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