Fluctuation symmetries for work and heat

We consider a particle dragged through a medium at constant temperature as described by a Langevin equation with a time-dependent potential. The time dependence is specified by an external protocol. We give conditions on potential and protocol under which the fluctuations of the dissipative work satisfy an exact symmetry for all times. We also present counterexamples to that fluctuation theorem when our conditions are not satisfied. Finally, we consider the dissipated heat, which differs from the work by a temporal boundary term. We explain why there is a correction to the standard fluctuation theorem due to the unboundedness of that temporal boundary. However, the corrected fluctuation relation has again a general validity.

Figure 1

Extension of the standard fluctuation theorem for the heat per unit time $q$. The function $f\left(q\right)$ is defined in Eq. (17). For small values of $q$, $f\left(q\right)$ is linear so the standard fluctuation theorem is recovered. Between $\overline{w}$ and $w^{\star }$, the behavior is determined by the large deviation rate function $I\left(q\right)$ of the work. The function $f\left(q\right)$ saturates for large $q$.

Figure 2

Plot of the deviations from the EFT (15), for several values of the exponents ${\alpha }_{+}$ and ${\alpha }_{-}$ in (18). A potential that is dragged with constant velocity $v=1$ is considered: the EFT is verified for the symmetric potential (here we chose ${\alpha }_{+}={\alpha }_{-}=3$), while it is not observed for asymmetric potentials. Parameters are $\tau =1$ and $dt={10}^{-3}$.

Figure 3

Plot of the deviations from the EFT (15), for symmetric potentials [${\alpha }_{+}={\alpha }_{-}\equiv \alpha$ in Eq. (18)] and spatially translated with protocol ${\gamma }_t=t+t^4$. We chose $\tau =1$ and $dt={10}^{-4}$. The fluctuation theorem is verified for the harmonic potential, while it is not valid for a symmetric potential with exponent ${\alpha }_{+}={\alpha }_{-}=1.2$. For the latter potential, a simulation with $dt={10}^{-3}$ shows that the numerical approximation is negligible.

Figure 4

Plot of the deviations from the EFT (15), for the symmetric potential $U\left(x\right)={\mid x\mid }^3∕3$ [${\alpha }_{+}={\alpha }_{-}=3$ in Eq. (18)] and spatially translated with protocol ${\gamma }_t=t^2$ and ${\gamma }_t=t^3$. We chose $\tau =1$ and $dt={10}^{-4}$. Since both protocols are neither symmetric or antisymmetric, the EFT is indeed not expected to hold.