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Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to L=3456. Our estimated values of the correction-to-scaling exponent omega tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e., omega about 0.7 and omega = 0.75(5). We discuss the possibilities that omega is either really smaller than usually expected or these values of omega describe some transient behavior. We propose refining MCRG simulations and analysis to resolve this issue. In distinction from omega, our actual MC estimations of the critical exponents eta and nu provide stable values eta=0.03632(13) and nu=0.63017(31), which well agree with those of the conformal bootstrap method, i.e., eta=0.0362978(20) and nu=0.6299709(40)., Comment: + Extended discussion and new references. 16 pages, 6 tables, 3 figures

Corrections to scaling in the 3D Ising model are studied based on non-perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L. Analytical arguments show the existence of corrections with the exponent (gamma-1)/nu (approximately 0.38), the leading correction-to-scaling exponent being omega =< (gamma-1)/nu. A numerical estimation of omega from the susceptibility data within 40 =< L =< 2048 yields omega=0.25(33). It is consistent with the statement omega =< (gamma-1)/nu, as well as with the value omega = 1/8 of the GFD theory. We reconsider the MC estimation of omega from smaller lattice sizes to show that it does not lead to conclusive results, since the obtained values of omega depend on the particular method chosen. In particular, estimates ranging from omega =1.274(72) to omega=0.18(37) are obtained by four different finite-size scaling methods, using MC data for thermodynamic average quantities, as well as for partition function zeros. We discuss the influence of omega on the estimation of exponents eta and nu., Comment: 18 pages, 2 figures, 13 tables

Corrections to scaling in the two-dimensional scalar phi^4 model are studied based on non-perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes L (from 4 to 1536) and different values of the phi^4 coupling constant lambda, i.~e., lambda = 0.1, 1, 10. According to our analysis, amplitudes of the nontrivial correction terms with the correction-to-scaling exponents omega_l < 1 become small when approaching the Ising limit (lambda --> infinity), but such corrections generally exist in the 2D phi^4 model. Analytical arguments show the existence of corrections with the exponent 3/4. The numerical analysis suggests that there exist also corrections with the exponent 1/2 and, very likely, also corrections with the exponent about 1/4, which are detectable at lambda = 0.1. The numerical tests clearly show that the structure of corrections to scaling in the 2D phi^4 model differs from the usually expected one in the 2D Ising model., Comment: 25 pages, 8 figures, 8 tables. This is a completed version, including a new theorem and new numerical data and analysis

Correlation functions in the O(n) models below the critical temperature are considered. Based on Monte Carlo (MC) data, we confirm the fact stated earlier by Engels and Vogt, that the transverse two-plane correlation function of the O(4) model for lattice sizes about L=120 and small external fields h is very well described by a Gaussian approximation. However, we show that fits of not lower quality are provided by certain non-Gaussian approximation. We have also tested larger lattice sizes, up to L=512. The Fourier-transformed transverse and longitudinal two-point correlation functions have Goldstone mode singularities in the thermodynamic limit at k --> 0 and h=+0, i.e., G_perp(k) = a k^{-lambda_perp} and G_parallel(k) = b k^{-lambda_parallel}, respectively. Here a and b are the amplitudes, k is the magnitude of the wave vector. The exponents lambda_perp, lambda_parallel and the ratio b M^2/a^2, where M is the spontaneous magnetization, are universal according to the GFD (grouping of Feynman diagrams) approach. Here we find that the universality follows also from the standard (Gaussian) theory, yielding b M^2/a^2 = (n-1)/16. Our MC estimates of this ratio are 0.06 +/- 0.01 for n=2, 0.17 +/- 0.01 for n=4 and 0.498 +/- 0.010 for n=10. According to these and our earlier MC results, the asymptotic behavior and Goldstone mode singularities are not exactly described by the standard theory. This is expected from the GFD theory. We have found appropriate analytic approximations for G_perp(k) and G_parallel(k), well fitting the simulation data for small k. We have used them to test the Patashinski--Pokrovski relation and have found that it holds approximately., Comment: 19 pages, 9 figures, 5 tables

Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G_{\perp}(k) \simeq ak^{-\lambda_{\perp}} and G_{\parallel}(k) \simeq bk^{-\lambda_{\parallel}} in the ordered phase at k -> 0, is performed in the three-dimensional O(n) models with n=2, 4, 10. Our aim is to test some challenging theoretical predictions, according to which the exponents \lambda_{\perp} and \lambda_{\parallel} are non-trivial (3/2<\lambda_{\perp}<2 and 0<\lambda_{\parallel}<1 in three dimensions) and the ratio bM^2/a^2 (where M is a spontaneous magnetization) is universal. The trivial standard-theoretical values are \lambda_{\perp}=2 and \lambda_{\parallel}=1. Our earlier MC analysis gives \lambda_{\perp}=1.955 \pm 0.020 and \lambda_{\parallel} about 0.9 for the O(4) model. A recent MC estimation of \lambda_{\parallel}, assuming corrections to scaling of the standard theory, yields \lambda_{\parallel} = 0.69 \pm 0.10 for the O(2) model. Currently, we have performed a similar MC estimation for the O(10) model, yielding \lambda_{\perp} = 1.9723(90). We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by \Delta \lambda_{\perp} = 0.0121(52). It is consistent with the idea that 2-\lambda_{\perp} decreases for large $n$ and tends to zero at n -> \infty. We have also verified and confirmed the expected universality of bM^2/a^2 for the O(4) model, where simulations at two different temperatures (couplings) have been performed., Comment: 8 pages, 5 figures

Critical phenomena and Goldstone mode effects in spin models with O(n) rotational symmetry are considered. Starting with the Goldstone mode singularities in the XY and O(4) models, we briefly review different theoretical concepts as well as state-of-the art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for very large lattices with linear sizes up to L=1536. These results are obtained, using a parallel OpenMP implementation of the Wolff single cluster algorithm. The finite-size scaling analysis of the critical exponent eta, assuming the usually accepted correction-to-scaling exponent omega=0.8, shows that eta is likely to be somewhat larger than the value 0.0335 +/- 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: eta=omega=1/8 and nu=2/3, found within the GFD theory., Comment: 15 pages, 6 figures

In this paper, we model the evolution and self-assembly of randomly oriented carbon nanotubes (CNTs), grown on a metallic substrate in the form of a thin film for field emission under diode configuration. Despite high output, the current in such a thin film device often decays drastically. The present paper is focused on understanding this problem. A systematic, multiphysics based modelling approach is proposed. First, a nucleation coupled model for degradation of the CNT thin film is derived, where the CNTs are assumed to decay by fragmentation and formation of clusters. The random orientation of the CNTs and the electromechanical interaction are then modeled to explain the self-assembly. The degraded state of the CNTs and the electromechanical force are employed to update the orientation of the CNTs. Field emission current at the device scale is finally obtained by using the Fowler-Nordheim equation and integration over the computational cell surfaces on the anode side. The simulated results are in close agreement with the experimental results. Based on the developed model, numerical simulations aimed at understanding the effects of various geometric parameters and their statistical features on the device current history are reported., Comment: 40 pages, 13 figures; keywords: field emission, carbon nanotubes, degradation, electrodynamics, self-assembly

Owing to their distinct properties, carbon nanotubes (CNTs) have emerged as promising candidate for field emission devices. It has been found experimentally that the results related to the field emission performance show variability. The design of an efficient field emitting device requires the analysis of the variabilities with a systematic and multiphysics based modeling approach. In this paper, we develop a model of randomly oriented CNTs in a thin film by coupling the field emission phenomena, the electron-phonon transport and the mechanics of single isolated CNT. A computational scheme is developed by which the states of CNTs are updated in time incremental manner. The device current is calculated by using Fowler-Nordheim equation for field emission to study the performance at the device scale., Comment: 4 pages, 5 figures

We consider problems of dynamic viscoelasticity taking into account the coupling of elastic and thermal fields. Efficient approximate models are developed and computational results on thermomechanical behaviour of shape-memory-alloy structures are presented., Comment: 15 pages, 2 figures