Injected Power Fluctuations in Langevin Equation

Using path integral method, we compute exactly the probability density function of the power (averaged over a large time interval of length τ) injected (and dissipated) by the random force into a Brownian particle driven by a Langevin equation. The resulting distribution, as well as the associated large deviation function, display strong asymmetry, whose origin is discussed. Moreover, the so-called “Fluctuation Theorem” is shown not to be verified in this dissipative system, although any numerical or experimental evidence of this violation would be extremely difficult to establish. Finally, we show, using numerical simulations, that this distribution seems to be insensitive to the presence of a pinning potential, despite a large disparity of typical behaviours of the Brownian particle motions; this addresses the question of possible universality of the large deviation function of the injected power in the 1D Langevin context.