Anjan Kumar Chandra

A small world network of prime numbers

According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers: n=p+q. We construct a network where each node is a prime number and corresponding to every even number n, we put a link between the component primes p and q. In most cases, an even number can be broken up in many ways, and then we chose one decomposition with a probability |p-q|α. Through computation of average shortest distance and clustering coefficient, we conclude that for α>-1.8 the network is of small world type and for α<-1.8 it is of regular type. We also present a theoretical justification for such behaviour.

Quantum quenching, annealing and computation

Quantum Approach to Classical Thermodynamics and Optimization.- Non-equilibrium Dynamics of Quantum Systems: Order Parameter Evolution, Defect Generation, and Qubit Transfer.- Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line.- Adiabatic Perturbation Theory: From Landau-Zener Problem to Quenching Through a Quantum Critical Point.- Quench Dynamics of Quantum and Classical Ising Chains: From the Viewpoint of the Kibble-Zurek Mechanism.- Quantum Phase Transition in the Spin Boson Model.- Influence of Local Moment Fluctuations on the Mott Transition.- Signatures of Quantum Phase Transitions via Quantum Information Theoretic Measures.- How Entangled Is a Many-Electron State?.- Roles of Quantum Fluctuation in Frustrated Systems - Order by Disorder and Reentrant Phase Transition.- Exploring Ground States of Quantum Spin Glasses by Quantum Monte Carlo Method.- Phase Transition in a Quantum Ising Model with Long-Range Interaction.- Length Scale-Dependent Superconductor-Insulator Quantum Phase Transitions in One Dimension: Renormalization Group Theory of Mesoscopic SQUIDs Array.- Quantum-Mechanical Variant of the Thouless-Anderson-Palmer Equation for Error-Correcting Codes.- Probabilistic Model of Fault Detection in Quantum Circuits.

Phase transitions and non-equilibrium relaxation in kinetic models of opinion formation

We review in details some recently proposed kinetic models of opinion dynamics. We discuss several variants including a generalised model. We provide mean field estimates for the critical points, which are numerically supported with reasonable accuracy. Using non-equilibrium relaxation techniques, we also investigate the nature of phase transitions observed in these models. We also study the nature of correlations as the critical points are approached.

Percolation in a kinetic opinion exchange model.

We study the percolation transition of the geometrical clusters in the square-lattice LCCC model [a kinetic opinion exchange model introduced by Lallouache, Chakrabarti, Chakraborti, and Chakrabarti, Phys. Rev. E 82, 056112 (2010)] with the change in conviction and influencing parameter. The cluster is comprised of the adjacent sites having an opinion value greater than or equal to a prefixed threshold value of opinion (Ω). The transition point is different from that obtained for the transition of the order parameter (average opinion value) found by Lallouache et al. Although the transition point varies with the change in the threshold value of the opinion, the critical exponents for the percolation transition obtained from the data collapses of the maximum cluster size, the cluster size distribution, and the Binder cumulant remain the same. The exponents are also independent of the values of conviction and influencing parameters, indicating the robustness of this transition. The exponents do not match any other known percolation exponents (e.g., the static Ising, dynamic Ising, and standard percolation). This means that the LCCC model belongs to a separate universality class.

MODELING TEMPORAL AND SPATIAL FEATURES OF COLLABORATION NETWORK

The collaboration network is an example of a social network which has both non-trivial temporal and spatial dependence. Based on the observations of collaborations in Physical Review Letters, a model of collaboration network is proposed which correctly reproduces the time evolution of the link length distributions, clustering coefficients, degree distributions, and assortative property of real data to a large extent.

Quantum phase transition in a disordered long-range transverse Ising antiferromagnet

We consider a long-range Ising antiferromagnet put in a transverse field (LRTIAF) with disorder. We have obtained the phase diagrams for both the classical and quantum cases. For the pure case applying quantum Monte Carlo method, we study the variation in order parameter (spin correlation in the Trotter direction), susceptibility, and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We discuss generally the phase diagram for the same LRTIAF model with perturbative Sherrington-Kirkpatrick-type disorder. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the pure LRTIAF system, an infinitesimal SK spin-glass disorder is enough to induce a stable glass order in the LRTIAF. This glass order eventually gets destroyed as the thermal or quantum fluctuations are increased beyond their threshold values and the transition to paramagnetic phase occurs. Analytical studies for the phase transitions are discussed in details in each case. These transitions have been confirmed by applying classical and quantum Monte Carlo methods. We show here that the disordered LRTIAF has a surrogate incubation property of the SK spin glass phase.

Themes and Applications of Kinetic Exchange Models: Redux

In this article, we briefly discuss the general formalism of kinetic exchange models and their various applications in economics and sociology. Inspired from the kinetic theory of gases in statistical physics, the kinetic exchange model for closed economic systems were first proposed by simply considering the agents as gas molecules, and wealth of agents as kinetic energy exchanged amongst the gas molecules. The formalism had been successfully applied to modeling of wealth distributions in 2000s. This has further spurred new research in recent times in various areas of soft sciences—firm dynamics, opinion formation in the society, etc.

Dynamical percolation transition in the Ising model studied using a pulsed magnetic field.

We study the dynamical percolation transition of the geometrical clusters in the two-dimensional Ising model when it is subjected to a pulsed field below the critical temperature. The critical exponents are independent of the temperature and pulse width and are different from the (static) percolation transition associated with the thermal transition. For a different model that belongs to the Ising universality class, the exponents are found to be same, confirming that the behavior is a common feature of the Ising class. These observations, along with a universal critical Binder cumulant value, characterize the dynamical percolation of the Ising universality class.

Noise-induced rupture process: phase boundary and scaling of waiting time distribution.

A bundle of fibers has been considered here as a model for composite materials, where breaking of the fibers occur due to a combined influence of applied load (stress) and external noise. Through numerical simulation and a mean-field calculation we show that there exists a robust phase boundary between continuous (no waiting time) and intermittent fracturing regimes. In the intermittent regime, throughout the entire rupture process avalanches of different sizes are produced and there is a waiting time between two consecutive avalanches. The statistics of waiting times follows a Γ distribution and the avalanche distribution shows power-law scaling, similar to what has been observed in the case of earthquake events and bursts in fracture experiments. We propose a prediction scheme that can tell when the system is expected to reach the continuous fracturing point from the intermittent phase.

Jamming of directed traffic on a square lattice

The phase transition from a free-flow phase to a jammed phase is an important feature of traffic networks. We study this transition in the case of a simple square lattice network for different values of data posting rate (ρ) by introducing a parameter p which selects a neighbour for onward data transfer depending on queued traffic. For every ρ there is a critical value of p above which the system becomes jammed. The ρ–p phase diagram shows some interesting features. We also show that the average load diverges logarithmically as p approaches pc and the queue length distribution exhibits exponential and algebraic nature in different regions of the phase diagram.