Work fluctuations for a Brownian particle driven by a correlated external random force

We have considered the underdamped motion of a Brownian particle in the presence of a correlated external random force. The force is modeled by an Ornstein-Uhlenbeck process. We investigate the fluctuations of the work done by the external force on the Brownian particle in a given time interval in the steady state. We calculate the large deviation functions as well as the complete asymptotic form of the probability density function of the performed work. We also discuss the symmetry properties of the large deviation functions for this system. Finally we perform numerical simulations and they are in a very good agreement with the analytic results.

Figure 1

The behavior of ${\lambda }^{*}$ is shown (solid line) as a function of $w$, for a set of parameters $\theta =4,\delta =2$, which merges to ${\lambda }_{\pm }$ (dashed lines) as $w\to \mp \infty$.

Figure 2

This plot depicts the analytic properties of $g\left(\lambda \right)$. In the shaded region of the $(\theta ,\delta )$ plane, $g\left(\lambda \right)$ possesses a singularity, where $f_2({\lambda }_{+},\theta ,\delta )<0$. On the other hand, in the unshaded region $g\left(\lambda \right)$ does not have any singularities, where $f_2({\lambda }_{+},\theta ,\delta )>0$. These two domains are separated by the boundary given by the equation $f_2({\lambda }_{+},\theta ,\delta )=0$.

Figure 3

The (red) dashed line plots the analytical result of $P\left(W_{\tau }\right)$ against the scaled variable $w=W_{\tau }/(\tau /{\tau }_{\gamma })$, while the (blue) points are numerical simulation results.

Figure 4

The (red) dashed line plots the analytical result for $P\left(W_{\tau }\right)$, while the (blue) points are numerical simulation results. The vertical dashed line marks the position of the singularity $w^{*}=-0.801661...$ for the values of $\theta =7,\delta =1$.

Figure 5

The (red) dashed lines plot analytical results for $P\left(W_{\tau }\right)$, while the (blue) points are numerical simulation results, for the $\delta =0$ case. The vertical dashed line in (b) marks the position of the singularity which is $w^{*}=0.037...$ in this case.