Pratip Bhattacharyya

Dynamic critical behavior of failure and plastic deformation in the random fiber bundle model

The random fiber bundle (RFB) model, with the strength of the fibers distributed uniformly within a finite interval, is studied under the assumption of global load sharing among all unbroken fibers of the bundle. At any fixed value of the applied stress sigma (load per fiber initially present in the bundle), the fraction U(t)(sigma) of fibers that remain unbroken at successive time steps t is shown to follow simple recurrence relations. The model is found to have stable fixed point U*, filled (sigma) for applied stress in the range 0 < or = sigma < or = sigma(c), beyond which total failure of the bundle takes place discontinuously [abruptly from U*, filled (sigma(c)) to 0]. The dynamic critical behavior near this sigma(c) has been studied for this model analyzing the recurrence relations. We also investigated the finite size scaling behavior near sigma(c). At the critical point sigma = sigma(c), one finds strict power law decay (with time t) of the fraction of unbroken fibers U(t)(sigma(c)) (as t--> infinity). The avalanche size distribution for this mean-field dynamics of failure at sigma < sigma(c) has been studied. The elastic response of the RFB model has also been studied analytically for a specific probability distribution of fiber strengths, where the bundle shows plastic behavior before complete failure, following an initial linear response.

The Mean Distance to the nth Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory.

We study different ways of determining the mean distance rn between a reference point and its nth neighbour among random points distributed with uniform density in a D-dimensional Euclidean space. First, we present a heuristic method; though this method provides only a crude mathematical result, it shows a simple way of estimating rn. Next, we describe two alternative means of deriving the exact expression of rn: we review the method using absolute probability and develop an alternative method using conditional probability. Finally, we obtain an approximation to rn from the mean volume between the reference point and its nth neighbour and compare it with the heuristic and exact results.

Phase transition in fiber bundle models with recursive dynamics

We study the phase transition in a class of fiber bundle models in which the fiber strengths are distributed randomly within a finite interval and global load sharing is assumed. The dynamics is expressed as recursion relations for the redistribution of the applied stress and the evolution of the surviving fraction of fibers. We show that an irreversible phase transition of second-order occurs, from a phase of partial failure to a phase of total failure, when the initial applied stress just exceeds a critical value. The phase transition is characterized by static and dynamic critical properties. We calculate exactly the critical value of the initial stress for three models of this kind, each with a different distribution of fiber strengths. We derive exact expressions for the order parameter, the susceptibility to changes in the initial applied stress, and the critical relaxation of the surviving fraction of fibers for all the three models. The static and dynamic critical exponents obtained from these expressions are found to be universal.

Time series of stock price and of two fractal overlap: Anticipating market crashes?

The features of the time series for the overlap of two Cantor sets when one set moves with uniform relative velocity over the other looks somewhat similar to the time series of stock prices. We analyze both and explore the possibilities of anticipating a large (change in Cantor set) overlap or a large change in stock price. An anticipation method for some of the crashes has been proposed here, based on these observations.

Critical phenomena in an one-dimensional probabilistic cellular automation

A one-dimensional probabilistic cellular automaton that models a transition from elementary rule 4 to elementary rule 22 (following Wolframs nomenclature scheme) is studied here. The evolution of the automaton follows rule 4 with probability 1 − p and rule 22 with probability p. In course of the transition the system shows two critical points, a trivial pc1 = 0 and a nontrivial pc2 ⋍ 0.75, at which the relaxation time of the system is observed to diverge in the form of a power law τ ∼ (p − pc1)−z1 and τ ≈ (pc2 − p)−z2 with z1 ≅ 0.86 and z2 ≅ 0.92. The point pc2 is also a point of phase transition with the density of occupied sites in the equilibrium state as the order parameter; the order parameter goes to zero as n ≈ (p − pc2)s, s ≅ 0.32 for p → pc2+. The possible cause of the observed behaviour is discussed.

Dynamic critical properties of a one-dimensional probabilistic cellular automaton

Abstract:Dynamic properties of a one-dimensional probabilistic cellular automaton are studied by Monte Carlo simulation near a critical point which marks a second-order phase transition from an active state to an effectively unique absorbing state. Values obtained for the dynamic critical exponents indicate that the transition belongs to the universality class of directed percolation. Finally the model is compared with a previously studied one to show that a difference in the nature of the absorbing states places them in different universality classes.

Two Fractal Overlap Time Series and Anticipation of Market Crashes

We find prominent similarities in the features of the time series for the (model earthquakes or) overlap of two Cantor sets when one set moves with uniform relative velocity over the other and time series of stock prices. An anticipation method for some of the crashes have been proposed here, based on these observations.

The nature of most probable paths at finite temperatures

We determine the most probable length of paths at finite temperatures, with a preassigned end-to-end distance and a unit of energy assigned to every step on a D-dimensional hypercubic lattice. The asymptotic form of the most probable path-length shows a transition from the directed walk nature at low temperatures to the random walk nature as the temperature is raised to a critical value Tc. We find Tc = 1/(ln 2 + ln D). Below Tc the most probable path-length shows a crossover from the random walk nature for small end-to-end distance to the directed walk nature for large end-to-end distance; the crossover length diverges as the temperature approaches Tc. For every temperature above Tc we find that there is a maximum end-to-end distance beyond which a most probable path-length does not exist.

TRANSIENTS IN A ONE-DIMENSIONAL PROBABILISTIC CELLULAR AUTOMATON

The characteristics of the distribution of transient times of a one-dimensional probabilistic cellular automaton are studied by computer simulation. The mean and width of the distribution are found to diverge by the same power-law at each of the two critical points, pc1 = 0 and pc2 0.75, of the model. Critical exponents obtained from finite-size scaling indicate that pc1 belongs to the universality class of directed random walks whereas pc2 belongs to the universality class of directed percolation. Between the two critical points there exists a point of minimum transient length at pm 0.23 where the mean transient time scales logarithmically with the system size.