Large deviation function for a driven underdamped particle in a periodic potential

Employing large deviation theory, we explore current fluctuations of underdamped Brownian motion for the paradigmatic example of a single particle in a one-dimensional periodic potential. Two different approaches to the large deviation function of the particle current are presented. First, we derive an explicit expression for the large deviation functional of the empirical phase space density, which replaces the level 2.5 functional used for overdamped dynamics. Using this approach, we obtain several bounds on the large deviation function of the particle current. We compare these to bounds for overdamped dynamics that have recently been derived, motivated by the thermodynamic uncertainty relation. Second, we provide a method to calculate the large deviation function via the cumulant generating function. We use this method to assess the tightness of the bounds in a numerical case study for a cosine potential.

Figure 1

Numerical results for the underdamped motion in the cosine potential. The upper panel shows the generating function $\alpha \left(\lambda \right)$ for different masses with $T=1,\gamma =1$ and $V_0=2,F_{\text{ext}}=1$. The second derivative is shown in the inset, with the points of largest curvature marked by colored dots. The lower panel shows the corresponding LDF $I\left(J\right)$ calculated via the Legendre-Fenchel transform (29).

Figure 2

Typical densities for small mass $m=0.01$ (top row) and large mass $m=3$ (bottom row) with $T=1,\gamma =1$, and $V_0=2$. Darker color corresponds to a small probability density. The tilting $\lambda$ of the operator (31) and with it the empirical current $J$ is varied along the columns as specified at the bottom of each plot. The contour lines of the energy landscape are plotted as black lines. The first column corresponds to the state with vanishing current $J=0$.

Figure 3

LDF for the cosine potential for different masses $m$ showing the transition from underdamped to overdamped motion. The inset displays the functions in the vicinity of $J=0$ as indicated by the gray box. The dashed lines correspond to the asymptotic bounds $I_{\text{a}}$ and $I_{\text{c}}$, where the former one is valid for all masses and the latter for the extreme underdamped case $m\to \infty$. The other parameters are $V_0=2.0,T=1,\gamma =1,F_{\text{ext}}=1$.

Figure 4

Hierarchy of the large deviation functions and bounds discussed in this paper. Straight arrows denote equalities due to full contraction or specification for limiting cases; wavy arrows point to upper bounds due to nonoptimal trial functions for the contraction.