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 (2020 J. Stat. Mech. 05206)). Treating the scalar field ϕ (and its time derivative ϕ̇ ) 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 (2020 J. Stat. Mech. 05206). We then present a similar EPR calculation for model AB with the conservative and non-conservative contributions to ϕ̇=ϕ̇A+ϕ̇B 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.

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