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 characterized 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 $L_{\pm}(t)$, that move with a constant velocity $v$ and are initially located at $L_{\pm}(0) = \pm L$. 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 $\mom{W} \sim L^{1-n}~\exp \left[ \left( {k L^{n}}/{n}-v L \right)/D \right]$ for large $L$, we get an algebraic scaling of the form $\mom{W} \sim L^n$ for the $n<1$ case. The marginal case of $n=1$ is exactly solvable and our analysis unravels three distinct scaling behaviours: (i) $\mom{W} \sim L$ for $v>k$, (ii) $\mom{W} \sim L^2$ for $v=k$ and (iii) $\mom{W} \sim \exp\left[{-(v-k)L}\right]$ 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.