Static Exponents Research Articles

Directed percolation criticality due to stochastic switching between attractive and repulsive coupling in coupled circle maps

We study a lattice model where the coupling stochastically switches between repulsive (subtractive) and attractive (additive) at each site with probability p at every time instant. We observe that such a kind of coupling stabilizes the local fixed point of a circle map, with the resultant globally stable attractor providing a unique absorbing state. Interestingly, a continuous phase transition is observed from the absorbing state to spatiotemporal chaos via spatiotemporal intermittency for a range of values of p . It is interesting to note that the transition falls in class of directed percolation. Static and spreading exponents along with relevant scaling laws are found to be obeyed confirming the directed percolation universality class in spatiotemporal intermittency regime.

Scaling analysis of the static and dynamic critical exponents in underdoped, overdoped, and optimally dopedPr2−xCexCuO4−yfilms

We report on current-voltage measurements of the zero-field normal-superconducting phase transition in thin films of Pr$_{2-x}$Ce$_x$CuO$_{4-y}$ as a function of doping. We find that the small size of the critical regime in these materials ($\approx 25$ mK) gives rise to mean-field behavior at the phase transition with a static exponent of $\nu \approx 0.5$ for all dopings (in contrast to hole-doped $\mathrm{YBa_{2}Cu_{3}O_{7-\delta}}$). We also find mean-field behavior in the dynamic exponent $z$. This indicates that Pr$_{2-x}$Ce$_x$CuO$_{4-y}$ behaves similarly to conventional superconductors in contrast to other cuprate superconductors. However, as the transition width in our samples decreases, the dynamic critical exponent approaches $z=1.5$, similar to the critical exponent found in hole-doped $\mathrm{YBa_{2}Cu_{3}O_{7-\delta}}$.

Roughness evolution of Si surfaces upon Ar ion erosion

We studied the roughness evolution of Si surfaces upon Ar ion erosion in real time. Following the theory of surface kinetic roughening, a model proposed by Majaniemi was used to obtain the value of the dynamic scaling exponent β from our data. The model was found to explain both the observed roughening and the smoothening of the surfaces. The values of the scaling exponents α and β, important for establishing a universal model for ion erosion of (Si) surfaces, have been determined. The value of β proved to increase with decreasing ion energy, while the static scaling exponent α was found to be ion energy independent.

Short-time domain-wall dynamics in the random-field Ising model with a driving field

With Monte Carlo methods, we investigate the relaxation dynamics of a domain wall in the two-dimensional random-field Ising model with a driving field. The short-time dynamic behavior at the depinning transition is carefully examined, and the roughening process of the domain wall is observed. Based on the short-time dynamic scaling form, we accurately determine the transition field, static and dynamic exponents, and local and global roughness exponents. In contrast to the usual assumption, the results indicate that the domain interface does not belong to the universality class of the Edwards-Wilkinson equation. In particular, due to the dynamic effect of overhangs, the domain interface exhibits intrinsic anomalous scaling and spatial multiscaling behaviors, compatible with the experiments

Solvent size effect on the static and dynamic properties of polymer chains in athermal solvents

The static and dynamic properties of polymer chains in athermal solvents with different sizes are studied by molecular dynamics method. With increasing solvent size, the radius of gyration and the diffusion coefficient of the polymer decay fast until a critical solvent size is reached. For the polymer diffusion coefficients, this decay only depends on the solvent size; while for the radius of gyration of polymers, this decay depends on both solvent size and the length of the polymers. The increase of solvent size also makes the polymer tend to be thicker ellipsoid until a critical solvent size is reached. The static scaling exponent of the polymer also shows the solvent size dependence. Moreover, four regions are identified where the polymers show different dynamic behaviors according to the dynamic structure factors of the polymer.

Size distributions of shocks and static avalanches from the functional renormalization group

Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d=4 this yields the mean-field distribution P(S) approximately S;{-3/2}e;{-S4S_{m}} , where S_{m} is a large-scale cutoff, in some cases calculable. Resumming all one-loop contributions, we find P(S) approximately S;{-tau}exp(C(SS_{m});{1/2}-B/4(S/S_{m});{delta}) , where B , C , delta , and tau are obtained to first order in =4-d . Our result is consistent to O() with the relation tau=tau_{zeta}:=2-2/d+zeta , where zeta is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field, and random-periodic disorders. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with tau=2-1/d+zeta to O(=2-d) . We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the relation tau=tau_{zeta} be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyperplane of codimension one is in mean field (valid close to and above d=4 ) given by P(S) approximately K_{13}(S)S , where K is the Bessel- K function, thus tau_{hyperplane}=4/3 .

ANALYSIS OF CRITICAL SHORT-TIME LANGEVIN DYNAMICS IN TWO-DIMENSIONAL ϕ4 THEORY ON THE BASIS OF A HIGHER-ORDER ALGORITHM

We investigate the critical short-time scaling of the two-dimensional lattice ϕ4 field theory with a Langevin dynamics. Starting from a "hot" initial configuration but with a small magnetization, the critical initial increase of the magnetization is observed, through which we determine the critical point. From the short-time relaxation dynamics of various quantities at the critical point obtained, the dynamic critical exponents θ, z and the static exponent β/ν are evaluated. In executing a Langevin simulation, an appropriate discretization method with respect to the time degree of freedom becomes essentially important to be adopted and we show that the well-known second-order form of discretized Langevin equation works well to investigate the short-time dynamics.

Diffusion scaling through structural templates given by the 3d dynamic Ising model

We present new results describing diffusion through networks exhibiting dynamic disorder where diffusion ‘pathways’ are found from dynamic structures obtained from kinetic Monte Carlo (KMC) computer simulations consistent with Kawasaki dynamics in the 3d Ising lattice model.Our simulation results for the diffusion/conductance scaling exponent s are in excellent accord with the most widely quoted theoretical value s=2ν-β, where ν and β are 3d static percolation exponents in cubic lattices. This is in contrast to experimentally obtained values for this scaling exponent where there appears to be substantial disparity between many of these prior published results.

Contact process in disordered and periodic binary two-dimensional lattices

The critical behavior of the contact process (CP) in disordered and periodic binary two-dimensional (2D) lattices is investigated numerically by means of Monte Carlo simulations as well as via an analytical approximation and standard mean field theory. Phase-separation lines calculated numerically are found to agree well with analytical predictions around the homogeneous point. For the disordered case, values of static scaling exponents obtained via quasistationary simulations are found to change with disorder strength. In particular, the finite-size scaling exponent of the density of infected sites approaches a value consistent with the existence of an infinite-randomness fixed point as conjectured before for the 2D disordered CP. At the same time, both dynamical and static scaling exponents are found to coincide with the values established for the homogeneous case thus confirming that the contact process in a heterogeneous environment belongs to the directed percolation universality class.

Growth behavior of amorphous Ca and La modified PbTiO3 thin film

The growth behavior of amorphous Ca and La modified PbTiO3 (PLCT) thin film has been studied in terms of the dynamic scaling approach, and the various scaling exponents associated with self-affine film surface have been characterized by small angle x-ray scattering technique. The results indicate that the amorphous relaxation has much influence on the growth behavior of films. For the light amorphous relaxation, the static scaling exponent α is 0.57, which suggests the surface diffusion dominates the surface morphology. As the effect of amorphous relaxation increasing, α is 0.30, suggests the desorption process is controlled mainly by the atomistic relaxation.

Nonequilibrium critical relaxation of the order parameter and energy in the two-dimensional ferromagnetic Potts model

The static and dynamic critical properties of the ferromagnetic q -state Potts models on a square lattice with q=2 and 3 are numerically studied via the nonequilibrium relaxation method. The relaxation behavior of both the order parameter and energy as well as that of the second moments are investigated, from which static and dynamic critical exponents can be obtained. We find that the static exponents thus obtained from the relaxation of the order parameter and energy together with the second moments of the order parameter exhibit a close agreement with the exact exponents, especially for the case of the q=2 (Ising) model, when care is taken in the choice of the initial states for the relaxation of the second moments. As for the case of q=3 , the estimates for the static exponents become less accurate, but still exhibit reasonable agreement with the exactly known static exponents. The dynamic critical exponent for the q=2 (Ising) model is estimated from the relaxation of the second moments of the order parameter with mixed initial conditions to give z(q=2) approximately 2.1668(19) .

Flexible chain molecules in the marginal and concentrated regimes: Universal static scaling laws and cross-over predictions

We present predictions for the static scaling exponents and for the cross-over polymer volumetric fractions in the marginal and concentrated solution regimes. Corrections for finite chain length are made. Predictions are based on an analysis of correlated fluctuations in density and chain length, in a semigrand ensemble in which mers and solvent sites exchange identities. Cross-over volumetric fractions are found to be chain length independent to first order, although reciprocal-N corrections are also estimated. Predicted scaling exponents and cross-over regimes are compared with available data from extensive off-lattice Monte Carlo simulations [Karayiannis and Laso, Phys. Rev. Lett. 100, 050602 (2008)] on freely jointed, hard-sphere chains of average lengths from N=12-500 and at packing densities from dilute ones up to the maximally random jammed state.

Influence of molecular topology on the static and dynamic properties of single polymer chain in solution

The influence of molecular topology on the structural and dynamic properties of polymer chain in solution with ring structure, three-arm branched structure, and linear structure are studied by molecular dynamics simulation. At the same degree of polymerization (N), the ring-shaped chain possesses the smallest size and largest diffusion coefficient. With increasing N, the difference of the radii of gyration between the three types of polymer chains increases, whereas the difference of the diffusion coefficients among them decreases. However, the influence of the molecular topology on the static and the dynamic scaling exponents is small. The static scaling exponents decrease slightly, and the dynamic scaling exponents increase slightly, when the topology of the polymer chain is changed from linear to ring-shaped or three-arm branched architecture. The dynamics of these three types of polymer chain in solution is Zimm-like according to the dynamic scaling exponents and the dynamic structure factors.

Effects of Constitutive Parameters on Thickness of Phase Transformed Adiabatic Shear Band for Ductile Metal Based on Johnson-Cook and Gradient Plasticity Models

Gradient-dependent plasticity where a characteristic length is involved to consider the microstructural effect (interactions and interplaying among microstructures due to the heterogeneous texture) is introduced into Johnson-Cook model considering the effects of strain-hardening, thermal softening and strain rate sensitivity. Effects of initial static yield stress, strain-hardening coefficient and exponent, strain-rate and thermal-softening parameters on the occurrence of phase transformation and the thickness of phase transformed adiabatic shear band (ASB) in deformed ASB are numerically investigated. Higher initial static yield stress, strain-hardening coefficient, strain-rate parameter and lower strain-hardening exponent lead to earlier occurrence of phase transformation (lower plastic shear strain). Effect of thermal-softening parameter on plastic shear strain corresponding to the onset of phase transformation is not monotonous. Transformed ASB is located at the center of deformed ASB since the position has higher temperature exceeding the temperature of phase transformation. The thickness of transformed ASB increases with decreasing flow shear stress and the increasing tendency becomes slow. For the same flow shear stress, the thickness of transformed ASB is wider for higher initial static yield stress, strain-hardening coefficient and exponent, strain-rate and thermal-softening parameters. Compared with classical elastoplastic theory applicable to completely homogenous material, gradient-dependent plasticity considering the microstructural effect predicts that phase transformation occurs earlier and that the thickness of transformed ASB changes with flow shear stress.

CRITICAL SHORT-TIME DYNAMICS IN LATTICE GAUGE THEORY

We investigated a critical short-time relaxation in a lattice gauge theory. A systematic procedure of estimating critical point based on the "short-time scaling" is formulated. It is applied to the (2+1)-dimensional SU(2) lattice gauge theory at finite temperature to deduce its critical point. Finally, we studied the short-time relaxation behavior at the critical temperature starting either from "cold" and "hot" initial configuration, and calculated the dynamic critical exponents θ and z, as well as the static exponents β/ν.

Short-time scaling via Monte Carlo single spin-flip algorithms for the Baxter–Wu model

The short-time critical dynamics of the Baxter–Wu model is investigated via Monte Carlo simulations using single spin-flip algorithms. The critical dynamic exponents z and θ are estimated and it is shown that the N-fold way provides a reliable estimate for the ratio of the static exponents ( 2 β / ν ) and a good confirmation of the short-time scheme. However, we find that the Metropolis and heat-bath algorithms are less satisfactory when used in the Baxter–Wu model to observe the short-time scaling.

Detailed structural analysis of epitaxial MBE-grown Fe/Cr superlattices by x-ray diffraction and transmission-electron spectroscopy

We have performed a detailed quantitative structural analysis of epitaxial ${[\mathrm{Fe}(3\phantom{\rule{0.3em}{0ex}}\mathrm{nm})∕\mathrm{Cr}(1.2\phantom{\rule{0.3em}{0ex}}\mathrm{nm})]}_{20}$ superlattices by low- and high-angle x-ray diffraction, and energy-filtered transmission electron microscopy on cross-section samples. The interface roughness was changed systematically by varying the substrate temperature (150-250 \ifmmode^\circ\else\textdegree\fi{}C) maintaining all other growth parameters fixed. Direct imaging of the interfaces allows examining the roughness of the individual interfaces and its evolution with thickness. A statistical analysis of the local interface width for the individual layers supplies the roughness static and dynamic exponents. High-temperature samples (250 \ifmmode^\circ\else\textdegree\fi{}C) show roughness decreasing with thickness as a result of surface-diffusion-dominated growth. Low-temperature samples (150 \ifmmode^\circ\else\textdegree\fi{}C) show anomalous non-self-affine roughness characterized by a time-dependent local interface width.

Colloidal processing, sintering and static grain growth behaviour of alumina-doped cubic zirconia

In order to determine the influence of alumina content on sintering and static grain growth behaviour of cubic zirconia, high purity commercial powders of 8 mol% yttria-stabilized cubic zirconia (8YSCZ) doped with 0, 1, 5, 10 wt.% Al 2O 3 were used. Colloidal processing was used for the mixing of powders in order to achieve a uniform distribution and homogeneous microstructure. The specimens were sintered at different temperatures between 1250 and 1400 °C for 1 h. The density increased with Al 2O 3 content up to 0.5 wt.% and a further increase in Al 2O 3 content led to a decrease in density. The enhanced density with increasing Al 2O 3 content up to 0.5 wt.% could be due to the rearrangement of Al 2O 3 particles during the early stages of sintering. The decreased density and grain growth in higher amount of Al 2O 3-doped specimens were due to the lower grain boundary diffusivity and mobility. X-ray diffraction results showed that Al 2O 3 had very limited solubility of 0.3 wt.% into 8YSCZ. The grain growth in the 0–10 wt.% Al 2O 3-doped 8YSCZ was studied. The experimental results showed that the grain growth in Al 2O 3-doped 8YSCZ occurred slowly and was more sluggish than that in undoped 8YSCZ. Also, the grain growth rate decreased with increasing Al 2O 3 content. The static grain growth exponent value and the activation energy for undoped 8YSCZ were found to be 2 and 289 kJ/mol, respectively. The addition of Al 2O 3 raised the grain growth exponent value to 3 and activation energy for the grain growth process was increased from 289 to 410 kJ/mol.

Critical behavior of the two-dimensional random-bond Potts model: A short-time dynamic approach

The short-time critical dynamics of the two-dimensional eight-state random-bond Potts model is investigated with large-scale Monte Carlo simulations. Dynamic relaxation starting from a disordered and an ordered state is carefully analyzed. The continuous phase transition induced by disorder is studied, and both the dynamic and static critical exponents are estimated. The static exponent beta/nu shows little dependence on the disorder amplitude r, while the dynamic exponent z and static exponent 1/nu vary with the strength of disorder.

Critical dynamics of the Potts model: short-time Monte Carlo simulations

We calculate the new dynamic exponent θ of the 4-state Potts model, using short-time simulations. Our estimates θ 1 = − 0.0471 ( 33 ) and θ 2 = − 0.0429 ( 11 ) obtained by following the behavior of the magnetization or measuring the evolution of the time correlation function of the magnetization corroborate the conjecture by Okano et al. [Nucl. Phys. B 485 (1997) 727]. In addition, these values agree with previous estimate of the same dynamic exponent for the two-dimensional Ising model with three-spin interactions in one direction, that is known to belong to the same universality class as the 4-state Potts model. The anomalous dimension of initial magnetization x 0 = z θ + β / ν is calculated by an alternative way that mixes two different initial conditions. We have also estimated the values of the static exponents β and ν. They are in complete agreement with the pertinent results of the literature.