Work statistics at first-passage times

Abstract

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_{\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 $\langle W\rangle \sim L^{1-n}~\text{exp} \left[ \left( {k L^{n}}/{n}-v L \right)/D \right] $ for large L , we get an algebraic scaling of the form $\langle W\rangle \sim L^n$ for the n k , (ii) $\langle W\rangle \sim L^2$ for v = k and (iii) $\langle W\rangle \sim \text{exp}\left[{-(v-k)L}\right]$ 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.

Keywords

• stochastic thermodynamics
• first passage time
• work statistics
• Feynman–Kac path integral method