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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.
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