Work statistics at first-passage times

We investigate the work fluctuations in an overdamped non-equilibrium process that is stopped at a stochastic time. The latter is characterised by a first passage event that marks the completion of the non-equilibrium process. In particular, we consider a particle diffusing in one dimension in the presence of a time-dependent potential U ( x , t ) = k | x − v t | n / n , where k > 0 is the stiffness and n > 0 is the order of the potential. Moreover, the particle is confined between two absorbing walls, located at L ± ( t ) , that move with a constant velocity v and are initially located at L ± ( 0 ) = ± L . As soon as the particle reaches any of the boundaries, the process is said to be completed and here, we compute the work done W by the particle in the modulated trap upto this random time. Employing the Feynman–Kac path integral approach, we find that the typical values of the work scale with L with a crucial dependence on the order n . While for n > 1, we show that ⟨ W ⟩ ∼ L 1 − n exp k L n / n − v L / D for large L , we get an algebraic scaling of the form ⟨ W ⟩ ∼ L n for the n k , (ii) ⟨ W ⟩ ∼ L 2 for v = k and (iii) ⟨ W ⟩ ∼ exp − ( v − k ) L for v