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Information transfer in coupled Langevin equations

We provide a general formula, based on stochastic thermodynamics, that describes the flow of information between an arbitrary number of coupled complex-valued Langevin equations. This permits to describe the transfer of information in complex networks of oscillators out of thermal equilibrium, that can model a multitude of physical, biological and man made systems. The information flow contains an incoherent component proportional to the amplitude difference and a coherent one proportional to the phase difference between the oscillators, which depends on their synchronisation. We illustrate the theory by simulating the dynamics of a spin-Seebeck diode, described by two coupled oscillators, that can rectify the flow of information, energy and spin. Remarkably, the system can operate in a regime where the synchronisation is broken and there is a flow of incoherent information without net transfer of energy.