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-vt|(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 >similar to L1-n exp{(kL(n)/n-v)L/D] for large L, we get an algebraic scaling of the form < W >similar to L-n for the n < 1 case. The marginal case of n = 1 is exactly solvable and our analysis unravels three distinct scaling behaviours: (i) < W >similar to L for v > k, (ii) < W >similar to L-2 for v = k and (iii) < W >similar to exp[-(v-k)L for v < k. For all cases, we also obtain the probability distribution associated with the typical values of W. Finally, we observe an interesting set of relations between the relative fluctuations of the work done and the first-passage time for different n-which we argue physically. Our results are well supported by the numerical simulations.