Broadly the subject area of my research is Statistical Physics and Complex Systems. By complex systems I mean systems involving many degrees of freedom and generally driven out of equilibrium. I am particularly interested in understanding complex systems as manifestations of some general principles of statistical physics.  In the Biological world we often encounter prototypical complex systems which are large (although not in the thermodynamic limit) and are driven. My present work and interest is broadly based on systems like 1. Resistive Regime of Superconductors, 2. Transport under Equilibrium and Nonequilibrium conditions, 3. Bose-Einstein Condensation 4. Pattern Formation 5.  Hydraulic Jump.

The work that I have done so far and am continuing at present can be divided into the following subgroups.


Directed transport under equilibrium conditions:

My works on driven diffusive transport has led me to having some results, on the basis of the model I am working on, that predicts the possibility of achieving directed motion in equilibrium. The general textbook statistical physics (of course without a general proof) teaches (preaches) that it is not possible to have directed motions in equilibrium. The Somoluchowski-Feynman ratchet is considered the prototypical system to show there is no possibility of directed motion in equilibrium. This consideration is clearly flawed because the Smoluchowski-Feynman ratchet breaks symmetry by temperature and eventually under equilibrium conditions there is no broken symmetry and hence no tirected motion. I am pursuing over a year now some calculations based on one of my models to establish the possibility of having directed motion in equilibrium. Thses results, if experimentally can thoroughly change our understanding of equilibrium statistical physics and can prove important to generally understand many biological transports as well which I believe not all happen under nonequilibrium circumstances.

Pattern formation: 

The subject of pattern formation deals with understanding how order emerges and evolves in space and time from a disordered or uniform state. To understand some of the basic aspects of pattern formation we have worked on Reaction-Diffusion systems at a mean field level where one is mainly concerned with the kinetic equation of reacting species which are transported by diffusion. We mostly have done weakly nonlinear analysis on Gierer-Meinhardt model near an instability threshold and have also analyzed some instabilities to the nonlinear states using amplitude equations near Turing and Hopf instability boundaries. These amplitude equations are generally of complex Ginzburg-Landau type and are universal in structure near an instability threshold of particular type. The main objectives of research, on my part, have so far been investigating various localized structures like spiral and target patterns, one dimensional asynchronous waves, some phase transitions between rolls to squares and hexagons, identifying oscillatory (Hopf) instabilities to stationary states etc. 

Recently, I have had some interest in analysing One-dimensional superconducting systems at a regime (resistive state) where a superconducting state co-exists with the the normal state. The goal of my study is to understand the morphology of the superconducting order parameter, the associated chemical potential and their stabilities and interactions etc. using the analytic tools of pattern formation and hydrodynamics. Based on these tools, a lot can be revealed within the range of validity of time dependent complex Ginzburg-Landau models for such systems.

A special type of reaction-diffusion systems are those where the reacting mass is conserved. These systems have some characteristics that might shed some light on the general processes of chemotaxis and find applications in the biological context. These are the systems I am also looking at.

One dimensional systems of interacting particles transported by diffusion and driven is an area of my interest. These systems are very strongly correlated due to volume exclusion constraint and hydrodynamic models have been shown to fail at this regime. Strong correlation makes application of kinetic theory a difficult job as well. The problem is challenging and is a subject of active research these days. 

Biophysics: 

The biophysics part of my work involves protein folding, quantitative genetics (sympatric speciation), cell spreading, cell-motility etc.

We developed an efficient method of measuring the solvent accessible surface are of  amino acids and have designed an effective hydrophobic interaction potential based on this. Hydrophobicity is the dominant long range force that makes the protein collapse onto a conformation close to its native fold and then other interactions are believed to take up the fine tuning process to lead to the final native fold. In minimization of the free energy corresponding to the native fold of the protein, the hydrophobic interaction is expected to have a lot of contributions. This fact is strongly supported through the potential function we have designed. Depending on hydrophobic interaction alone we could discriminate between the native fold and several thousands of decoys for about 20 small proteins. I intend to carry on this research to understand the statistical mechanics of the folding process.

In the quantitative genetics part of our work we focused on understanding sympatric speciation ( competition driven speciation) under the action of a Lotka-Volterra dynamics. The competition drives species apart on a trait space in time so that they eventually become reproductively isolated leading to speciation. Considering a coupled set of species, strong competition in the population of one of them can even induce sympatric speciation onto the other species which themselves do not have much of competition within their populations. This is an interesting statistical problem under non equilibrium conditions.

The cell spreading part of my work deals basically so far with understanding the dynamic phase transitions of a spreading cell. A cell spreads on a suitable surface to perform various activities, the process is basically polymerization driven (although there are other complexities at the bio-molecular level). The dynamic phases (phenotypes) being large scale observation can be captured on some mean field level phenomenological modeling and thats what I have so far studied and presently working on to extend it.

 Molecular self-assembly: 

We have put forward an efficient Monte Carlo cluster algorithm to capture the equilibrium statistics of a self assembling molecular system. Self assembly of molecules of various microscopic symmetries give rise to exotic macroscopic structures. This is a fascinating statistical problem involving multiple scales. Processes are generally hierarchical and to capture the statistics of such self-assembling transitions one has to simulate the system at multiple space and time scales. Applying a cluster move Monte Carlo keeping the detailed balance intact is always tricky. We have proposed a general scheme in continuum modifying the famous Swendsen-Wang-Woolf method for spins on a lattice which applies to a wide class of self assembling systems. Our method enables one have a lot of control over the simulation and is extremely powerful. Carrying on this research further is a priority to me.

Networks and synchronization:

Collective dynamics of interacting nodes on a network, emergent properties, synchronization etc. are the issues that I am presently having interest on. At present we would like to look at systems where the time scales of the nodal dynamics matches that of the spreading of the connections between nodes or the time scale of the network growth. Such processes are commonplace in biological contexts, like spreading of a viral disease where the lifetime of the virus is comparable to the time scale of its spreading in a community. Emergent properties in such web of interconnected dynamic units have so far been mostly observed at a scale where the local and global time scales are much different. Its an active field of research with a lot of scope for applications and understanding.