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Entropy production for partially observed harmonic systems

The probability distribution of the total entropy production in the non-equilibrium steady state follows a symmetry relation called the fluctuation theorem. When a certain part of the system is masked or hidden, it is difficult to infer the exact estimate of the total entropy production. Entropy produced from the observed part of the system shows a significant deviation from the steady state fluctuation theorem. This deviation occurs due to the interaction between the observed and the masked part of the system. A naive guess would be that the deviation from the steady state fluctuation theorem may disappear in the limit of small interaction between both parts of the system. In contrast, we investigate the entropy production of a particle in a harmonically coupled Brownian particle system (say, particle A and B) in a heat reservoir at a constant temperature. The system is maintained in the non-equilibrium steady state using stochastic driving. When the coupling between particle A and B is infinitesimally weak, the deviation from the steady state fluctuation theorem for the entropy production of a partial system of a coupled system is studied. Furthermore, we consider a harmonically confined system (i.e., a harmonically coupled system of particle A and B in harmonic confinement). In the weak coupling limit, the entropy produced by the partial system (e.g., particle A) of the coupled system in a harmonic trap satisfies the steady state fluctuation theorem. Numerical simulations are performed to support the analytical results. Part of these results was announced in a recent letter, Europhys. Lett. 115, 60003 (2016).