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A percolation model is applied to the explanation of some of the qualitative and quantitative aspects associated with the recent observations of earthquake ‘nucleation’. An additional assumption is introduced that nucleation starts at the critical point of percolation. The model explains the order of magnitude of the seismic moment release during the nucleation, the dependence of the seismic moment of the main shock on the duration of the nucleation process, and the observation that the fraction of the moment release during the nucleation has no systematic variation with the size of the main shock. The model also suggests that the source time function of the nucleation phase may be complex, and also that not all earthquakes are accompanied by a nucleation process, which is supported by observational results. By assuming that there exists a scale invariance associated with the criticality, a Widom scaling model is proposed to describe the electromagnetic emission during earthquake rupture.

We find numerical evidence for the scaling laws of the moments of the free path length in 3D Sinai's billiard systems with a simple hexahedron lattice involving an ellipsoid scatterer. We also find scaling exponents of the moments by regarding the mean free path length as the relevant scaling variable.

We propose a simple method of Monte Carlo renormalization group. Thermodynamic quantities of the block-spin system are analyzed phenomenologically on the basis of scaling concept_ The field exponent YH is obtained from the ratio of the square of block-spin magnetization at the fixed point, while the temperature exponent YT is obtained as a consequence of nonlinearity of the scaling fields_ In order to check the effectiveness of the method, we perform calculation on the three-dimensional Ising model. The critical temperature and the two critical exponents are estimated precisely_ Finite-size effect is shown to be weak, whose origin is the regular part of the free energy density. Statistical errors of the ratios of the block-spin quantities are proved to be surprisingly small compared to those of the block-spin quantities themselves.

The kinetic Ising model on a square lattice is studied by means of the finite· size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical exponent is universal for certain types of transition pr:obabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z~2.2±O.1.

Fisher's static finite-size scaling law is derived on the basis of the renormalization group theory and it is extended to dynamic critical phenomena in a finite system. This dynamic finite-size scaling law yields a cross-over effect with respect to the size and time-region. This effect is useful in analyzing computer simulations and also in studying the scaling property of the Kondo effect near the absolute zero temperature.

Here is proposed a new (or weak) universality that reduced critical exponents r=r/v, fi=[j/v', 1=£1/v, ¢= (2-a,) /11, etc., as well as r; and iJ are independent of the details of the system Hamiltonian. Two concepts of scaling laws and universality1l have been main motives for recent investigations of critical phenomena. The scaling law has been confirmed in many examples. We find, however, several counter-examples against the ordinary universality which claims that critical exponents themselves depend only upon such fundamental parameters as dimensionality d, symmetry (or the degree of freedom n) and potential-range parameter a. One of such typical counter-examples will be the eight-vertex modeFl in which critical exponents vary continuously as the strength of interaction changes. Here we propose a new concept of universality which covers "apparently" exceptional cases such as the eight-vertex model. The old universality is concerned with critical exponents defined by the power of the singularities of physical quantities with respect to the temperature difference from the critical point Tc. However, does the temperature difference have an absolute meaning in critical phenomena? There exist many examples indicating that it is not the case, as will be shown later. Instead of the temperature difference itself, the inverse correlation length tC plays an essential role in critical phenomena. Thus, our new proposal is that critical exponents defined through IC instead of the variable TTc should be universal. That is, our new universality claims that reduced critical exponents