arXiv subjects

Bikas K. Chakrabarti

Publications and source records attributed to Bikas K. Chakrabarti.

At least 127 records · Page 7Linked to original sources

arXiv2003

Magnitude distribution of earthquakes: Two fractal contact area distribution

The `plate tectonics' is an observed fact and most models of earthquake incorporate that through the frictional dynamics (stick-slip) of two surfaces where one surface moves over the other. These models are more or less successful to reproduce the well known Gutenberg-Richter type power law in the (released) energy distribution of earthquakes. During sticking period, the elastic energy gets stored at the contact area of the surfaces and is released when a slip occurs. Therefore, the extent of the contact area between two surfaces plays an important role in the earthquake dynamics and the power law in energy distribution might imply a similar law for the contact area distribution. Since, fractured surfaces are fractals and tectonic plate- earth's crust interface can be considered to have fractal nature, we study here the contact area distribution between two fractal surfaces. We consider the overlap set of two self-similar fractals, characterised by the same fractal dimensions, and look for their distribution. We have studied numerically the specific cases of both regular and random Cantor sets in one dimension and gaskets and percolation fractals in two dimension. We find that in all the cases the distributions show an universal finite size scaling behavior. The contact area distributions have got a power law decay for both regular and random Cantor sets and also for gaskets. However, for percolation clusters the distribution shows Gaussian variation.

cond-mat.stat-mech↗

arXiv2003

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 characterised 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 the exact expressions for the order parameter, the susceptibility to changes in the initial applied sress 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.

cond-mat.stat-mech↗

arXiv2003

Failure due to fatigue in fiber bundles and solids

We consider first a homogeneous fiber bundle model where all the fibers have got the same stress threshold beyond which all fail simultaneously in absence of noise. At finite noise, the bundle acquires a fatigue behavior due to the noise-induced failure probability at any stress. We solve this dynamics of failure analytically and show that the average failure time of the bundle decreases exponentially as the stress increases. We also determine the avalanche size distribution during such failure and find a power law decay. We compare this fatigue behavior with that obtained phenomenologically for the nucleation of Griffith cracks. Next we study numerically the fatigue behavior of random fiber bundles having simple distributions of individual fiber strengths, at stress less than the bundle's strength (beyond which it fails instantly). The average failure time is again seen to decrease exponentially as the stress increases and the avalanche size distribution shows similar power law decay. These results are also in broad agreement with experimental observations on fatigue in solids. We believe, these observations regarding the failure time are useful for quantum breakdown phenomena in disordered systems.

cond-mat.stat-mech↗

arXiv2003

Fluctuation Cumulant Behavior for the Field-Pulse Induced Magnetisation-Reversal Transition in Ising Models

The universality class of the dynamic magnetisation-reversal transition, induced by a competing field pulse, in an Ising model on a square lattice, below its static ordering temperature, is studied here using Monte Carlo simulations. Fourth order cumulant of the order parameter distribution is studied for different system sizes around the phase boundary region. The crossing point of the cumulant (for different system sizes) gives the transition point and the value of the cumulant at the transition point indicates the universality class of the transition. The cumulant value at the crossing point for low temperature and pulse width range is observed to be significantly less than that for the static transition in the same two-dimensional Ising model. The finite size scaling behaviour in this range also indicates a higher correlation length exponent value. For higher temperature and pulse width range, the transition seems to fall in a mean-field like universality class.

cond-mat.stat-mech↗

arXiv2003

Ideal Gas-Like Distributions in Economics: Effects of Saving Propensity

We consider the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving (two-body) collision. Unlike in the ideal gas, we introduce saving propensity $λ$ of agents, such that each agent saves a fraction $λ$ of its money and trades with the rest. We show the steady-state money or wealth distribution in a market is Gibbs-like for $λ=0$, has got a non-vanishing most-probable value for $λ\ne 0$ and Pareto-like when $λ$ is widely distributed among the agents. We compare these results with observations on wealth distributions of various countries.

cond-mat.stat-mech↗

arXiv2002

An Electrical Network Model of Plant Intelligence

A simple electrical network model, having logical gate capacities, is proposed here for computations in plant cells. It is compared and contrasted with the animal brain network structure and functions.

cond-mat.dis-nn↗

arXiv2002

Mathematics, Brain Modelling and Indian Concept of Mind

We describe briefly the recent advances in understanding the distributed nature of computations in the (neural) network structure of the brain. We discuss if such artificial networks will be able to perform mathematics and natural sciences. The problem of consciousness in such machines is addressed. Ancient Indian ideas regarding mind-body relations and J. C. Bose's experimental observations regarding the highly distributed computations in the plant body is discussed.

cond-mat.dis-nn↗

arXiv2001

Dynamics of linear polymers in random media

We study phenomenological scaling theories of the polymer dynamics in random media, employing the existing scaling theories of polymer chains and the percolation statistics. We investigate both the Rouse and the Zimm model for Brownian dynamics and estimate the diffusion constant of the center-of-mass of the chain in such disordered media. For internal dynamics of the chain, we estimate the dynamic exponents. We propose similar scaling theory for the reptation dynamics of the chain in the framework of Flory theory for the disordered medium. The modifications in the case of correlated disordered are also discussed.

cond-mat.soft↗

arXiv2001

Precursors of catastrophe in the BTW, Manna and random fiber bundle models of failure

We have studied precursors of the global failure in some self-organised critical models of sand-pile (in BTW and Manna models) and in the random fiber bundle model (RFB). In both BTW and Manna model, as one adds a small but fixed number of sand grains (heights) to any central site of the stable pile, the local dynamics starts and continues for an average relaxation time (τ) and an average number of topplings (Δ) spread over a radial distance (ξ). We find that these quantities all depend on the average height (h_{av}) of the pile and they all diverge as (h_{av}) approaches the critical height (h_{c}) from below: (Δ) (\sim (h_{c}-h_{av}))(^{-δ}), (τ\sim (h_{c}-h_{av})^{-γ}) and (ξ) (\sim) ((h_{c}-h_{av})^{-ν}). Numerically we find (δ\simeq 2.0), (γ\simeq 1.2) and (ν\simeq 1.0) for both BTW and Manna model in two dimensions. In the strained RFB model we find that the breakdown susceptibility (χ) (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time (τ), both diverge as the applied load or stress (σ) approaches the network failure threshold (σ_{c}) from below: (χ) (\sim) ((σ_{c}) (-)(σ)^{-1/2}) and (τ) (\sim) ((σ_{c}) (-)(σ)^{-1/2}). These self-organised dynamical models of failure therefore show some definite precursors with robust power laws long before the failure point. Such well-characterised precursors should help predicting the global failure point of the systems in advance.

cond-mat.stat-mech↗

arXiv2001

Dynamic Transitions in Pure Ising Magnets under Pulsed and Oscillating Fields

Response of pure Ising systems to time-dependent external magnetic fields, like pulsed and oscillating fields, are discussed and compared here. Because of the two time scales involved, namely the thermodynamic relaxation time of the system and the pulse width or the time period of the external field, dynamically broken symmetric phases appear spontaneously when both become comparable. A particularly simple case is that of an Ising ferromagnet below its static critical temperature, when it is perturbed for a short duration by a pulsed magnetic field competing with the existing order in the system. If the field strength and duration is more than the threshold (dependent on the temperature), the system, and consequently the magnetization, switches from one minimum to the other of the static free energy. This magnetization reversal transition here shows intriguing dynamic transition behaviour, similar to those for oscillating fields. Monte Carlo studies for such dynamic transitions are discussed and compared with the mean field results for the same and the Monte Carlo results for the oscillating field case. In particular, we discuss about the Monte Carlo results for the fluctuations and their growth behaviour near this magnetization reversal (dynamic) transition point.

cond-mat.stat-mech↗

arXiv2001

Small-world phenomena and the statistics of linear polymer networks

A regular lattice in which the sites can have long range connections at a distance l with a probabilty $P(l) \sim l^{-δ}$, in addition to the short range nearest neighbour connections, shows small-world behaviour for $0 \le δ< δ_c$. In the most appropriate physical example of such a system, namely the linear polymer network, the exponent $δ$ is related to the exponents of the corresponding n-vector model in the $n \to 0$ limit, and its value is less than $δ_c$. Still, the polymer networks do not show small-world behaviour. Here, we show that this is due a (small value) constraint on the number q of long range connections per monomer in the network. In the general $δ- q$ space, we obtain a phase boundary separating regions with and without small-world behaviour, and show that the polymer network falls marginally in the regular lattice region.

cond-mat.stat-mech↗

arXiv2001

The average distance of the n-th neighbour in a uniform distribution of random points

We first review the derivation of the exact expression for the average distance $ $ of the n-th neighbour of a reference point among a set of N random points distributed uniformly in a unit volume of a D-dimensional geometric space. Next we propose a `mean-field\rq theory of $ $ and compare it with the exact result. The result of the `mean-field\rq theory is found to agree with the exact expression only in the limit $D \to \infty$ and $n \to \infty$. Thus the `mean-field\rq approximation is useless in this context.

cond-mat.stat-mech↗

arXiv2000

A Self-organising Model of Market with Single Commodity

We have studied here the self-organising features of the dynamics of a model market, where the agents `trade' for a single commodity with their money. The model market consists of fixed numbers of economic agents, money supply and commodity. We demonstrate that the model, apart from showing a self-organising behaviour, indicates a crucial role for the money supply in the market and also its self-organising behaviour is seen to be significantly affected when the money supply becomes less than the optimum. We also observed that this optimal money supply level of the market depends on the amount of `frustration' or scarcity in the commodity market.

cond-mat.stat-mech↗

arXiv2000

Statistical mechanics of money: How saving propensity affects its distribution

We consider a simple model of a closed economic system where the total money is conserved and the number of economic agents is fixed. In analogy to statistical systems in equilibrium, money and the average money per economic agent are equivalent to energy and temperature, respectively. We investigate the effect of the saving propensity of the agents on the stationary or equilibrium money distribution.The equilibrium probablity distribution of money becomes the usual Gibb's distribution, characteristic of non-interacting agents, when the agents do not save. However with saving, even for local or individual self-interest, the dynamics become cooperative and the resulting asymmetric Gaussian-like stationary distribution acquires global ordering properties. Intriguing singularities are observed in the stationary money distribution in the market, as function of the ``marginal saving propensity'' of the agents.

cond-mat.stat-mech↗

arXiv2000

The travelling salesman problem on randomly diluted lattices: results for small-size systems

If one places N cities randomly on a lattice of size L, we find that the normalized optimal travel distances per city in the Euclidean and Manhattan metrics vary monotonically with the city concentration p. We have studied such optimal tours for visiting all the cities using a branch and bound algorithm, giving exact optimized tours for small system sizes (N<100). Extrapolating the results for N tending to infinity, we find that the normalized optimal travel distances per city in the Euclidean and Manhattan metrics both equal unity for p=1, and they reduce to about 0.74 and 0.94, respectively, as p tends to zero. Although the problem is trivial for p=1, it certainly reduces to the standard TSP on continuum (NP-hard problem) for p tending to zero. We did not observe any irregular behaviour at any intermediate point. The crossover from the triviality to the NP-hard problem seems to occur at p=1.

cond-mat.stat-mech↗

arXiv2000

Nucleation theory and the phase diagram of the magnetization-reversal transition

The phase diagram of the dynamic magnetization-reversal transition in pure Ising systems under a pulsed field competing with the existing order can be explained satisfactorily using the classical nucleation theory. Indications of single-domain and multi-domain nucleation and of the corresponding changes in the nucleation rates are clearly observed. The nature of the second time scale of relaxation, apart from the field driven nucleation time, and the origin of its unusual large values at the phase boundary are explained from the disappearing tendency of kinks on the domain wall surfaces after the withdrawal of the pulse. The possibility of scaling behaviour in the multi-domain regime is identified and compared with the earlier observations.

cond-mat.stat-mech↗

arXiv2000

Statistical Physics of the Travelling Salesman Problem

If one places N cities on a continuum in an unit area, extensive numerical results and their analysis (scaling, etc.) suggest that the best normalized optimal travel distance becomes 0.72 for the Euclidean metric and 0.92 for the Manhattan metric. The analytic bounds, we discuss here, give 0.5 and 0.92 as the lower and upper bounds for the Euclidean metric, and 0.64 and 1.17 for the Manhattan metric. When the cities are randomly placed on a lattice with concentration p, we find that the normalized optimal travel distance vary monotonically with p. For p=1, the values in both Euclidean and Manhattan metric are 1, and as p tends to zero, the values are 0.72 and 0.92 in the Euclidean and Manhattan metrics respectively.The problem is trivial for p=1 but it reduces to the continuum TSP as p tends to zero. We do not get any irregular behaviour at any intermediate point, e.g., the percolation point. The crossover from the triviality to the NP- hard problem seems to occur at p<1.

cond-mat.stat-mech↗

arXiv1999

Stick-slip statistics for two fractal surfaces: A model for earthquakes

Following the observations of the self-similarity in various length scales in the roughness of the fractured solid surfaces, we propose here a new model for the earthquake. We demonstrate rigorously that the contact area distribution between two fractal surfaces follows an unique power law. This is then utilised to show that the elastic energy releases for slips between two rough fractal surfaces indeed follow a Guttenberg-Richter like power law.

cond-mat.stat-mech↗