We present analytical and numerical studies of fluctuations about and transitions from the nonequi- librium dissipative steady state to the stable equilibrium state of a damped physical pendulum driven by constant torque and small white noise. We find the probability density function of the local flucutations about the nonequilibrium steady state and the transition rate from the nonequilibrium state to the equilibrium state by constructing an asymptotic solution to the Fokker-Planck equation by the WKB method. The solution to the eikonal (Hamilton-Jacobi) equation is constructed both analytically by an asymptotic series expansion, and numerically by integrating the set of characteristic equations. We compare the numerical results to the analytical calculations and determine the limits of validity of the asymptotic approximation. We apply the results to the case of the hysteretic Josephson junction and discuss a generalization of the numerical and analytical methods to other systems.