Stochastic Line Integrals as Metrics of Irreversibility and Heat Transfer

Stochastic line integrals allow quantitative characterization of irreversibility and detailed balance violation in noise-driven dynamical systems. One example of such integrals, the stochastic area, was introduced for linear systems and tested experimentally in coupled linear electrical circuits [1,2]. Here we establish the general properties of stochastic line integrals and clarify their implementation for experiments and simulations as well as their utility for quantifying non-equilibrium behavior. Theoretical results are supported by numerical studies of an overdamped, two-dimensional mass-spring system driven out of equilibrium. In this case, the stochastic area can be concisely expressed in terms of a streamfunction the sign of which determines the orientation of probability current loops. The streamfunction provides analytical insight to the dependence of stochastic area on parameters such as the noise strength for both nonlinear and linear springs; in particular, we find distinct scaling regimes for stochastic area versus noise amplitude depending on the character of nonlinearity.
[1] A. Ghanta, J. Neu, and S. Teitsworth, Phys. Rev. E 95, 032128 (2017).
[2] J. P. Gonzalez, J. Neu, and S. Teitsworth, Phy. Rev. E 99, 022143 (2019).