Exact results for the finite time thermodynamic uncertainty relation

We obtain exact results for the recently discovered finite-time thermodynamic uncertainty relation in a stochastically driven system with non-Gaussian work statistics, both in the steady state and transient regimes, by obtaining exact expressions for any moment of the dissipated work at arbitrary times. The uncertainty function (the Fano factor of the dissipated work) is bounded from below by $2k_BT$ as expected, for all times $\tau$, in both steady state and transient regimes. The lower bound is reached at $\tau=0$ as well as when certain system parameters vanish (corresponding to an equilibrium state). Surprisingly, we find that the uncertainty function also reaches a constant value at large $\tau$ for all the cases we have looked at. For a system starting and remaining in steady state, the uncertainty function increases monotonically, as a function of $\tau$ as well as other system parameters, implying that the large $\tau$ value is also an upper bound. For the same system in the transient regime, however, we find that the uncertainty function can have a local minimum at an accessible time $\tau_m$, for a range of parameter values. The non-monotonicity suggests, rather counter-intuitively, that there might be an optimal time for the working of microscopic machines, as well as an optimal configuration in the phase space of parameter values. Our solutions show that the ratios of higher moments of the dissipated work are also bounded from below by $2k_BT$. For another model, also solvable by our methods, which never reaches a steady state, the uncertainty function, is in some cases, bounded from below by a value less than $2k_BT$.