Abstract
We present an integral theory for the fluctuation-dissipation relations embedded in Markov switching diffusion processes, which mimic the fluctuations induced simultaneously by noise originating from different time-scale levels. The formulas for the extended Einstein’s relation in terms of second-order moments, as well as the auto and cross spectra are derived. Several universal properties for two types of fluctuations are proved: The thermally induced one remains globally constant for isothermal systems and its spectrum is Lorentzian; the fluctuation induced by the switching noise has distinctive scaling laws that are embedded in the auto and cross spectra.
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