Singular scaling functions in clustering phenomena

We study clustering in a stochastic system of particles sliding down a fluctuating surface in one and two dimensions. In steady state, the density-density correlation function is a scaling function of separation and system size. This scaling function is singular for small argument — it exhibits a cusp singularity for particles with mutual exclusion, and a divergence for noninteracting particles. The steady state is characterized by giant fluctuations which do not damp down in the thermodynamic limit. The autocorrelation function is a singular scaling function of time and system size. The scaling properties are surprisingly similar to those for particles moving in a quenched disordered environment that results if the surface is frozen.     

Finite data-size scaling of clustering in earthquake networks

An earthquake network is known to be of the small-world type. The values of the network’s characteristics, however, depend not only on the cell size (i.e., the scale of coarse graining needed for constructing the network) but also on the size of a seismic data set. Here, discovery of a scaling law for the clustering coefficient in terms of the data size, which is referred to here as finite data-size scaling, is reported. Its universality is shown to be supported by the detailed analysis of the data taken from California, Japan and Iran. Effects of setting a threshold of magnitude are also discussed.     

Dynamic scaling of non-euclidean interfaces

The dynamic scaling of curved interfaces presents features that are strikingly different from those of the planar ones. Spherical surfaces above one dimension are flat because the noise is irrelevant in such cases. Kinetic roughening is thus a one-dimensional phenomenon characterized by a marginal logarithmic amplitude of the fluctuations. Models characterized by a planar dynamical exponent z > 1, which include the most common stochastic growth equations, suffer a loss of correlation along the interface, and their dynamics reduce to that of the radial random deposition model in the long time limit. The consequences in several applications are discussed, and we conclude that it is necessary to reexamine some experimental results in which standard scaling analysis was applied.     

Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes

Analyzing diverse seismic catalogs, we have determined that the probability densities of the earthquake recurrence times for different spatial areas and magnitude ranges can be described by a unique universal distribution if the time is rescaled with the rate of seismic occurrence, which therefore fully governs seismicity. The shape of the distribution shows the existence of clustering beyond the duration of aftershock bursts, and scaling reveals the self-similarity of the clustering structure in the space-time-magnitude domain. This holds from worldwide to local scales, for quite different tectonic environments and for all the magnitude ranges considered.     

Dynamic scaling of migration-driven aggregate growth

We study the kinetic behaviour of the growth of aggregates driven by reversible migration between any two aggregates. For a simple model with the migration rate K(i;j)=K′(i;j)∝i^uj^v at which the monomers migrate from the aggregates of size i to those of size j, we find that the aggregate size distribution in the system with u+v≤3 and u<2 approaches a conventional scaling form, which reduces to the Smoluchovski form in the u=1 case. On the other hand, for the system with u<2, the average aggregate size S(t) grows exponentially in the u+v=3 case and as (tlnt)^{1/(5-2u)} in another special case of v=u-2. Moreover, this typical size S(t) grows as t^{1/(3-u-v)} in the general u-2相似文献   

Dynamic scaling approach to glass formation

Experimental data for the temperature dependence of relaxation times are used to argue that the dynamic scaling form, with relaxation time diverging at the critical temperature T(c) as (T-T(c))(-nuz), is superior to the classical Vogel form. This observation leads us to propose that glass formation can be described by a simple mean-field limit of a phase transition. The order parameter is the fraction of all space that has sufficient free volume to allow substantial motion, and grows logarithmically above T(c). Diffusion of this free volume creates random walk clusters that have cooperatively rearranged. We show that the distribution of cooperatively moving clusters must have a Fisher exponent tau=2. Dynamic scaling predicts a power law for the relaxation modulus G(t) approximately t(-2/z), where z is the dynamic critical exponent relating the relaxation time of a cluster to its size. Andrade creep, universally observed for all glass-forming materials, suggests z=6. Experimental data on the temperature dependence of viscosity and relaxation time of glass-forming liquids suggest that the exponent nu describing the correlation length divergence in this simple scaling picture is not always universal. Polymers appear to universally have nuz=9 (making nu=3 / 2). However, other glass-formers have unphysically large values of nuz, suggesting that the availability of free volume is a necessary, but not sufficient, condition for motion in these liquids. Such considerations lead us to assert that nuz=9 is in fact universal for all glass- forming liquids, but an energetic barrier to motion must also be overcome for strong glasses.     

Dynamic transition and hysteresis scaling in Heisenberg ferromagnet

Based on the mean-field treatment and Monte Carlo simulation, we studied the nature of the dynamic phase transition of two and three-dimensional magnetic films in Heisenberg model. The time averaged magnetization components ( mx , my, mz),the average hysteresis-loop area components A for magnetic films with different thickness have been calculated. The dynamic transition phase diagrams from to Q=0 for the 2D and 3D cases have been obtained. The relaxation times for different values of magnetic field, temperature, thickness of the films and the orientation number of spin have been simulated. It is found that the loop area follows the scaling relation, A-A0H0 T-, and the exponents and increase with increasing thickness, while the exponent decreases with increasing thickness. It was observed that the phase boundary line shrinks inward in the H0-T plane with decreasing value of the frequency of the magnetic field and thickness of multilayer film. The phase diagrams were explained by the competition between the relaxation time and the period of the external magnetic field. Moreover, it has been indicated that the dynamical behaviors for 2D and 3D cases derived by both mean-field method and Monte Carlo method in this work are consistent.