There are two other expressions in the literature: Gallovoti-Cohen fluctuation theorem or GCFT of 1995 and the extended heat theorem of van Zon and Cohen of 2004. These theorems are different from that discussed here. The GCFT has the same functional form as the FT, Eq. (1), however, the argument of the FT is the phase-space compression factor. The left- and right-hand sides of the extended heat theorem are also similar in form to Eq. (1) although the extended theorem is a nonequivalence of these terms and the argument is the accumulated heat exchange.

The Boltzmann and propagator distributions were expressed in the fixed coordinate frame $r$, in accord with the laboratory reference frame in the Wang experiment. However, we could easily have expressed these distributions, $P_B$ and $G$, the trajectory distributions, $P$, and the steady-state dissipation function in terms of the moving coordinate frame $\mathbf{x}$. When cast in $\mathbf{x}$, the resulting expression for the steady-state dissipation function is ${\Omega }_t\left(\mathbf{x}\right)=0$, confirming that, under steady-state conditions, the particle is in equilibrium in the $\mathbf{x}$ frame. There is no dissipation in $\mathbf{x}$ and the FT is trivially satisfied. The full, stochastically derived dissipation function, i.e., the dissipation function evaluated along the trajectory starting from its equilibrium initial condition, can be derived in either the moving or laboratory coordinate frame. In the moving $\mathbf{x}$ coordinate frame, it is$${\Omega }_t\left(\mathbf{x}\right)=-\frac{\xi {\mathbf{v}}_{\mathit{opt}}}{k_{B}T}\bullet ({\mathbf{x}}_t-{\mathbf{x}}_0),$$i.e., only the initial, transients contribute to ${\Omega }_t\left(\mathbf{x}\right)$. In the $\mathbf{r}$ stationary laboratory frame, the full dissipation function is$${\Omega }_t\left(\mathbf{r}\right)=(\frac{k{\mathbf{v}}_{\mathit{opt}}t}{k_{B}T[1-\exp (-t∕\tau )]}-\frac{\xi {\mathbf{v}}_{\mathit{opt}}}{k_{B}T})\bullet ({\mathbf{r}}_t-{\mathbf{r}}_0).$$The first term on the RHS represents the transient contribution associated with displacing the particle with the bead from the center of the trap to the lag distance. The second term on the RHS is the steady-state contribution and is equivalent to the stochastically derived steady-state dissipation function, Eq. (25), investigated in this paper.