Steady state entropy production rate for scalar Langevin field theories

The entropy production rate (EPR) offers a quantitative measure of time reversal symmetry breaking in non-equilibrium systems. It can be defined either at particle level or the coarse-grained fields such as density; EPR for latter quantifies extent to which these behave irreversibly. In this work, we first develop general method compute scalar Langevin field theories with additive noise. This large class includes active versions Model A (non-conserved density dynamics) and B (conserved) also models where both types dynamics are simultaneously present (such AB). Treating $\phi$ (and its derivative $\dot\phi$) sole observable(s), arrive an expression that is non-negative every configuration quadratic time-antisymmetric component dynamics. Our function quasipotential, determines full probability distribution configurations, not generally calculable. To alleviate difficulty, small-noise expansion EPR, only requires knowledge deterministic (mean-field) solution steady state, calculable, least numerically. We demonstrate calculation case AB. then similar AB conservative non-conservative contributions $\dot\phi = \dot\phi_{\rm A} + B}$ viewed separately observable quantities. results qualitatively different, confirming field-level depends on choice information retained within dynamical description.