Probability current loops in non-equilibrium steady states and statistical properties of angular momenta in configuration space*

Unlike systems in thermal equilibrium, steady probability current loops persist in non-equilibrium stationary states. One of the consequences is that, in the space of two or more observable quantities ($q_{i}$), the average ``angular momentum'' ($\left\langle L_{ij}\right\rangle \equiv \left\langle q_{i}\times \dot{q}_{j}\right\rangle $) is typically non-trivial. In addition, the full distribution of $L$ often display remarkable properties. We will provide a general framework for the study of $% L$, as well as specific examples -- in the context of both exactly solvable models (based on linear Langevin equations with additive white noise) and physical data of global ocean heat content.