Without violating causality, we allow performing measurements in time reverse process of a feedback manipulated stochastic system. As a result we come across an entropy production due to the measurement process. This entropy production, in addition t
o the usual system and medium entropy production, constitutes the total entropy roduction of the combined system of the reservoir, the system and the feedback controller. We show that this total entropy production of full system satisfies an integrated fluctuation theorem as well as a detailed fluctuation theorem as expected. We illustrate and verify this idea through explicit calculation and direct simulation in two examples.
We have used an atomistic {it ab initio} approach with no adjustable parameters to compute the lattice thermal conductivity of Si$_{0.5}$Ge$_{0.5}$ with a low concentration of embedded Si or Ge nanoparticles of diameters up to 4.4 nm. Through exact G
reens function calculation of the nanoparticle scattering rates, we find that embedding Ge nanoparticles in $text{Si}_{0.5}text{Ge}_{0.5}$ provides 20% lower thermal conductivities than embedding Si nanoparticles. This contrasts with the Born approximation which predicts an equal amount of reduction for the two cases, irrespective of the sign of the mass difference. Despite these differences, we find that the Born approximation still performs remarkably well, and it permits investigation of larger nanoparticle sizes, up to 60 nm in diameter, not feasible with the exact approach.
