  Stochastic line integrals provide a useful tool 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 steady-state probability currents.
Additionally, the stream function permits analytical understanding of the
scaling dependence of stochastic area on key parameters such as the noise
strength for both nonlinear and linear springs. Theoretical results are
supported by numerical studies of an overdamped, two-dimensional mass-spring
system driven out of equilibrium.
