Heat fluctuations and initial ensembles

Time-integrated quantities such as work and heat increase incessantly in time during nonequilibrium processes near steady states. In the long-time limit, the average values of work and heat become asymptotically equivalent to each other, since they only differ by a finite energy change in average. However, the fluctuation theorem (FT) for the heat is found not to hold with the equilibrium initial ensemble, while the FT for the work holds. This reveals an intriguing effect of everlasting initial memory stored in rare events. We revisit the problem of a Brownian particle in a harmonic potential dragged with a constant velocity, which is in contact with a thermal reservoir. The heat and work fluctuations are investigated with initial Boltzmann ensembles at temperatures generally different from the reservoir temperature. We find that, in the infinite-time limit, the FT for the work is fully recovered for arbitrary initial temperatures, while the heat fluctuations significantly deviate from the FT characteristics except for the infinite initial-temperature limit (a uniform initial ensemble). Furthermore, we succeed in calculating finite-time corrections to the heat and work distributions analytically, using the modified saddle point integral method recently developed by us. Interestingly, we find noncommutativity between the infinite-time limit and the infinite-initial-temperature limit for the probability distribution function (PDF) of the heat.

Figure 1

$f_{\infty }\left(p\right)$ as in Eq. (38) is plotted as a function of $p$ and ${\beta }^{\prime }$.

Figure 2

$f_{\infty }\left(p\right)$ for ${\beta }^{\prime }=0$, $1/2$, 1, and 2.

Figure 3

$f_{\tau }\left(p\right)=\frac{1}{w\tau }ln\frac{P_{\tau }\left(w\tau p\right)}{P_{\tau }(-w\tau p)}$ is drawn for ${\beta }^{\prime }=0.1$, $\tau =20$, and $d=3$.

Figure 4

$f_{\tau }\left(p\right)$ is drawn for ${\beta }^{\prime }=2$, $\tau =20$, and $d=3$.