Supervisor: Dr. Aniruddha Chakraborty

Multi-state models appear in wide areas of science which include systems ranging from physics, chemistry, and even in economics. The theory of such complex systems can be basically explained using statistical physics or quantum physics. The governing equations of multi-state problems are usually coupled-Smoluchowski/Schrödinger equations. The descriptions using Smoluchowski (or) Schrödinger equations appropriately can explain various molecular processes inside gaseous phase and condensed phases respectively. However, such systems have mathematical solutions only for few simple cases. And some solutions are available in the time-transformed Laplace variable (s), which can only give kinetic picture regarding such processes. By solving such systems in time-domain, the exact dynamics of the molecules can be given. In this thesis, we develop mathematical methods for such equations to derive time-domain solution. The presented time-domain profiles are useful in understanding such processes as a function of molecular and system parameters [1, 2, 3, 4, 5, 6].

Potential traps decorated by delta interaction potentials constitute dimple potential models. Quantum properties of such models has been useful in studying the properties of a Bose-Einstein condensate. We derived exact time-dependent wavefunctions for the potential cases V(x)=a|x|, 1/2*a*x^2 decorated by a delta-potential analytically. We investigate the effect of dimpleness on the energy density and derive properties such as chemical potential, critical temperature, atomic density and condensate fraction of a harmonic trap with respect to the various strength, and location of Dirac- δ functions.

(preprints to be updated in the publications section)

We aim to develop efficient algorithms to solve multi-channel quantum scattering problems. We introduced a new model potential having a finite short width and arbitrary height which has the same advantages as the Dirac delta model potential. A collection scheme is introduced using the model potential to represent an arbitrary potential. The scheme is seen to be efficient and is analytical with the use of transfer-matrix technique [7]. The model potential is also another exactly solvable model in the time-dependent framework other than the Dirac delta scattering model [8]. We aim to extend the algorithm for solving arbitrary multi-channel quantum scattering problems.

Several life-making processes such as cell division, DNA replication, RNA folding are either essentially a polymer looping process or the looping are their rate determining steps. We study the looping dynamics of an polymer chain inside a solvent bath by using the Langevin equation. We develop python codes for studying the statistical properties of the process and to study the effect of system and monomer parameters on the looping rates. The code is being extended to study the effect in considering co-polymers, block polymers using various types of solvent forces, etc..

(preprints to be updated in the publications section)

7) Deriving stochastic governing equations for non-Markovian processes

8) Quantum-mechanical solvent effects in condensed-phase chemical dynamics