Naeem Jan

Social percolation models

We here relate the occurrence of extreme market shares, close to either 0 or 100%, in the media industry to a percolation phenomenon across the social network of customers. We further discuss the possibility of observing self-organized criticality when customers and cinema producers adjust their preferences and the quality of the produced films according to previous experience. Comprehensive computer simulations on square lattices do indeed exhibit self-organized criticality towards the usual percolation threshold and related scaling behaviour.

Computer simulation of the main gel–fluid phase transition of lipid bilayers

Monte Carlo techniques have been applied to a study of two related quasi‐two‐dimensional microscopic interaction models which describe the phase behavior of phospholipid bilayers. The two models are Ising‐like lattice models in which (a) the acyl chains of the phospholipids interact via anisotropic van der Waals forces and (b) the rotational isomerism of the chains is accounted for by two and ten selected conformational states, respectively. Monte Carlo experiments are performed on both models so as to determine whether the static thermodynamic properties of lipid bilayers are most accurately represented by a simple two state gel–fluid concept or whether a more complicated melting process involving intermediate states takes place. To this purpose, the temperature dependence of several static thermodynamic properties has been calculated for both models. This includes the chain cross‐sectional area, the internal and free energies, the coherence length, the lateral compressibility, and the specific heat. Par...

Exact Relations Between Damage Spreading and Thermodynamical Properties

We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d−β/ν at Tc and whose size distribution is also related to static exponents.

Polymer conformations through 'wiggling'

A new Monte Carlo method is proposed which allows for the efficient generation of equilibrium conformations of polymer chains in two and three dimensions. The method treats each site (monomer) as a potential pivot around which a new conformation may be generated by rotating a portion of the chain. The method does not suffer from the severe attrition associated with the simple sampling of self-avoiding walks and may be extended to treat the interacting polymer chain. The authors find in two dimensions that nu =0.748+or-0.005 (exact=0.750) and in three dimensions nu =0.595+or-0.005 (series expansion and renormalisation group predict nu approximately 0.588). The end-end distances calculated for shorter chains are in good agreement with the exact values from enumeration techniques.

Kinetic Growth Walk: A New Model for Linear Polymers

To describe the irreversible growth of linear polymers, we introduce a new type of perturbed random walk, related to the zero initiator concentration limit of the kinetic gelation model. Our model simulates real polymer growth by permitting the initiator (walker) to form the next bond with an unsaturated monomer at one of the neighbouring sites of its present location. A heuristic kinetic self-consistent field argument along the lines introduced by Pietronero suggests a fractal dimensionality, df = (d + 1)/2, in agreement with our Monte Carlo and series expansion results (including the usually expected logarithmic correction at the upper critical dimension dc = 3.

Self-avoiding walks on the simple cubic lattice

We have substantially extended the series for the number of self-avoiding walks and the mean-square end-to-end distance on the simple cubic lattice. Our analysis of the series gives refined estimates for the critical point and critical exponents. Our estimates of the exponents γ and ν are in good agreement with recent high-precision Monte Carlo estimates, and also with recent renormalization group estimates. Critical amplitude estimates are also given. A new, improved rigorous upper bound for the connective constant µ<4.7114 is obtained.

Sharp peaks in the percolation model for stock markets

The empirically observed asymmetry of sharp peaks and flat troughs in stock market fluctuations is recovered by a feedback mechanism in the Cont–Bouchaud model, changing the trader activity proportionally to the price change.

Large lattice random site percolation

We simulate the two dimensional (2d), simple square and three-dimensional (3d), simple cubic random site percolation systems for L=2000000 (2d) and L=10001 (3d) at the percolation thresholds for these systems. We report excellent agreement with the Fisher exponent, τ, in 2d, with the proposed exact value 187/91 and good agreement with other good high quality simulation results in 3d of 2.186. We have also computed how the first, second, third,…, largest clusters scale with L at the percolation threshold. These clusters all scale with the same fractal dimensionality as the largest cluster.

MICELLE FORMATION, RELAXATION TIME, AND THREE-PHASE COEXISTENCE IN A MICROEMULSION MODEL

Our Larson‐type microemulsion model for surfactant chains in oil–water solvents leads to long relaxation times as well as, for essential modifications, to a stable peak in the chain‐cluster size distribution. Transfer energies for surfactant chains moving to the oil–water interface, and characteristic micelle concentrations (CMC) as a function of chain length are compared with experiment.

Random Site Percolation in Three Dimensions

We simulate sytems up to 10 001 × 10 001 × 10 001 in three dimensions at the percolation threshold pc of 0.31160. We find that the fractal dimension is ≃2.53±0.02, the cluster size distribution exponent, τ, is 2.186±0.002 and an exponent of 0.85 describing how the mass of the clusters scale with rank. Corrections-to-scaling exponents of ≃-0.7 are observed for ns and for the mass of the largest cluster. We also check the percolation threshold and report good agreement with recent values.