K P N Murthy

Brief summary of some of my scientific contributions

I am interested in thermodynamics, statistical mechanics, Boltzmann and non-Boltzmann Monte Carlo simulation methods, Monte Carlo simulation of radiation transport, random waks in regular and disordered lattices, fractal geometry, multifractal measures, non-linear dynamics, and chaos.

A common thread that runs through most of my research work carried out over more than four decades is on fluctuation phenomena in model non-equilibrium systems, investigated employing random walk formulations and Monte Carlo simulations.
In the early part of my career, I carried out extensive work on fast breeder reactor program. Important project work carried out include

● bulk and complementary shielding of the Fast Breeder Test Reactor (FBTR) and to some extent bulk shielding of Prototype Fast Breeder Reactor;
● Research and development in the area of Monte Carlo simulation of neutron and gamma transport through thick shields;
● extensive work on the generation of fission products in fast reactor cores; a FORTRAN code called CHANDY was developed and a library of data on fission products were assimilate;
● extensive calculations on the decay power from spent fast reactor fuel were carried out. The code CHANDY and the associated data library are being used routinely in fission product and decay power calculations in IGCAR.

The research component of my work, while at the reactor group, mainly consisted of theoretical and computational Monte Carlo simulation of neutral particle transport through a thick scattering - cum - absorbing medium. The fluctuations of transmission are large. Analogue Monte Carlo simulation is impossible. Techniques like exponential biasing that reduce fluctuations but preserve average are required.

● Employing random walk and first passage time formulations, we could demonstrate analytically that optimal exponential biasing depends on the scattering probability and scattering anisotropy of the medium.
● Exponential biasing, if employed even slightly above an optimal measure (i.e. over-biasing), introduces a long tail in the distribution of the transmission rendering error estimates difficult and often meaningless.

Then I took up a study of random walks on a disordered lattice.

● An exact analytical expression for the Mean First Passage Time (MFPT) in terms of quenched disorder at the lattice sites, was derived. This work, carried out in collaboration with Prof. Klaus W. Kehr, is the most cited of my work; for, the analytical expression derived has found application in the study of fluctuations of transport in disordered systems in a large variety of contexts.
●We also showed that the fluctuations of the MFPT over Sinai disorder are self similar and quantified them through a multi-fractal analysis.
● We could construct an Iterated Functions System (IFS) from Sinai disorder and study the stochastic dynamical evolution of escape probability; we found interestingly that the global dynamics exhibits rich intermittency.
● In a subsequent study which formed a part of the Ph.D. thesis of my student R. Harish, we showed that intermittency obtains even when Sinai conditions are not obeyed.

An early work on Monte Carlo simulation carried out in collaboration with S. Dattagupta, was on the statistics of switching time of a two mode laser. More recently extensive Monte Carlo simulation of Isotropic - Nematic transition in liquid crystals confined to the pores of a random network were carried out in collaboration with V. S. S. Sastry and K. Venu. We modelled a porous network in terms of quenched disorder in a three dimensional cubic lattice with periodic boundaries.

● We proposed a phenomenological relation that expresses the strength of disorder in terms of the Monte Carlo system size, the average size of a pore in the porous medium and the size of a liquid crystal molecule.
● For the first time, we could reproduce the experimentally observed suppression of transition temperature with decreasing pore size and the enthalpy of transition.
● We could also find signatures of softening of the transition from weakly first order to second order when the average pore size decreases to very small values.

Employing multi-canonical Monte Carlo and Wang-Landau algorithm, we investigated the phase behaviour and phase transition in liquid crystals under different homeotropic boundary conditions. An algorithm to improve the Wang-Landau Monte Carlo, for problems with continuous energy was developed in a collaborative work with D. Jayasri, and V. S. S. Sastry. Employing multi-canonical Monte Carlo simulation, the competition between the radial order imposed by homeotropic spherical boundary and the axial order preferred by the mutual interaction mediated by the disordering entropy were investigated in a collaborative study with Sairam, N. Satyavathi and V. S. S. Sastry,
S. L. Narasimhan and I proposed an Interacting Growth Walk (IGW) to generate a compact and long, lattice polymer especially at specified growth temperatures TG.

● An attractive feature of IGW is that it does not suffer much from attrition especially at low TG. In fact at TG=0 there is no attrition at all on a two dimensional square lattice.
● IGW model exhibits a transition from a collapsed phase at low TG to an extended phase at high TG.
●We found that IGW belongs to the same universality class as the Interacting Self Avoiding Walks (ISAW).
● Interpreting IGW as an approximate multicanonical ensemble, we could relate the non-equilibrium TG to equilibrium temperature T.
● More importantly, from the fluctuations of T for a given TG we could extract specific heat which exhibits a broad hump besides a sharp transition suggesting that IGW ensemble may be used for modeling glassy homopolymer.

An IGW does not generate an equilibrium ensemble polymer configurations describing a closed system. In this sense we can view IGW as non-equilibrium analogue of PERM-B walks of Grasberger. It is similar to Kinetic Growth Walk (KGW) being analogue of Rosenbluth-Rosenbluth walk that visits an unvisited nearest neighbour site with equal probability.

● The Rosenbluth-Rosenbluth (RR) weight can be interpreted as representing the atmosphere of a self avoiding walk.
● Essentially the value of the atmosphere is the product of the unvisited nearest neighbour sites seen by the growing self avoiding walk.
● This quantity differs from one walk to the other and hence is a random variable. It has been shown that the average of this random variable over a suitably augmented ensemble of Kinetic Growth Walks gives an estimate of the microcanonical entropy of Interacting self avoiding walk (ISAW).
● This is an important finding since from microcanonical entropy one can determine all other thermodynamic properties of ISAW.
● Ponmurugan, Sridhar, Narasimhan and me generalized the notion of atmosphere to IGW and showed that this could also lead to a good estimate of microcanonical entropy.
● More importantly we could show that IGW can indeed provide an alternative to pruning and enriching required to study the thermodynamics of ISAW as a function of temperature.
● Klaus W. Kehr, Hailermariam Ambaye and I showed that extreme value statistics could explain an intriguing experimental finding of thickness dependent mobility of charge carriers across a thin film of amorphous material.
● In the context of coarsening phenomena me and my colleagues V. Sridhar and M. C. Valsakumar showed that the life time distribution obeys an asymptotic scaling; the scaling function exhibits singularities at either ends of the support due to persistence and slow dynamics.

A piece of work on nonlinear dynamics carried out in collaboration with S. Venkadesan, M. C. Valsakumar and S. Rajasekar consisted of embedding a scalar time series in a low dimensional phase space by constructing time delay vectors and studying the underlying dynamics through exponential divergence plots and largest Lyapunov exponent. We could show that a noisy looking experimental time series from a tensile test of Al-Mg alloy in plastic flow domain has originated from a low dimensional chaotic dynamics.
Klaus W. Kehr and I investigated an entropy barrier model and obtained analytically the slow glassy relaxation times. Such a glassy relaxation also obtains in a reaction diffusion model as shown by me and Prof. Gunter Schuetz : particles diffuse on a lattice; when two or more of them meet at a lattice site, they annihilate each other; the dynamics becomes slower and slower as time proceeds; the density - density correlations exhibit ageing scaling: The more you wait, longer does it take, to equilibrate.
An oft-cited work carried out in collaboration with G. Ananthakrishna and M. C. Valsakumar, pertained to linearization of non-linear Langevin equations; we proposed single and double Gaussian decoupling of higher moments and tested the model with results of exact Monte Carlo simulation of the corresponding non-linear Langevin process.
An early work carried out in collaboration with M. C. Valsakumar was on diffusion controlled multiplicative processes. We showed that the typical number of particles in the system increases with time and the time dependence is different from that of the average. Also the fluctuations of the growth process diverge exponentially with time.
A piece of work carried out in collaboration with S. L. Narasimhan pertains to correlations in stochastically generated sequence of symbols. We found that coarse graining introduces spurious long range correlations.
Algorithms, inspired by statistical mechanics, for restoration of digital images were investigated. These are based on Bayesian methodology and employ Ising / Potts spin Priors. Cluster algorithms can be easily incorporated in these models to improve the efficiency of the restoration process.
A monograph entitled Monte Carlo: Basics, published by the Indian Society of Radiation Physics (ISRP/TD-3, 2000) and a book on Monte Carlo methods in statistical physics, published by the Universities Press, Hyderabad (2004) are based on my work and lectures on Monte Carlo theory and practice. Both these books and the versions archived in cond-mat e-print service are being used as a teaching material in several places within our country and abroad.
Another review of mine that has caught wide attention is arxiv: cond-mat/0601566 on Ludwig Boltzmann, Transport equation and the Second Law. This review is essentially based on the invited talk I gave at the Sixteenth National Symposium on Radiation Physics held at the Meenakshi College for Women, Chennai during 16-18, January 2006. An extended version of this was written up as a book and distributed to the students of the Summer Training in Physics - 2006 (STIP-2006), held at Materials Science Division, IGCAR, Kalpakkam during 29 May - 7 July 2006, under the title Trends in thermodynamics and statistical mechanics. These notes were further edited with description of more recent work and has since been published by the Universities Press in the year 2009 under the title Excursions in Thermodynamics and Statistical Mechanics.
A piece of work carried out with Anjan Prasad and M Suman Klayan relate to non-equilibrium work fluctuations and their relation to equilibrium free energies. We find that dissipation defined as average work minus the reversible work goes to zero in the reversible limit. However the area of the work distributions from its lower limit to reversible work increases when the process becomes more and more quasi-static. This quantity, often called as the probability of violation of the second law, goes to one-half in the reversible limit. We have found a reason for this counter-intuitive behaviour : the standard deviation which is proportional to the squre root of the mean dissipation goes to zero slower in the quasi-static limit.
Another piece of work carried out in collaboration with my students P. Ch Sandhya and Hima Bindu Kolli relates to thermodynamics and statistical mechanics of bond fluctuating lattice polymers. We have calculated Landau-Ginzberg free energy as a function of order parameter and shown that the collapse transition is discontinuous. R Bharath, M Suman Kalyan and me found an efficient way of calculating joint density of states by averaging over an entropic ensemble employing suitable un-weightng and re-weighting factors. In recent work carried out in collaboration with M Suman Kalyan, we developed an algorithm to calculate free energy difference by combining Wang-Landau Monte Carlo, Jarzynski work fluctuation relation and instantaneous switching.
In a collaborative work carried out with Siva Nasarayya Chari and Inguva Rama Rao, we proposed a scheme for truncating the BBGKY hierarchy of equations, at the level of three particle distribution. This scheme results in a set of coupled equations for single particle distribution and the two point correlation. More importantly we have shown analytically that, irrespective of the truncation of the hierarchy, the closure scheme preserves the time reversibility. Therefore, we added a BGK term to kinetic equation to bring in irreversibility. We have demonstrated the Crooks' fluctuation theorem over both interacting and non-interacting systems, by employing fluctuating lattice Boltzmann model.
In a very recent work carried out in collaboration with M Suman Kalyan, we developed an algorithm to calculate free energy difference by combining Wang-Landau Monte Carlo, Jarzynski work fluctuation relation and instantaneous switching. Preliminary results show that the algorithm is fast and predicts free energy differences accurately.