  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
non-commutativity between the infinite-time limit and the
infinite-initial-temperature limit for the probability distribution function
(PDF) of the heat.
