Time-dependent trajectory of a one-dimensional Gaussian non-Markovian observable does not reveal its nonequilibrium...

When analyzing experimental or simulation time-series data, the question arises whether it is possible to tell from the mere observation of the time-dependent trajectory of a one-dimensional observable whether the system is in equilibrium or not. We here consider the nonequilibrium version of the generalized Langevin equation for a Gaussian non-Markovian observable and show that (i) the multipoint joint distribution solely depends on the two-point correlation function and that (ii) for any nonequilibrium process an equilibrium process with uniquely determined parameters can be found that produces the same two-point correlation function. Since the multipoint joint distribution completely characterizes the dynamics of an observable, we conclude that the nonequilibrium character of a system, in contrast to its non-Markovianity, cannot be read off from the one-dimensional trajectory of a Gaussian observable. These findings are relevant for human cancer and algae cells, whose single-cell velocity distributions have been found to be Gaussian within the data accuracy.