent and relate this behavior to the physical interpretation of the crossover exponent.