Work fluctuations for a Brownian particle in a harmonic trap with fluctuating locations

We consider a Brownian particle in a harmonic trap. The location of the trap is modulated according to an Ornstein-Uhlenbeck process. We investigate the fluctuation of the work done by the modulated trap on the Brownian particle in a given time interval in the steady state. We compute the large deviation as well as the complete asymptotic form of the probability density function of the work done. The theoretical asymptotic forms of the probability density function are in very good agreement with the numerics. We also discuss the validity of the fluctuation theorem for this system.

Figure 1

The regions in the (a) $\theta ,\delta$ and (b) $\alpha ,\delta$ spaces, where $g\left(\lambda \right)$ has the number of singularities mentioned in the figure. The equations of the boundary lines separating different regions are given in Appendix B. ${\alpha }_{c_1}=\frac{1}{3}$ for $\delta =0$ and ${\theta }_{c_2}\to \frac{1}{3}$ as $\delta \to \infty$. Each of the phase boundaries meet at $\theta =0$ ($\alpha =0$), $\delta =1$.

Figure 2

$P\left(W_{\tau }\right)$ against the scaled variable $w=W_{\tau }/\tau$ for $\tau =10$, ${\tau }_c=1$. The points (blue) are obtained from numerical simulation, and the dashed solid lines (red) plot the analytical asymptotic forms given by Eq. (27). ${\theta }_{c_1}=\frac{9}{35}=0.257...$ for $\delta =\frac{1}{2}$.

Figure 3

$P\left(W_{\tau }\right)$ against the scaled variable $w=W_{\tau }/\tau$ for $\tau =10$, ${\tau }_c=1$. The points (blue) are obtained from numerical simulation, and the dashed solid lines (red) plot the analytical asymptotic forms given by Eq. (29) for $w>w_{\mathrm{a}}^{*}$ and Eqs. (32, 33, 34, 35, 36) for $w<w_{\mathrm{a}}^{*}$, where $w_{\mathrm{a}}^{*}=0$ for $\delta =1,\theta =4$, and $w_{\mathrm{a}}^{*}=-0.0135...$ for $\delta =0.5,\theta =13.5$.

Figure 4

Schematic steepest descent contours for the case when there are three branch points at ${\lambda }_{\mathrm{a}}$, ${\lambda }_{\mathrm{c}}$, and ${\lambda }_{\mathrm{d}}$, where ${\lambda }_{-}<{\lambda }_{\mathrm{c}}<0<{\lambda }_{\mathrm{a}}<{\lambda }_{\mathrm{d}}<{\lambda }_{+}$; and the saddle point ${\lambda }^{*}$ lies between (a) ${\lambda }_{-}$ and ${\lambda }_{\mathrm{c}}$, (b) ${\lambda }_{\mathrm{c}}$ and ${\lambda }_{\mathrm{a}}$, (c) ${\lambda }_{\mathrm{a}}$ and ${\lambda }_{\mathrm{d}}$, and (d) ${\lambda }_{\mathrm{d}}$ and ${\lambda }_{+}$, respectively.

Figure 5

$P\left(W_{\tau }\right)$ against the scaled variable $w=W_{\tau }/\tau$ for $\tau =10$, ${\tau }_c=1$, $\delta =5$, and $\theta =0.5$. The points (blue) are obtained from numerical simulation, and the dashed solid line (red) plots the analytical asymptotic forms given in the text. The vertical dashed lines mark the positions $w_{\mathrm{c}}^{*}=0.8398...$, $w_{\mathrm{a}}^{*}=0.06269...$, and $w_{\mathrm{d}}^{*}=-0.8398...$ .

Figure 6

$P\left(W_{\tau }\right)$ against the scaled variable $w=W_{\tau }/\tau$ for $\tau =10$, ${\tau }_c=1$, $\delta =5$, and $\theta =0.1$. The points (blue) are obtained from numerical simulation, and the dashed solid line (red) plots the analytical asymptotic forms given in the text. The vertical dashed lines mark the positions $w_{\mathrm{b}}^{*}=1.4062...$, $w_{\mathrm{c}}^{*}=0.3125$, $w_{\mathrm{a}}^{*}=-0.0976...$, and $w_{\mathrm{d}}^{*}=-0.3125$.

Figure 7

The phase diagrams in the parameter space showing the state of the validity of the fluctuation theorem. Apart from the $\theta ={\theta }_c$ and $\alpha ={\alpha }_c$ (orange) lines, the other lines are the same as in Fig. 1. In the (light orange) regions above the (orange) lines $\theta ={\theta }_c$ in (a) and $\alpha ={\alpha }_c$ in (b), the fluctuation theorem is not valid at all, whereas it is always valid in the white regions below the (blue) lines $\theta ={\theta }_{c_1}$ in (a) and $\alpha ={\alpha }_{c_1}$ in (b). In the intermediate (light blue) region, the fluctuation theorem is valid only within a limited range of $w$ given by $w_{\mathrm{a}}^{*}<w<-w_{\mathrm{a}}^{*}$ for $\delta <1$ and $\max (w_{\mathrm{a}}^{*},-w_{\mathrm{c}}^{*})<w<\min (-w_{\mathrm{a}}^{*},w_{\mathrm{c}}^{*})$ for $\delta >1$. For $\delta =0$, we have ${\alpha }_{c_1}=\frac{1}{3}$ and ${\alpha }_c=3$.

Figure 8

In the shaded regions of the $(\theta ,\delta )$ plane in (a), (b), (c), and (d), the respective mathematical conditions given there are satisfied and consequently $g\left(\lambda \right)$ possesses singularities at ${\lambda }_{\mathrm{a}}$, ${\lambda }_{\mathrm{b}}$, ${\lambda }_{\mathrm{c}}$, and ${\lambda }_{\mathrm{d}}$, respectively, given by Eqs. (B1, B2, B3, B4).

Figure 9

Schematic steepest descent contour (in red) for the case when the branch point ${\lambda }_{\mathrm{a}}$ is not between the origin and the saddle point ${\lambda }^{*}$. Here, it is shown for ${\lambda }^{*}<0$, however, one can also have ${\lambda }^{*}\in (0,{\lambda }_{\mathrm{a}})$. The direction towards which the contour bends depends on the value of $w$. Here, it is shown for $w<0$. For $w>0$, the contour bends towards the positive $\mathrm{Re}\left(\lambda \right)$ axis, whereas for $w=0$, the steepest descent contour is parallel to the $\mathrm{Im}\left(\lambda \right)$ axis. The thick solid (orange) line along the $\mathrm{Re}\left(\lambda \right)$ axis from ${\lambda }_{\mathrm{a}}$ represents the branch cut.

Figure 10

Schematic steepest descent contour for the case when the branch point ${\lambda }_{\mathrm{a}}$ is between the origin and the saddle point ${\lambda }^{*}$. The thick solid (orange) line along the $\mathrm{Re}\left(\lambda \right)$ axis from ${\lambda }_{\mathrm{a}}$ represents the branch cut. The steepest descent contour goes around the branch cut as shown by $C_2$ and $C_3$ (in blue). The contribution coming from the circular contour (in magenta) $C_{\epsilon }$ around the branch point becomes zero in the the limit of the radius $\epsilon \to 0$. The direction towards which the contours $C_1$ and $C_4$ (shown in red) bend depends on the value of $w$. Here, it is shown for $w>0$. For $w<0$, the $C_1$ and $C_4$ bend towards the negative $\mathrm{Re}\left(\lambda \right)$ axis, whereas for $w=0$, they are parallel to the $\mathrm{Im}\left(\lambda \right)$ axis.