Area of Research

I am presently involved in the research of Seismic Hazard Assessment. A seismic hazard is the probability that an earthquake will occur in a given geographic area, within a given window of time, and with ground motion intensity exceeding a given threshold. The estimation of the expected ground motion, which can occur at a particular site, is vital to the design of important structures such as dams, bridges and nuclear power plants. The process of evaluating the design parameters of earthquakes ground motion is called seismic hazard assessment. The design parameters are Target Response Spectra (TRS), Design Acceleration Time Histories compatible with the TRS, Peak Ground Acceleration (PGA), Design Response Spectra, Seismic Coefficients etc. There exists two approaches of seismic hazard assessment - (i) Deterministic Seismic Hazard Assessment (DSHA) (ii) Probabilistic Seismic Hazard Assessment (PSHA). The process of DSHA involves the initial assessment of the maximum possible earthquake magnitude for each of the various seismic sources. Assuming that each of these earthquakes will occur at the minimum possible distance from the site, the ground motion is calculated using appropriate Ground Motion Prediction Equations (GMPEs). Unfortunately, this straightforward and intuitive procedure is overshadowed by the complexity and the uncertainty in selecting the appropriate earthquake scenario, creating the need for an alternative methodology (probabilistic), which is free from discrete selection of ’scenario earthquakes’. We do not know when earthquakes will occur, we do not know where they will occur and we do not know how big they will be. PSHA takes the randomness of earthquake occurrences in space, time and magnitude into account. Thus, by incorporating the effects of various random uncertainties in the input parameters, the PSHA approach provides an avenue to arrive at a more objective and cost-effective engineering decisions.

Monte Carlo simulation has been performed on lattice spin models with continuous energy spectrum. For the purpose of investigation, we have chosen a two-dimensional XY model where the classical spins (of unit length), located at the sites of a square lattice and free to rotate in a plane, say the XY plane, interact with nearest neighbours through a modified potential

V (θ) = 2 [1 − (cos^2 (θ/2))^ (p^2)] (1)

where θ is the angle between the nearest neighbor spins and p^2 is a parameter used to alter the shape of the potential, or in other words, p^2 controls the nonlinearity of the potential well, although variation in p^2 does not disturb the essential symmetry of the Hamiltonian. For p^2 = 1, the potential reduces to that of a conventional XY model while for large values of p^2 , the potential well gets narrower and exhibits a strongly first order phase transition.

Performance of Wang-Landau (WL) algorithm in these class of models has been tested by determining the fluctuations in energy histogram. We have found that the fluctuations in energy histogram, after an initial increase, saturates to a value proportional to 1/√f, where f is the modification factor in the WL sampling. Difficulties faced in simulating relatively large continuous systems using WL algorithm have been investigated. The main problem while simulating a continuous model using WL algorithm is that configurations near the minimum energy take a very long time to be sampled during the random walk and it becomes impractical to simulate continuous models of even moderate size because of the huge CPU time that becomes necessary. Conventional single spin flip Metropolis algorithm, as we have seen, does not work well in this model for higher values of p^2 . This work was published in Phys. Lett. A 373, 308 (2009).

Since modified XY model exhibiting first order phase transition turned out to be a very interesting problem, we have investigated it in greater detail. The motivation is to resolve the question on the nature of the phase transition in this model and the contradictions among the views put forward by different investigators for the last quarter of a century. We have applied the cluster algorithm of Wolff which turned out to be very convenient to simulate this model. Besides this, we have used Ferrenberg-Swendsen multiple histogram reweighting technique and finite size scaling rules of Lee and Kosterlitz. Our observation is that the transition is indeed first order for large values of p^2 as all the finite size scaling rules for the first order phase transition are nicely obeyed. We also observe that while the lowest order correlation function decays to zero, long range order prevails in the system via higher order correlation function. A great virtue of the Wolff algorithm is that it does not contain any adjustable parameter even while simulating a continuous model. This work was published in Phys. Rev. E 81, 022102 (2010).

We have also investigated the role played by the topological defects and the factors which are responsible for the change over the nature of the phase transition from a continuous one in the conventional XY model to a strongly first order one in the modified XY model. The average defect density is found to increase sharply as the temperature T increases through the transition temperature. The behaviour of the topological excitations with the parameter p^2 has also been studied. We observe that the role played by the vortices changes qualitatively with change in p^2 (which increases the nonlinearity of the potential well). Qualitatively, one may therefore think that the modified XY model for large values of p^2 , behaves like a dense defect system and gives rise to a first order phase transition. In simulations in a restricted ensemble in which configurations containing topological defects are not allowed, the system appears to remain ordered at all temperatures. Suppression of topological defects on the square plaquettes in the modified XY model leads to complete elimination of the phase transition observed in this model. This work was published in Phys. Rev. E 81, 041120 (2010).

The two-dimensional (2D) classical XY model with a modified form of interaction potential has been the subject of interest for a long time. The shape of the potential is modified with the introduction of an additional parameter which controls the amount of nonlinearity in the potential well. The introduction of the nonlinearity in the potential leads to a first order phase transition. Earlier we have shown that this model, now known as modified XY model, behaves like a dense defect system for larger values of the nonlinearity parameter. An important issue is the study of helicity modulus (γ) and its temperature derivative (τ ) in this model for different values of the nonlinearity parameter. This is a rather elegant concept and a particular way to locate the phase transition. Helicity modulus is a thermodynamic function which measures the “rigidity” of an isotropic system under an imposed phase twist. Although (τ ) behaves very characteristically for a continuous phase transition, its behaviour is clearly different for a first order phase transition. We have studied the connection of spin-stiffness (the helicity modulus) to topological defects in this system. This work, studying this aspect of phase transition, allows one to check whether the universal jump predicted in the KT picture is also valid in systems governed by the modified form of potential. We have also explored how disorder influences the properties of the phase transition in these 2D systems. We have performed an extensive Monte Carlo simulation on this system to estimate the helicity modulus and its temperature derivative. We found that instead of the subtle and smooth KT transition, the transition coincides with a nonuniversal jump in γ. Topological defects are shown to introduce disorder in the system, which makes the helicity jump non universal. The results corroborate the experimental observation of a non universal jump of the superconducting density in highTc superconducting films. This work has been published in Phys. Rev. E 84, 010102(R) (2011).

We also made estimates of thermodynamic quantities other than energy, specific heat etc (such as magnetization, susceptibility and fourth order cumulant of magnetization) which have not been estimated earlier in this system. These thermodynamic quantities are measured using multiple histogram reweighting of the data obtained from simulations. We employ an approach which eliminates the need to construct two-dimensional histograms, even when applied to a lattice spin model with continuous energy spectra, exhibiting a sharp first order phase transition. This approach makes judicious use of computer memory as well as CPU time. Lee-Kosterlitz method of finite size scaling for a first order transition and analysis using Binder’s cumulant method allow us to make an accurate determination of transition temperature. This work has been published in Phys. Rev. E 87, 054102 (2013).

We made a study on the performance of Wang-Landau (WL) algorithm in a lattice model of liquid crystals. For the purpose of investigation, we chose a one-dimensional array of three dimensional spins interacting with nearest neighbours via a potential Vij = −P2(cos θij ) where P2 is the second Legendre polynomial and θij is the angle between the nearest neighbour spins i and j. The spins are three-dimensional and headless, i.e, the system has the O(3) as well as the local Z2 symmetry, characteristic of a nematic liquid crystal. The model, known as the Lebwohl-Lasher (LL) model, is the lattice version of the Maier-Saupe (MS) model which describes a nematic liquid crystal in the mean field approximation. We proposed a novel method of the spin update scheme in this continuous lattice spin model and had shown that the proposed scheme reduces the auto-correlation time of the simulation and results in faster convergence. We hope that this spin update method will be of general interest in the area of research in Monte Carlo simulations of continuous lattice spin models. This work has been published in Computer Physics Communications, 183, 2616 (2012).

I have keen interests on disordered spin systems and their relaxation in the non equilibrium regime. We studied the disorder induced changes in the properties of a disordered spin system, namely the random field Ising model (RFIM). RFIM belongs to a class of disordered spin models in which disorder is coupled to the order parameter of the system. The time evolution of the domain growth, the order parameter and the spin spin correlation functions were studied in the non equilibrium regime at a non zero temperature. It was seen that the RFIM exhibits a variety of behaviours depending on the strength of the random fields. It was also observed that, except for weak random fields, exchange interaction never wins over pinning interaction to establish long range order in the system. This work has been published in Phys. Rev. E 87, 022121 (2013).

In recent years there has been a trend toward the applications of Statistical Physics to Social Sciences and interdisciplinary fields as diverse as biology, information technology, computer science etc. Statistical Physics has proved to be very fruitful to describe phenomena outside the domain of traditional physics. I got interested in opinion dynamics and currently I am involved in the statistical physical modeling of opinion dynamics. The aim is to define the opinion states of a population and the processes that determine transitions between such states. The important issue is if this is possible and whether this approach could shed new light on the process of opinion formation. We introduce a stochastic model of binary opinion dynamics in which the opinions are determined by the size of the neighbouring domains. The exit probability shows a step function behaviour indicating a phase transition even in one dimension in contrast to other well known opinion dynamics models where no such phase transition takes place. The coarsening study of the model also yields novel exponent values. A lower value of persistence exponent is obtained in the present model, which involves stochastic dynamics, when compared to that in a similar type of model with deterministic dynamics. This apparently counter-intuitive result is justified using further analysis. Based on these results it is concluded that the proposed model belongs to a unique dynamical class. This work has been published in Phys. Rev. E 88, 022152 (2013).

We report a study of nonequilibrium relaxation in a two-dimensional random field Ising model at a nonzero temperature. We attempt to observe the coarsening from a different perspective with a particular focus on three dynamical quantities that characterize the kinetic coarsening. We provide a simple generalized scaling relation of coarsening supported by numerical results. The excellent data collapse of the dynamical quantities justifies our proposition. The scaling relation corroborates the recent observation that the average linear domain size satisfies different scaling behavior in different time regimes. This work has been published Phys. Rev. E 89, 042144 (2014).