Developing analytical methods to solve multi-state problems
Thesis title: Generalized multistate systems in statistical and quantum mechanics: Analytical and semi-analytical method development
Supervisor: Dr. Aniruddha Chakraborty
Statistical physics
Reaction-diffusion problems of diffusing entities are of great interest in a broad range of physical systems in physics, chemistry, and biology. In statistical mechanics, we employ the Smoluchowski equation to model the general reaction-diffusion phenomena. The statistical description of such processes explains different statistical observables like the rate of the reaction, the survival probability before the reaction, and also the first passage time profile, etc. of the system. We employ different potential energy curves (including flat, linear, piecewise linear, parabolic), acting as an external field along with the arbitrary sink function (including localized Dirac delta function, non localized exponential and Gaussian function), used to ensure the trapping or reaction of the particle executing random walk. Since many systems (electronic relaxation in solution, electron transfer processes, processes occurring in molecular aggregates and immunological systems, electrostatic steering in enzyme ligand binding in biological processes ) could be modeled by this equation, the primary aim is to get the solution of the equation analytically in time and Laplace domain. Therefore, we try to develop a different method of solving the equation by analytical or semi-analytical approach.
Quantum mechanics
The primary goal is to solve the time-dependent Schrödinger equation for different time-dependent Hamiltonian. Very few cases of the equation with a time-dependent potential can be solved exactly. The application of the equation is well known and mainly the equation is engaged to describe different microscopic phenomena, where the classical approach is not sufficient to describe the phenomenon. In the multistate quantum problems, we try to develop the method of solving the equation when states are coupled by arbitrary function. We also are interested to calculate the transition time among the quantum states.