Field theories of active particle systems and their entropy production

Active particles that translate chemical energy into self-propulsion can maintain a far-from-equilibrium steady state and perform work. The entropy production measures how far from equilibrium such a particle system operates and serves as a proxy for the work performed. Field theory offers a promising route to calculating entropy production, as it allows for many interacting particles to be considered simultaneously. Approximate field theories obtained by coarse-graining or smoothing that draw on additive noise can capture densities and correlations well, but they generally ignore the microscopic particle nature of the constituents, thereby producing spurious results for the entropy production. As an alternative we demonstrate how to use Doi-Peliti field theories, which capture the microscopic dynamics, including reactions and interactions with external and pair potentials. Such field theories are in principle exact, while offering a systematic approximation scheme, in the form of diagrammatics. We demonstrate how to construct them from a Fokker-Planck equation and show how to calculate entropy production of active matter from first principles. This framework is easily extended to include interaction. We use it to derive exact, compact and efficient general expressions for the entropy production for a vast range of interacting conserved particle systems. These expressions are independent of the underlying field theory and can be interpreted as the spatial average of the local entropy production. They are readily applicable to numerical and experimental data. In general, the entropy production due to any pair interaction draws at most on the three point, equal time density; and an $n$-point interaction on the $(2n-1)$-point density. We illustrate the technique in a number of exact, tractable examples, including some with pair-interaction as well as in a system of many interacting ABPs.