Stochastic resetting in underdamped Brownian motion

We consider a single Brownian particle in one dimension in a medium at a constant temperature in the underdamped regime. We stochastically reset the position of the Brownian particle to a fixed point in the space with a constant rate r whereas its velocity evolves irrespective of the position of the particle and the stochastic resetting mechanism. The nonequilibrium steady state of the position distribution is studied for this model system. Further, we study the distribution of the position of the particle in the finite time and the approach to the nonequilibrium steady state distribution with time. Numerical simulations are done to verify the analytical results. (paper: classical statistical mechanics, equilibrium and non-equilibrium)