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Open classical systems with balanced, spatially separated gain and loss, alsocalled $\mathcal{PT}$ symmetric systems, are a subject of intense, ongoingresearch. We investigate the properties of a classical chain with spatiallyseparated viscous loss and stochastic gain that are balanced only in astatistical sense. For a purely harmonic chain, we show that a split Langevinbath leads to either the absence of thermalization or non-equilibrium steadystates with inhomogeneous temperature profile. Both phenomena are understood interms of normal modes of the chain, where dissipation in one normal mode iscorrelated with the velocities of all other modes. We obtain closed-formexpressions for the mode temperatures and show that nonlinearities lead tosteady states due to mode mixing even in the presence of a split Langevin bath.

A split-Langevin bath: steady states from loss and a statistically balanced gain