Thermodynamic cost due to changing the initial distribution over states

We consider nonequilibrium systems that obey local detailed balance and are driven by an external system such that no work is dissipated for some initial distribution over states $x \in X$, $q(X)$. We show that in general work is dissipated under that driving if instead the initial distribution is some $r(X) \ne q(X)$, calculating that amount of dissipated work. We then use this result to investigate the thermodynamics of computation. Specifically, we suppose a Markov partition of $X$ into a set of coarse-grained "computational states" labelled by $v \in V$, and identify the dynamics over those computational states as a (possibly noisy) "computation" that runs on the system. We identify the initial distribution over computational states as the distribution over inputs to the computer. We calculate the work that such a computer dissipates, if it is designed to dissipate no work for some input distribution $q(V)$ but is instead used with a different input distribution $r(V)$. This dissipated work is an extra thermodynamic cost of computation, in addition to the well-known Landauer's cost. We also calculate the extra expected dissipated work if there is a probability distribution over the possible input distributions rather than a single one.