Stochastic line integrals and stream functions as metrics of irreversibility and heat transfer

Stochastic line integrals are presented as a useful metric for quantitatively characterizing irreversibility and detailed balance violation in noise-driven dynamical systems. A particular realization is the stochastic area, recently studied in coupled electrical circuits. Here we provide a general framework for understanding properties of stochastic line integrals and clarify their implementation for experiments and simulations. For two-dimensional systems, stochastic line integrals can be expressed in terms of a stream function, the sign of which determines the orientation of nonequilibrium steady-state probability currents. Theoretical results are supported by numerical studies of an overdamped two-dimensional mass-spring system driven out of equilibrium. Additionally, the stream function permits analytical understanding of the scaling dependence of stochastic area growth rate on key parameters such as the noise strength for both linear and nonlinear springs.