Probability distribution of work done on a two-level system during a nonequilibrium isothermal process

We present an exact calculation of the probability density for the work done by an external agent on a two-level system. Due to the external drive, both the transition rates between the two states and their energies depend on time. Within this setting we calculate the probability of every possible sample path of the system evolution and also the work done along any such path. The general procedure yields an evolution equation for the characteristic function of the work. Assuming that the energies change with constant rates, the properties of the work distribution are controlled by a single parameter representing the ratio of the time scales of the driving protocol, and of the internal dynamics, respectively. We calculate the mean work and characterize those sample paths which are not in agreement with the second law. In the slow driving limit, the probability density for the work collapses to a delta function localized at the reversible work. In the strongly nonequilibrium regime, the most probable work is smaller and the mean work is bigger than the reversible work.