Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Coagulation equations with gelation

Smoluchowski's equation for rapid coagulation is used to describe the kinetics of gelation, in which the coagulation kernelK _ij models the bonding mechanism. For different classes of kernels we derive criteria for the occurrence of gelation, and obtain critical exponents in the pre- and postgelation stage in terms of the model parameters; we calculate bounds on the time of gelationt _c , and give an exact postgelation solution for the modelK _ij =(ij ^ω ) (ω>1/2) andK _ij =a ^i+j (a>1). For the modelK _ij =i ^ω +j ^ω (ω<1, without gelation) initial solutions are given. It is argued that the kernelK _ij ～ij ^ω with ω?1?1/d (d is dimensionality) effectively models the sol-gel transformation in polymerizing systems and approximately accounts for the effects of cross-linking and steric hindrance neglected in the classical theory of Flory and Stockmayer (Ω=1). For allΩ the exponents,t=Ω+3/2 andσ=Ω?1/2,γ=(3/2?Ω)/(Ω ? 1/2) andΒ=1, characterize the size distribution, at and slightly below the gel point, under the assumption that scaling is valid.     

Efficient Monte Carlo algorithm and high-precision results for percolation

We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.     

Bioconcentration of zinc and its effect on the biochemical constituents of the gill tissues of Labeo rohita: An FT-IR study

In the present work, an attempt has been made to assess the bioconcentration and distribution of zinc on the selected organs of Labeo rohita and to study the effect of zinc exposure on the biochemical constitutions of gill tissues of L. rohita by using FT-IR Spectroscopy. The concentration pattern in the organs reveals that the liver is the prime site of metal binding and muscle accumulates least metal concentration. The accumulation profile is in the order: liver > gill > kidney > brain > bone > muscle. It has also been observed that the administration of chelating agent d-Penicillamine (DPA) reduces the zinc concentration in all tissues more effectively than the administration of the chelating agent Ethylene Diamine Tetra Acetic acid. The FT-IR spectra reveal that zinc exposure causes significant changes in the biochemical constitutions of the gill tissues. It causes an alteration in the protein secondary structures by decreasing the α-helix and increasing the β-sheet contents. Further, it has been observed that the administration of chelating agent DPA improves the protein and lipid contents in the gill tissues compared to zinc exposed tissues. This result shows that DPA is the effective chelator of zinc in reducing the body burden of L. rohita fingerlings. In conclusion, the findings of the current study suggest that zinc exposure causes significant changes in both lipids and proteins of the gill tissues, and changes the protein profile in favour of β-sheet structure.