We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.

We consider a system-reservoir model where the reservoir is modulated by an external noise. Both the internal noise of the reservoir and the external noise are stationary, Gaussian and are characterized by arbitrary decaying correlation functions. Based on a relation between the dissipation of the system and the response function of the reservoir driven by external noise we numerically examine the model using a full bistable potential to show that one can recover the turn-over features of the usual Kramers' dynamics when the external noise modulates the reservoir rather than the system directly.We derive the generalized Kramers' rate for this nonequilibrium open system.

Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed [Phys. Rev. E 65, 021109 (2002); 66, 051106 (2002)] to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers' turnover and the quantum Arrhenius plot.

In the present study, we have proposed the Fokker-Planck equation in a simple way for a Langevin equation of motion having ordinary derivative (OD), the Gaussian random force and a generalized frictional memory kernel. The equation may be associated with or without conservative force field from harmonic potential. We extend this method for a charged Brownian particle in the presence of a magnetic field. Thus, the present method is applicable for a Langevin equation of motion with OD, the Gaussian colored thermal noise and any kind of linear force field that may be conservative or not. It is also simple to apply this method for the colored Gaussian noise that is not related to the damping strength.

Based on a Fokker-Planck description of external Ornstein-Uhlenbeck noise and cross-correlated noise processes driving a dynamical system we examine the interplay of the properties of noise processes and the dissipative characteristic of the dynamical system in the steady state entropy production and flux. Our analysis is illustrated with appropriate examples.