Some techniques for free energy calculations are discussed, both in equilibrium case and nonequilibrium case. The first part deals with a temperature accelerated method that reduces the simulation time. A two scale system is designed to get rid of the error introduced by taking an artificially higher temperature for the variables of interest. Some convergence results are obtained by studying behavior of corresponding Poisson equation in the whole space R ^N. First strong convergence is established for any finite time interval. Then this result is used to study the long time behavior with the help of a spectral gap under certain weighted norm of a function and its derivative. In particular, the convergence rate of the invariant measure of the perturbed system to that of its limit is given in terms of time separation. Then we comment on its implications for sampling with examples.

The second part discusses some methods in free energy reconstruction from nonequilibrium trajectories, which is of much interest from an experimental point of view. A one dimensional diffusion model is introduced to describe the dynamic of collective variable. This model is used to compute the equilibrium free energy from a set of short nonequilibrium trajectories that are terminated as soon as they hit a predetermined point. Working under the assumption that the variable of interest satisfies an overdamped Langevin equation, it is shown that the nonequilibrium trajectories sample a nonequilibrium stationary distribution which can be calculated in closed form. This distribution can be used to estimate the free energy via an inversion procedure that is similar in spirit to the one used in the equilibrium context but differs in the details. Then we discuss several generalizations of this method to include external force. Results are compared to that those on Jarzynski equality.